Optimal. Leaf size=93 \[ \frac {\sin ^3(e+f x) \tan (e+f x) \cos ^2(e+f x)^{\frac {1}{2} (n p+1)} \, _2F_1\left (\frac {1}{2} (n p+1),\frac {1}{2} (n p+4);\frac {1}{2} (n p+6);\sin ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+4)} \]
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Rubi [A] time = 0.14, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3659, 2602, 2577} \[ \frac {\sin ^3(e+f x) \tan (e+f x) \cos ^2(e+f x)^{\frac {1}{2} (n p+1)} \, _2F_1\left (\frac {1}{2} (n p+1),\frac {1}{2} (n p+4);\frac {1}{2} (n p+6);\sin ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+4)} \]
Antiderivative was successfully verified.
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Rule 2577
Rule 2602
Rule 3659
Rubi steps
\begin {align*} \int \sin ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx &=\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \sin ^3(e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\left (\cos ^{n p}(e+f x) \sin ^{-n p}(e+f x) \left (b (c \tan (e+f x))^n\right )^p\right ) \int \cos ^{-n p}(e+f x) \sin ^{3+n p}(e+f x) \, dx\\ &=\frac {\cos ^2(e+f x)^{\frac {1}{2} (1+n p)} \, _2F_1\left (\frac {1}{2} (1+n p),\frac {1}{2} (4+n p);\frac {1}{2} (6+n p);\sin ^2(e+f x)\right ) \sin ^3(e+f x) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (4+n p)}\\ \end {align*}
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Mathematica [C] time = 2.84, size = 506, normalized size = 5.44 \[ \frac {4 (n p+4) \sin \left (\frac {1}{2} (e+f x)\right ) \sin ^3(e+f x) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \left (F_1\left (\frac {n p}{2}+1;n p,3;\frac {n p}{2}+2;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-F_1\left (\frac {n p}{2}+1;n p,4;\frac {n p}{2}+2;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+2) \left (2 (n p+4) \cos ^2\left (\frac {1}{2} (e+f x)\right ) F_1\left (\frac {n p}{2}+1;n p,3;\frac {n p}{2}+2;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 (n p+4) \cos ^2\left (\frac {1}{2} (e+f x)\right ) F_1\left (\frac {n p}{2}+1;n p,4;\frac {n p}{2}+2;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 (\cos (e+f x)-1) \left (3 F_1\left (\frac {n p}{2}+2;n p,4;\frac {n p}{2}+3;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 F_1\left (\frac {n p}{2}+2;n p,5;\frac {n p}{2}+3;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+n p \left (F_1\left (\frac {n p}{2}+2;n p+1,4;\frac {n p}{2}+3;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-F_1\left (\frac {n p}{2}+2;n p+1,3;\frac {n p}{2}+3;\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (f x + e\right )^{2} - 1\right )} \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \sin \left (f x + e\right ), x\right ) \]
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 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \sin \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 33.92, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{3}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \sin \left (f x + e\right )^{3}\,{d x} \]
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 \[ \int {\sin \left (e+f\,x\right )}^3\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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