Optimal. Leaf size=22 \[ \frac {1}{2} \tan (x) \sqrt {\sec ^2(x)}+\frac {1}{2} \sinh ^{-1}(\tan (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3657, 4122, 195, 215} \[ \frac {1}{2} \tan (x) \sqrt {\sec ^2(x)}+\frac {1}{2} \sinh ^{-1}(\tan (x)) \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 3657
Rule 4122
Rubi steps
\begin {align*} \int \left (1+\tan ^2(x)\right )^{3/2} \, dx &=\int \sec ^2(x)^{3/2} \, dx\\ &=\operatorname {Subst}\left (\int \sqrt {1+x^2} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \sqrt {\sec ^2(x)} \tan (x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \sinh ^{-1}(\tan (x))+\frac {1}{2} \sqrt {\sec ^2(x)} \tan (x)\\ \end {align*}
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Mathematica [B] time = 0.07, size = 52, normalized size = 2.36 \[ \frac {1}{2} \cos (x) \sqrt {\sec ^2(x)} \left (\tan (x) \sec (x)-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 72, normalized size = 3.27 \[ \frac {1}{2} \, \sqrt {\tan \relax (x)^{2} + 1} \tan \relax (x) + \frac {1}{4} \, \log \left (\frac {\tan \relax (x)^{2} + \sqrt {\tan \relax (x)^{2} + 1} \tan \relax (x) + 1}{\tan \relax (x)^{2} + 1}\right ) - \frac {1}{4} \, \log \left (\frac {\tan \relax (x)^{2} - \sqrt {\tan \relax (x)^{2} + 1} \tan \relax (x) + 1}{\tan \relax (x)^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 29, normalized size = 1.32 \[ \frac {1}{2} \, \sqrt {\tan \relax (x)^{2} + 1} \tan \relax (x) - \frac {1}{2} \, \log \left (\sqrt {\tan \relax (x)^{2} + 1} - \tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 19, normalized size = 0.86 \[ \frac {\tan \relax (x ) \sqrt {1+\tan ^{2}\relax (x )}}{2}+\frac {\arcsinh \left (\tan \relax (x )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 18, normalized size = 0.82 \[ \frac {1}{2} \, \sqrt {\tan \relax (x)^{2} + 1} \tan \relax (x) + \frac {1}{2} \, \operatorname {arsinh}\left (\tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 18, normalized size = 0.82 \[ \frac {\mathrm {asinh}\left (\mathrm {tan}\relax (x)\right )}{2}+\frac {\mathrm {tan}\relax (x)\,\sqrt {{\mathrm {tan}\relax (x)}^2+1}}{2} \]
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 \[ \int \left (\tan ^{2}{\relax (x )} + 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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