Optimal. Leaf size=54 \[ \frac {1}{4} \tan ^3(x) \sqrt {a \sec ^2(x)}-\frac {3}{8} \tan (x) \sqrt {a \sec ^2(x)}+\frac {3}{8} \cos (x) \sqrt {a \sec ^2(x)} \tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.11, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3657, 4125, 2611, 3770} \[ \frac {1}{4} \tan ^3(x) \sqrt {a \sec ^2(x)}-\frac {3}{8} \tan (x) \sqrt {a \sec ^2(x)}+\frac {3}{8} \cos (x) \sqrt {a \sec ^2(x)} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3657
Rule 3770
Rule 4125
Rubi steps
\begin {align*} \int \tan ^4(x) \sqrt {a+a \tan ^2(x)} \, dx &=\int \sqrt {a \sec ^2(x)} \tan ^4(x) \, dx\\ &=\left (\cos (x) \sqrt {a \sec ^2(x)}\right ) \int \sec (x) \tan ^4(x) \, dx\\ &=\frac {1}{4} \sqrt {a \sec ^2(x)} \tan ^3(x)-\frac {1}{4} \left (3 \cos (x) \sqrt {a \sec ^2(x)}\right ) \int \sec (x) \tan ^2(x) \, dx\\ &=-\frac {3}{8} \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} \sqrt {a \sec ^2(x)} \tan ^3(x)+\frac {1}{8} \left (3 \cos (x) \sqrt {a \sec ^2(x)}\right ) \int \sec (x) \, dx\\ &=\frac {3}{8} \tanh ^{-1}(\sin (x)) \cos (x) \sqrt {a \sec ^2(x)}-\frac {3}{8} \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{4} \sqrt {a \sec ^2(x)} \tan ^3(x)\\ \end {align*}
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Mathematica [A] time = 0.08, size = 32, normalized size = 0.59 \[ \frac {1}{8} \sqrt {a \sec ^2(x)} \left (2 \tan ^3(x)-3 \tan (x)+3 \cos (x) \tanh ^{-1}(\sin (x))\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 56, normalized size = 1.04 \[ \frac {1}{8} \, \sqrt {a \tan \relax (x)^{2} + a} {\left (2 \, \tan \relax (x)^{3} - 3 \, \tan \relax (x)\right )} + \frac {3}{16} \, \sqrt {a} \log \left (2 \, a \tan \relax (x)^{2} + 2 \, \sqrt {a \tan \relax (x)^{2} + a} \sqrt {a} \tan \relax (x) + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 48, normalized size = 0.89 \[ \frac {1}{8} \, \sqrt {a \tan \relax (x)^{2} + a} {\left (2 \, \tan \relax (x)^{2} - 3\right )} \tan \relax (x) - \frac {3}{8} \, \sqrt {a} \log \left ({\left | -\sqrt {a} \tan \relax (x) + \sqrt {a \tan \relax (x)^{2} + a} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 56, normalized size = 1.04 \[ \frac {\tan \relax (x ) \left (a +a \left (\tan ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}{4 a}-\frac {5 \sqrt {a +a \left (\tan ^{2}\relax (x )\right )}\, \tan \relax (x )}{8}+\frac {3 \sqrt {a}\, \ln \left (\sqrt {a}\, \tan \relax (x )+\sqrt {a +a \left (\tan ^{2}\relax (x )\right )}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.55, size = 860, normalized size = 15.93 \[ \text {result too large to display} \]
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 \[ \int {\mathrm {tan}\relax (x)}^4\,\sqrt {a\,{\mathrm {tan}\relax (x)}^2+a} \,d x \]
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 \[ \int \sqrt {a \left (\tan ^{2}{\relax (x )} + 1\right )} \tan ^{4}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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