Optimal. Leaf size=63 \[ \frac {\, _2F_1\left (1,\frac {1}{2} (3+n p);\frac {1}{2} (5+n p);-\tan ^2(e+f x)\right ) \tan ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3+n p)} \]
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Rubi [A]
time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3740, 16, 3557,
371} \begin {gather*} \frac {\tan ^3(e+f x) \, _2F_1\left (1,\frac {1}{2} (n p+3);\frac {1}{2} (n p+5);-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 371
Rule 3557
Rule 3740
Rubi steps
\begin {align*} \int \tan ^2(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 \tan ^2(e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\frac {\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))^{2+n p} \, dx}{c^2}\\ &=\frac {\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \text {Subst}\left (\int \frac {x^{2+n p}}{c^2+x^2} \, dx,x,c \tan (e+f x)\right )}{c f}\\ &=\frac {\, _2F_1\left (1,\frac {1}{2} (3+n p);\frac {1}{2} (5+n p);-\tan ^2(e+f x)\right ) \tan ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3+n p)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 65, normalized size = 1.03 \begin {gather*} \frac {\, _2F_1\left (1,\frac {1}{2} (3+n p);1+\frac {1}{2} (3+n p);-\tan ^2(e+f x)\right ) \tan ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3+n p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (\tan ^{2}\left (f x +e \right )\right ) \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \tan ^{2}{\left (e + f x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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