Optimal. Leaf size=59 \[ \frac {1}{4} a \tan (x) \sec ^2(x) \sqrt {a \sec ^2(x)}-\frac {1}{8} a \tan (x) \sqrt {a \sec ^2(x)}-\frac {1}{8} a \cos (x) \sqrt {a \sec ^2(x)} \tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.12, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3657, 4125, 2611, 3768, 3770} \[ \frac {1}{4} a \tan (x) \sec ^2(x) \sqrt {a \sec ^2(x)}-\frac {1}{8} a \tan (x) \sqrt {a \sec ^2(x)}-\frac {1}{8} a \cos (x) \sqrt {a \sec ^2(x)} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3657
Rule 3768
Rule 3770
Rule 4125
Rubi steps
\begin {align*} \int \tan ^2(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx &=\int \left (a \sec ^2(x)\right )^{3/2} \tan ^2(x) \, dx\\ &=\left (a \cos (x) \sqrt {a \sec ^2(x)}\right ) \int \sec ^3(x) \tan ^2(x) \, dx\\ &=\frac {1}{4} a \sec ^2(x) \sqrt {a \sec ^2(x)} \tan (x)-\frac {1}{4} \left (a \cos (x) \sqrt {a \sec ^2(x)}\right ) \int \sec ^3(x) \, dx\\ &=-\frac {1}{8} a \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} a \sec ^2(x) \sqrt {a \sec ^2(x)} \tan (x)-\frac {1}{8} \left (a \cos (x) \sqrt {a \sec ^2(x)}\right ) \int \sec (x) \, dx\\ &=-\frac {1}{8} a \tanh ^{-1}(\sin (x)) \cos (x) \sqrt {a \sec ^2(x)}-\frac {1}{8} a \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} a \sec ^2(x) \sqrt {a \sec ^2(x)} \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 34, normalized size = 0.58 \[ \frac {1}{8} \left (a \sec ^2(x)\right )^{3/2} \left (2 \tan (x)-\sin (x) \cos (x)+\cos ^3(x) \left (-\tanh ^{-1}(\sin (x))\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 57, normalized size = 0.97 \[ \frac {1}{16} \, a^{\frac {3}{2}} \log \left (2 \, a \tan \relax (x)^{2} - 2 \, \sqrt {a \tan \relax (x)^{2} + a} \sqrt {a} \tan \relax (x) + a\right ) + \frac {1}{8} \, {\left (2 \, a \tan \relax (x)^{3} + a \tan \relax (x)\right )} \sqrt {a \tan \relax (x)^{2} + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 49, normalized size = 0.83 \[ \frac {1}{8} \, {\left (\sqrt {a \tan \relax (x)^{2} + a} {\left (2 \, \tan \relax (x)^{2} + 1\right )} \tan \relax (x) + \sqrt {a} \log \left ({\left | -\sqrt {a} \tan \relax (x) + \sqrt {a \tan \relax (x)^{2} + a} \right |}\right )\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 54, normalized size = 0.92 \[ \frac {\tan \relax (x ) \left (a +a \left (\tan ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}{4}-\frac {a \tan \relax (x ) \sqrt {a +a \left (\tan ^{2}\relax (x )\right )}}{8}-\frac {a^{\frac {3}{2}} \ln \left (\sqrt {a}\, \tan \relax (x )+\sqrt {a +a \left (\tan ^{2}\relax (x )\right )}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.59, size = 934, normalized size = 15.83 \[ \text {result too large to display} \]
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 \[ \int {\mathrm {tan}\relax (x)}^2\,{\left (a\,{\mathrm {tan}\relax (x)}^2+a\right )}^{3/2} \,d x \]
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 \[ \int \left (a \left (\tan ^{2}{\relax (x )} + 1\right )\right )^{\frac {3}{2}} \tan ^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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