Optimal. Leaf size=141 \[ \frac{2 e^{2 i (d+e x)} F^{c (a+b x)} (-b c \log (F)+2 i e) \text{Hypergeometric2F1}\left (2,1-\frac{i b c \log (F)}{2 e},2-\frac{i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )}{3 e^2}-\frac{b c \log (F) \csc ^2(d+e x) F^{c (a+b x)}}{6 e^2}-\frac{\cot (d+e x) \csc ^2(d+e x) F^{c (a+b x)}}{3 e} \]
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Rubi [A] time = 0.0549809, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4449, 4453} \[ \frac{2 e^{2 i (d+e x)} F^{c (a+b x)} (-b c \log (F)+2 i e) \, _2F_1\left (2,1-\frac{i b c \log (F)}{2 e};2-\frac{i b c \log (F)}{2 e};e^{2 i (d+e x)}\right )}{3 e^2}-\frac{b c \log (F) \csc ^2(d+e x) F^{c (a+b x)}}{6 e^2}-\frac{\cot (d+e x) \csc ^2(d+e x) F^{c (a+b x)}}{3 e} \]
Antiderivative was successfully verified.
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Rule 4449
Rule 4453
Rubi steps
\begin{align*} \int F^{c (a+b x)} \csc ^4(d+e x) \, dx &=-\frac{F^{c (a+b x)} \cot (d+e x) \csc ^2(d+e x)}{3 e}-\frac{b c F^{c (a+b x)} \csc ^2(d+e x) \log (F)}{6 e^2}+\frac{1}{6} \left (4+\frac{b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \csc ^2(d+e x) \, dx\\ &=-\frac{F^{c (a+b x)} \cot (d+e x) \csc ^2(d+e x)}{3 e}-\frac{b c F^{c (a+b x)} \csc ^2(d+e x) \log (F)}{6 e^2}+\frac{2 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))}{3 e^2}\\ \end{align*}
Mathematica [A] time = 2.98715, size = 173, normalized size = 1.23 \[ \frac{F^{c (a+b x)} \left (-\frac{2 i \left (b^2 c^2 \log ^2(F)+4 e^2\right ) \left (1+\left (-1+e^{2 i d}\right ) \text{Hypergeometric2F1}\left (1,-\frac{i b c \log (F)}{2 e},1-\frac{i b c \log (F)}{2 e},e^{2 i (d+e x)}\right )\right )}{-1+e^{2 i d}}+\csc (d) \sin (e x) \csc (d+e x) \left (b^2 c^2 \log ^2(F)+4 e^2\right )-e \csc ^2(d+e x) (b c \log (F)+2 e \cot (d))+2 e^2 \csc (d) \sin (e x) \csc ^3(d+e x)\right )}{6 e^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.134, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( \csc \left ( ex+d \right ) \right ) ^{4}\, 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 )^{4}, 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 )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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