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 = 0.0085539, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {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 4122
Rule 195
Rule 215
Rubi steps
\begin{align*} \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*}
Mathematica [B] time = 0.0565225, 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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Maple [B] time = 0.065, size = 53, normalized size = 2.4 \begin{align*}{\frac{\cos \left ( x \right ) }{2} \left ( \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) -\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) - \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +\sin \left ( x \right ) \right ) \left ( \left ( \cos \left ( x \right ) \right ) ^{-2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.7188, size = 24, normalized size = 1.09 \begin{align*} \frac{1}{2} \, \sqrt{\tan \left (x\right )^{2} + 1} \tan \left (x\right ) + \frac{1}{2} \, \operatorname{arsinh}\left (\tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.33071, size = 109, normalized size = 4.95 \begin{align*} -\frac{\cos \left (x\right )^{2} \log \left (\sin \left (x\right ) + 1\right ) - \cos \left (x\right )^{2} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \, \sin \left (x\right )}{4 \, \cos \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\sec ^{2}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.36668, size = 59, normalized size = 2.68 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{4 \, \mathrm{sgn}\left (\cos \left (x\right )\right )} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{4 \, \mathrm{sgn}\left (\cos \left (x\right )\right )} - \frac{\sin \left (x\right )}{2 \,{\left (\sin \left (x\right )^{2} - 1\right )} \mathrm{sgn}\left (\cos \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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