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 = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 364
Rule 3476
Rule 3659
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*}
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Mathematica [A] time = 0.05, 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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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\left (b \tan \left (d x + c\right )\right )^{p} a\right )^{n}, x\right ) \]
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 \[ \int \left (\left (b \tan \left (d x + c\right )\right )^{p} a\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (b \tan \left (d x +c \right )\right )^{p}\right )^{n}\, dx \]
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 \[ \int \left (\left (b \tan \left (d x + c\right )\right )^{p} a\right )^{n}\,{d x} \]
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 \[ \int {\left (a\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^p\right )}^n \,d x \]
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 \[ \int \left (a \left (b \tan {\left (c + d x \right )}\right )^{p}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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