Optimal. Leaf size=184 \[ -\frac {2 e^{i (d+e x)} F^{c (a+b x)} (e+i b c \log (F)) \, _2F_1\left (2,1-\frac {i b c \log (F)}{e};2-\frac {i b c \log (F)}{e};i e^{i (d+e x)}\right )}{3 e^2 f^2}-\frac {b c \log (F) \csc ^2\left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right ) F^{c (a+b x)}}{6 e^2 f^2}-\frac {\cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right ) F^{c (a+b x)}}{6 e f^2} \]
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Rubi [A] time = 0.10, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4456, 4448, 4450} \[ -\frac {2 e^{i (d+e x)} F^{c (a+b x)} (e+i b c \log (F)) \, _2F_1\left (2,1-\frac {i b c \log (F)}{e};2-\frac {i b c \log (F)}{e};i e^{i (d+e x)}\right )}{3 e^2 f^2}-\frac {b c \log (F) \csc ^2\left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right ) F^{c (a+b x)}}{6 e^2 f^2}-\frac {\cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right ) F^{c (a+b x)}}{6 e f^2} \]
Antiderivative was successfully verified.
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Rule 4448
Rule 4450
Rule 4456
Rubi steps
\begin {align*} \int \frac {F^{c (a+b x)}}{(f+f \sin (d+e x))^2} \, dx &=\frac {\int F^{c (a+b x)} \sec ^4\left (\frac {d}{2}-\frac {\pi }{4}+\frac {e x}{2}\right ) \, dx}{4 f^2}\\ &=-\frac {F^{c (a+b x)} \cot \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right ) \csc ^2\left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )}{6 e f^2}-\frac {b c F^{c (a+b x)} \csc ^2\left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right ) \log (F)}{6 e^2 f^2}+\frac {\left (1+\frac {b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \sec ^2\left (\frac {d}{2}-\frac {\pi }{4}+\frac {e x}{2}\right ) \, dx}{6 f^2}\\ &=-\frac {F^{c (a+b x)} \cot \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right ) \csc ^2\left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )}{6 e f^2}-\frac {b c F^{c (a+b x)} \csc ^2\left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right ) \log (F)}{6 e^2 f^2}-\frac {2 e^{i (d+e x)} F^{c (a+b x)} \, _2F_1\left (2,1-\frac {i b c \log (F)}{e};2-\frac {i b c \log (F)}{e};i e^{i (d+e x)}\right ) (e+i b c \log (F))}{3 e^2 f^2}\\ \end {align*}
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Mathematica [A] time = 3.20, size = 240, normalized size = 1.30 \[ \frac {F^{c (a+b x)} \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right ) \left ((-1+i) \left (b^2 c^2 \log ^2(F)+e^2\right ) \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )^3 \left (1-(1-i) \, _2F_1\left (1,-\frac {i b c \log (F)}{e};1-\frac {i b c \log (F)}{e};i \cos (d+e x)-\sin (d+e x)\right )\right )+2 \sin \left (\frac {1}{2} (d+e x)\right ) \left (b^2 c^2 \log ^2(F)+e^2\right ) \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )^2-e (b c \log (F)+e) \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )+2 e^2 \sin \left (\frac {1}{2} (d+e x)\right )\right )}{3 e^3 f^2 (\sin (d+e x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {F^{b c x + a c}}{f^{2} \cos \left (e x + d\right )^{2} - 2 \, f^{2} \sin \left (e x + d\right ) - 2 \, f^{2}}, x\right ) \]
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 \[ \int \frac {F^{{\left (b x + a\right )} c}}{{\left (f \sin \left (e x + d\right ) + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.95, size = 0, normalized size = 0.00 \[ \int \frac {F^{c \left (b x +a \right )}}{\left (f +f \sin \left (e x +d \right )\right )^{2}}\, dx \]
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 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (f+f\,\sin \left (d+e\,x\right )\right )}^2} \,d x \]
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 \[ \frac {\int \frac {F^{a c} F^{b c x}}{\sin ^{2}{\left (d + e x \right )} + 2 \sin {\left (d + e x \right )} + 1}\, dx}{f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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