Optimal. Leaf size=59 \[ \frac {\, _2F_1\left (1,\frac {1}{2} (1+2 n);\frac {1}{2} (3+2 n);-\tan ^2(c+d x)\right ) \tan (c+d x) \left (b \tan ^2(c+d x)\right )^n}{d (1+2 n)} \]
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Rubi [A]
time = 0.03, 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 = {3739, 3557,
371} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 3557
Rule 3739
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 ) \text {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*}
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Mathematica [A]
time = 0.06, size = 49, normalized size = 0.83 \begin {gather*} \frac {\, _2F_1\left (1,\frac {1}{2}+n;\frac {3}{2}+n;-\tan ^2(c+d x)\right ) \tan (c+d x) \left (b \tan ^2(c+d x)\right )^n}{d+2 d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \left (b \left (\tan ^{2}\left (d x +c \right )\right )\right )^{n}\, dx\]
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 \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 \tan ^{2}{\left (c + d x \right )}\right )^{n}\, 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 {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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