Optimal. Leaf size=61 \[ \frac{\tan (c+d x) \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(c+d x)\right ) \left (a (b \tan (c+d x))^p\right )^n}{d (n p+1)} \]
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Rubi [A] time = 0.0453724, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3659, 3476, 364} \[ \frac{\tan (c+d x) \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(c+d x)\right ) \left (a (b \tan (c+d x))^p\right )^n}{d (n p+1)} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \left (a (b \tan (c+d x))^p\right )^n \, dx &=\left ((b \tan (c+d x))^{-n p} \left (a (b \tan (c+d x))^p\right )^n\right ) \int (b \tan (c+d x))^{n p} \, dx\\ &=\frac{\left (b (b \tan (c+d x))^{-n p} \left (a (b \tan (c+d x))^p\right )^n\right ) \operatorname{Subst}\left (\int \frac{x^{n p}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{\, _2F_1\left (1,\frac{1}{2} (1+n p);\frac{1}{2} (3+n p);-\tan ^2(c+d x)\right ) \tan (c+d x) \left (a (b \tan (c+d x))^p\right )^n}{d (1+n p)}\\ \end{align*}
Mathematica [A] time = 0.0494637, size = 59, normalized size = 0.97 \[ \frac{\tan (c+d x) \, _2F_1\left (1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(c+d x)\right ) \left (a (b \tan (c+d x))^p\right )^n}{d n p+d} \]
Antiderivative was successfully verified.
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Maple [F] time = 5.166, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ( b\tan \left ( dx+c \right ) \right ) ^{p} \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (b \tan \left (d x + c\right )\right )^{p} a\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (\left (b \tan \left (d x + c\right )\right )^{p} a\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (b \tan{\left (c + d x \right )}\right )^{p}\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (b \tan \left (d x + c\right )\right )^{p} a\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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