What is this??!!!

\(\begin{align*} \prod _{ n=0 }^{ \infty }{ \left( 1-{ x }^{ n+1 } \right) =\prod _{ n=0 }^{ \infty }{ \left( 1-{ x }^{ 3n+1 } \right) \left( 1-{ x }^{ 3n+2 } \right) \left( 1-{ x }^{ 3n+3 } \right) } } &&&&&&&&&&&& \textcolor{green}{\textrm{Jacobi triple product}\sum _{ n=-\infty }^{ \infty }{ { q }^{ { n }^{ 2 } }{ z }^{ n }=\prod _{ n=0 }^{ \infty }{ \left( 1-{ q }^{ 2n+2 } \right) \left( 1+{ q }^{ 2n+1 }z \right) \left( 1+{ q }^{ 2n+1 }{ z }^{ -1 } \right) } } ;\quad q={ x }^{ \frac { 3 }{ 2 } }, z=-{ x }^{ \frac { -1 }{ 2 } }}\\

\end{align*}\) therefore we get $\begin{aligned}\quad \phi \left( x \right) ={\prod _{ n=0 }^{ \infty }{ \left( 1-{ x }^{ 3n+1 } \right) \left( 1-{ x }^{ 3n+2 } \right) \left( 1-{ x }^{ 3n+3 } \right) } }= &\sum _{ n=-\infty }^{ \infty }{ { \left( -1 \right) }^{ n }{ x }^{ \frac { n\left( 3n-1 \right) }{ 2 } } }\end{aligned}$ $\begin{aligned} \int _{ 0 }^{ 1 }{ \phi \left( x \right) } &=\int _{ 0 }^{ 1 }{ \sum _{ n=-\infty }^{ \infty }{ { \left( -1 \right) }^{ n }{ x }^{ \frac { n\left( 3n-1 \right) }{ 2 } } } }\\& =\sum _{ n=-\infty }^{ \infty }{ \frac { 2{ \left( -1 \right) }^{ n } }{ 3{ n }^{ 2 }-n+2 } }&&&&&&&&&&&&&&&&\textcolor{#BA33D6}{ \sum _{ n=-\infty }^{ \infty }{ \frac { { \left( -1 \right) }^{ n } }{ a{ n }^{ 2 }+bn+c } } =\frac { 1 }{ \sqrt { { b }^{ 2 }-4ac } } \frac { 4\pi \sin \left( \frac { \pi \sqrt { { b }^{ 2 }-4ac } }{ 2a } \right) \cos \left( \frac { \pi b }{ 2a } \right) }{ \cos \left( \frac { \pi \sqrt { { b }^{ 2 }-4ac } }{ a } \right) -\cos \left( \frac { \pi b }{ a } \right) }}\\&=\frac { 8\sqrt { \frac { 3 }{ 23 } } \pi \sinh \left( \frac { \pi \sqrt { 23 } }{ 6 } \right) }{ 2\cosh \left( \frac { \pi \sqrt { 23 } }{ 3 } \right) -1 } \end{aligned}$ so we get $a=8,b=3,c=23,d=2,f=1$ $\boxed{a+b+c+d+f=37}$