Optimal. Leaf size=79 \[ \frac {\sin (e+f x) \cos ^2(e+f x)^{\frac {n p}{2}} \, _2F_1\left (\frac {n p}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]
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Rubi [A] time = 0.08, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3659, 2617} \[ \frac {\sin (e+f x) \cos ^2(e+f x)^{\frac {n p}{2}} \, _2F_1\left (\frac {n p}{2},\frac {1}{2} (n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]
Antiderivative was successfully verified.
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Rule 2617
Rule 3659
Rubi steps
\begin {align*} \int \cos (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 \cos (e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\frac {\cos ^2(e+f x)^{\frac {n p}{2}} \, _2F_1\left (\frac {n p}{2},\frac {1}{2} (1+n p);\frac {1}{2} (3+n p);\sin ^2(e+f x)\right ) \sin (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)}\\ \end {align*}
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Mathematica [C] time = 3.61, size = 482, normalized size = 6.10 \[ \frac {(n p+3) \sin (2 (e+f x)) \left (F_1\left (\frac {1}{2} (n p+1);n p,1;\frac {1}{2} (n p+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 F_1\left (\frac {1}{2} (n p+1);n p,2;\frac {1}{2} (n p+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \left (b (c \tan (e+f x))^n\right )^p}{2 f (n p+1) \left ((n p+3) F_1\left (\frac {1}{2} (n p+1);n p,1;\frac {1}{2} (n p+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 \left (\tan ^2\left (\frac {1}{2} (e+f x)\right ) \left (F_1\left (\frac {1}{2} (n p+3);n p,2;\frac {1}{2} (n p+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 F_1\left (\frac {1}{2} (n p+3);n p,3;\frac {1}{2} (n p+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n p F_1\left (\frac {1}{2} (n p+3);n p+1,1;\frac {1}{2} (n p+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 n p F_1\left (\frac {1}{2} (n p+3);n p+1,2;\frac {1}{2} (n p+5);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )+(n p+3) F_1\left (\frac {1}{2} (n p+1);n p,2;\frac {1}{2} (n p+3);\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \cos \left (f x + e\right ), x\right ) \]
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 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \cos \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 15.82, size = 0, normalized size = 0.00 \[ \int \cos \left (f x +e \right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
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 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \cos \left (f x + e\right )\,{d x} \]
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 \[ \int \cos \left (e+f\,x\right )\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \cos {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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