Optimal. Leaf size=74 \[ \frac {\tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (1,\frac {1}{2} (m+n p+1);\frac {1}{2} (m+n p+3);-\tan ^2(e+f x)\right )}{f (m+n p+1)} \]
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Rubi [A] time = 0.10, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3578, 20, 3476, 364} \[ \frac {\tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (1,\frac {1}{2} (m+n p+1);\frac {1}{2} (m+n p+3);-\tan ^2(e+f x)\right )}{f (m+n p+1)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 364
Rule 3476
Rule 3578
Rubi steps
\begin {align*} \int (d \tan (e+f x))^m \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 (c \tan (e+f x))^{n p} (d \tan (e+f x))^m \, dx\\ &=\left ((c \tan (e+f x))^{-m-n p} (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p\right ) \int (c \tan (e+f x))^{m+n p} \, dx\\ &=\frac {\left (c (c \tan (e+f x))^{-m-n p} (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname {Subst}\left (\int \frac {x^{m+n p}}{c^2+x^2} \, dx,x,c \tan (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (1,\frac {1}{2} (1+m+n p);\frac {1}{2} (3+m+n p);-\tan ^2(e+f x)\right ) \tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p}{f (1+m+n p)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 76, normalized size = 1.03 \[ \frac {\tan (e+f x) (d \tan (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (1,\frac {1}{2} (m+n p+1);\frac {1}{2} (m+n p+1)+1;-\tan ^2(e+f x)\right )}{f (m+n p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \tan \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \tan \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 12.61, size = 0, normalized size = 0.00 \[ \int \left (d \tan \left (f x +e \right )\right )^{m} \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
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 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \tan \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]
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 \[ \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \left (d \tan {\left (e + f x \right )}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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