Optimal. Leaf size=137 \[ -\frac {e^{i (d+e x)} F^{c (a+b x)} (e+i b c \log (F)) \, _2F_1\left (1,\frac {e-i b c \log (F)}{2 e};\frac {1}{2} \left (3-\frac {i b c \log (F)}{e}\right );e^{2 i (d+e x)}\right )}{e^2}-\frac {b c \log (F) \csc (d+e x) F^{c (a+b x)}}{2 e^2}-\frac {\cot (d+e x) \csc (d+e x) F^{c (a+b x)}}{2 e} \]
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Rubi [A] time = 0.05, antiderivative size = 137, normalized size of antiderivative = 1.00, 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*}
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Mathematica [B] time = 7.67, size = 334, normalized size = 2.44 \[ \frac {F^{c (a+b x)} \left (-\frac {4 i \left (b^2 c^2 \log ^2(F)+e^2\right ) \left (1+(i \sin (d)+\cos (d)-1) \, _2F_1\left (1,-\frac {i b c \log (F)}{e};1-\frac {i b c \log (F)}{e};\cos (d+e x)+i \sin (d+e x)\right )\right )}{b c \log (F) (i \sin (d)+\cos (d)-1)}-\frac {4 i \left (b^2 c^2 \log ^2(F)+e^2\right ) \left (1-(i \sin (d)+\cos (d)+1) \, _2F_1\left (1,-\frac {i b c \log (F)}{e};1-\frac {i b c \log (F)}{e};-\cos (d+e x)-i \sin (d+e x)\right )\right )}{b c \log (F) (i \sin (d)+\cos (d)+1)}+\csc (d) \left (\frac {4 e^2}{b c \log (F)}+4 b c \log (F)\right )-2 b c \sec \left (\frac {d}{2}\right ) \log (F) \sin \left (\frac {e x}{2}\right ) \sec \left (\frac {1}{2} (d+e x)\right )+2 b c \csc \left (\frac {d}{2}\right ) \log (F) \sin \left (\frac {e x}{2}\right ) \csc \left (\frac {1}{2} (d+e x)\right )-4 b c \csc (d) \log (F)-e \csc ^2\left (\frac {1}{2} (d+e x)\right )+e \sec ^2\left (\frac {1}{2} (d+e x)\right )\right )}{8 e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{b c x + a c} \csc \left (e x + d\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{{\left (b x + a\right )} c} \csc \left (e x + d\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int F^{c \left (b x +a \right )} \left (\csc ^{3}\left (e x +d \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{{\sin \left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{c \left (a + b x\right )} \csc ^{3}{\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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