Optimal. Leaf size=137 \[ -\frac{e^{i (d+e x)} F^{c (a+b x)} (e+i b c \log (F)) \text{Hypergeometric2F1}\left (1,\frac{e-i b c \log (F)}{2 e},\frac{1}{2} \left (3-\frac{i b c \log (F)}{e}\right ),e^{2 i (d+e x)}\right )}{e^2}-\frac{b c \log (F) \csc (d+e x) F^{c (a+b x)}}{2 e^2}-\frac{\cot (d+e x) \csc (d+e x) F^{c (a+b x)}}{2 e} \]
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Rubi [A] time = 0.0490738, antiderivative size = 137, 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} \[ -\frac{e^{i (d+e x)} F^{c (a+b x)} (e+i b c \log (F)) \, _2F_1\left (1,\frac{e-i b c \log (F)}{2 e};\frac{1}{2} \left (3-\frac{i b c \log (F)}{e}\right );e^{2 i (d+e x)}\right )}{e^2}-\frac{b c \log (F) \csc (d+e x) F^{c (a+b x)}}{2 e^2}-\frac{\cot (d+e x) \csc (d+e x) F^{c (a+b x)}}{2 e} \]
Antiderivative was successfully verified.
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Rule 4449
Rule 4453
Rubi steps
\begin{align*} \int F^{c (a+b x)} \csc ^3(d+e x) \, dx &=-\frac{F^{c (a+b x)} \cot (d+e x) \csc (d+e x)}{2 e}-\frac{b c F^{c (a+b x)} \csc (d+e x) \log (F)}{2 e^2}+\frac{1}{2} \left (1+\frac{b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \csc (d+e x) \, dx\\ &=-\frac{F^{c (a+b x)} \cot (d+e x) \csc (d+e x)}{2 e}-\frac{b c F^{c (a+b x)} \csc (d+e x) \log (F)}{2 e^2}-\frac{e^{i (d+e x)} F^{c (a+b x)} \, _2F_1\left (1,\frac{e-i b c \log (F)}{2 e};\frac{1}{2} \left (3-\frac{i b c \log (F)}{e}\right );e^{2 i (d+e x)}\right ) (e+i b c \log (F))}{e^2}\\ \end{align*}
Mathematica [B] time = 7.89086, size = 334, normalized size = 2.44 \[ \frac{F^{c (a+b x)} \left (-\frac{4 i \left (b^2 c^2 \log ^2(F)+e^2\right ) \left (1+(i \sin (d)+\cos (d)-1) \text{Hypergeometric2F1}\left (1,-\frac{i b c \log (F)}{e},1-\frac{i b c \log (F)}{e},\cos (d+e x)+i \sin (d+e x)\right )\right )}{b c \log (F) (i \sin (d)+\cos (d)-1)}-\frac{4 i \left (b^2 c^2 \log ^2(F)+e^2\right ) \left (1-(i \sin (d)+\cos (d)+1) \text{Hypergeometric2F1}\left (1,-\frac{i b c \log (F)}{e},1-\frac{i b c \log (F)}{e},-\cos (d+e x)-i \sin (d+e x)\right )\right )}{b c \log (F) (i \sin (d)+\cos (d)+1)}+\csc (d) \left (\frac{4 e^2}{b c \log (F)}+4 b c \log (F)\right )-2 b c \sec \left (\frac{d}{2}\right ) \log (F) \sin \left (\frac{e x}{2}\right ) \sec \left (\frac{1}{2} (d+e x)\right )+2 b c \csc \left (\frac{d}{2}\right ) \log (F) \sin \left (\frac{e x}{2}\right ) \csc \left (\frac{1}{2} (d+e x)\right )-4 b c \csc (d) \log (F)-e \csc ^2\left (\frac{1}{2} (d+e x)\right )+e \sec ^2\left (\frac{1}{2} (d+e x)\right )\right )}{8 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.117, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( \csc \left ( ex+d \right ) \right ) ^{3}\, 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 )^{3}, 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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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