Optimal. Leaf size=59 \[ \frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p \, _2F_1\left (1,\frac {1}{2} (2 p+1);\frac {1}{2} (2 p+3);-\tan ^2(e+f x)\right )}{f (2 p+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3658, 3476, 364} \[ \frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p \, _2F_1\left (1,\frac {1}{2} (2 p+1);\frac {1}{2} (2 p+3);-\tan ^2(e+f x)\right )}{f (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \left (b \tan ^2(e+f x)\right )^p \, dx &=\left (\tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int \tan ^{2 p}(e+f x) \, dx\\ &=\frac {\left (\tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \operatorname {Subst}\left (\int \frac {x^{2 p}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (1,\frac {1}{2} (1+2 p);\frac {1}{2} (3+2 p);-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 49, normalized size = 0.83 \[ \frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p \, _2F_1\left (1,p+\frac {1}{2};p+\frac {3}{2};-\tan ^2(e+f x)\right )}{2 f p+f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \tan \left (f x + e\right )^{2}\right )^{p}, x\right ) \]
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 \[ \int \left (b \tan \left (f x + e\right )^{2}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.23, size = 0, normalized size = 0.00 \[ \int \left (b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx \]
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 \[ \int \left (b \tan \left (f x + e\right )^{2}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^p \,d x \]
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 \[ \int \left (b \tan ^{2}{\left (e + f x \right )}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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