Optimal. Leaf size=59 \[ \frac{\tan (c+d x) \left (b \tan ^2(c+d x)\right )^n \, _2F_1\left (1,\frac{1}{2} (2 n+1);\frac{1}{2} (2 n+3);-\tan ^2(c+d x)\right )}{d (2 n+1)} \]
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Rubi [A] time = 0.0379199, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3658, 3476, 364} \[ \frac{\tan (c+d x) \left (b \tan ^2(c+d x)\right )^n \, _2F_1\left (1,\frac{1}{2} (2 n+1);\frac{1}{2} (2 n+3);-\tan ^2(c+d x)\right )}{d (2 n+1)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \left (b \tan ^2(c+d x)\right )^n \, dx &=\left (\tan ^{-2 n}(c+d x) \left (b \tan ^2(c+d x)\right )^n\right ) \int \tan ^{2 n}(c+d x) \, dx\\ &=\frac{\left (\tan ^{-2 n}(c+d x) \left (b \tan ^2(c+d x)\right )^n\right ) \operatorname{Subst}\left (\int \frac{x^{2 n}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\, _2F_1\left (1,\frac{1}{2} (1+2 n);\frac{1}{2} (3+2 n);-\tan ^2(c+d x)\right ) \tan (c+d x) \left (b \tan ^2(c+d x)\right )^n}{d (1+2 n)}\\ \end{align*}
Mathematica [A] time = 0.0455909, size = 49, normalized size = 0.83 \[ \frac{\tan (c+d x) \left (b \tan ^2(c+d x)\right )^n \, _2F_1\left (1,n+\frac{1}{2};n+\frac{3}{2};-\tan ^2(c+d x)\right )}{2 d n+d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.265, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ( \tan \left ( dx+c \right ) \right ) ^{2} \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 (b \tan \left (d x + c\right )^{2}\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 (b \tan \left (d x + c\right )^{2}\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 (b \tan ^{2}{\left (c + d x \right )}\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 (b \tan \left (d x + c\right )^{2}\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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