Optimal. Leaf size=36 \[ \frac{1}{2} \tan (x) \sqrt{a \sec ^2(x)}-\frac{1}{2} \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.0926689, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3657, 4125, 2611, 3770} \[ \frac{1}{2} \tan (x) \sqrt{a \sec ^2(x)}-\frac{1}{2} \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4125
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \tan ^2(x) \sqrt{a+a \tan ^2(x)} \, dx &=\int \sqrt{a \sec ^2(x)} \tan ^2(x) \, dx\\ &=\left (\cos (x) \sqrt{a \sec ^2(x)}\right ) \int \sec (x) \tan ^2(x) \, dx\\ &=\frac{1}{2} \sqrt{a \sec ^2(x)} \tan (x)-\frac{1}{2} \left (\cos (x) \sqrt{a \sec ^2(x)}\right ) \int \sec (x) \, dx\\ &=-\frac{1}{2} \tanh ^{-1}(\sin (x)) \cos (x) \sqrt{a \sec ^2(x)}+\frac{1}{2} \sqrt{a \sec ^2(x)} \tan (x)\\ \end{align*}
Mathematica [A] time = 0.0490207, size = 24, normalized size = 0.67 \[ \frac{1}{2} \sqrt{a \sec ^2(x)} \left (\tan (x)-\cos (x) \tanh ^{-1}(\sin (x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 39, normalized size = 1.1 \begin{align*}{\frac{\tan \left ( x \right ) }{2}\sqrt{a+a \left ( \tan \left ( x \right ) \right ) ^{2}}}-{\frac{1}{2}\sqrt{a}\ln \left ( \sqrt{a}\tan \left ( x \right ) +\sqrt{a+a \left ( \tan \left ( x \right ) \right ) ^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.87742, size = 398, normalized size = 11.06 \begin{align*} \frac{{\left (4 \,{\left (\sin \left (3 \, x\right ) - \sin \left (x\right )\right )} \cos \left (4 \, x\right ) -{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) +{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) - 4 \,{\left (\cos \left (3 \, x\right ) - \cos \left (x\right )\right )} \sin \left (4 \, x\right ) + 4 \,{\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (3 \, x\right ) - 8 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 8 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \sin \left (x\right ) - 4 \, \sin \left (x\right )\right )} \sqrt{a}}{4 \,{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41434, size = 147, normalized size = 4.08 \begin{align*} \frac{1}{4} \, \sqrt{a} \log \left (2 \, a \tan \left (x\right )^{2} - 2 \, \sqrt{a \tan \left (x\right )^{2} + a} \sqrt{a} \tan \left (x\right ) + a\right ) + \frac{1}{2} \, \sqrt{a \tan \left (x\right )^{2} + a} \tan \left (x\right ) \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 \sqrt{a \left (\tan ^{2}{\left (x \right )} + 1\right )} \tan ^{2}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17488, size = 54, normalized size = 1.5 \begin{align*} \frac{1}{2} \, \sqrt{a} \log \left ({\left | -\sqrt{a} \tan \left (x\right ) + \sqrt{a \tan \left (x\right )^{2} + a} \right |}\right ) + \frac{1}{2} \, \sqrt{a \tan \left (x\right )^{2} + a} \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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