Optimal. Leaf size=78 \[ -\frac{4 e^{2 i (d+e x)} F^{c (a+b x)} \text{Hypergeometric2F1}\left (2,1-\frac{i b c \log (F)}{2 e},2-\frac{i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )}{b c \log (F)+2 i e} \]
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Rubi [A] time = 0.0286723, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {4453} \[ -\frac{4 e^{2 i (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1-\frac{i b c \log (F)}{2 e};2-\frac{i b c \log (F)}{2 e};e^{2 i (d+e x)}\right )}{b c \log (F)+2 i e} \]
Antiderivative was successfully verified.
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Rule 4453
Rubi steps
\begin{align*} \int F^{c (a+b x)} \csc ^2(d+e x) \, dx &=-\frac{4 e^{2 i (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1-\frac{i b c \log (F)}{2 e};2-\frac{i b c \log (F)}{2 e};e^{2 i (d+e x)}\right )}{2 i e+b c \log (F)}\\ \end{align*}
Mathematica [A] time = 1.50092, size = 101, normalized size = 1.29 \[ -\frac{2 i F^{c (a+b x)} \left (\left (-1+e^{2 i d}\right ) \text{Hypergeometric2F1}\left (1,-\frac{i b c \log (F)}{2 e},1-\frac{i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )+\sin (d) \csc (d+e x) (\cos (e x)-i \sin (e x))\right )}{\left (-1+e^{2 i d}\right ) e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( \csc \left ( ex+d \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (F^{b c x + a c} \csc \left (e x + d\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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