Optimal. Leaf size=84 \[ -\frac{x}{2}+\frac{\tan ^{-1}\left (\frac{1-\tan (x)}{\sqrt{2} \sqrt{\tan (x)}}\right )}{\sqrt{2}}+\frac{1}{1-\sqrt{\tan (x)}}+\log \left (1-\sqrt{\tan (x)}\right )+\frac{1}{2} \log (\cos (x))+\frac{\tanh ^{-1}\left (\frac{\tan (x)+1}{\sqrt{2} \sqrt{\tan (x)}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.371099, antiderivative size = 133, normalized size of antiderivative = 1.58, number of steps used = 19, number of rules used = 13, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {3670, 6725, 1831, 297, 1162, 617, 204, 1165, 628, 1248, 635, 203, 260} \[ -\frac{x}{2}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (x)}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (x)}+1\right )}{\sqrt{2}}+\frac{1}{1-\sqrt{\tan (x)}}+\log \left (1-\sqrt{\tan (x)}\right )-\frac{\log \left (\tan (x)-\sqrt{2} \sqrt{\tan (x)}+1\right )}{2 \sqrt{2}}+\frac{\log \left (\tan (x)+\sqrt{2} \sqrt{\tan (x)}+1\right )}{2 \sqrt{2}}+\frac{1}{2} \log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 3670
Rule 6725
Rule 1831
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 1248
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{\tan (x)}{\left (-1+\sqrt{\tan (x)}\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{x}{\left (-1+\sqrt{x}\right )^2 \left (1+x^2\right )} \, dx,x,\tan (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x^3}{(-1+x)^2 \left (1+x^4\right )} \, dx,x,\sqrt{\tan (x)}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{2 (-1+x)^2}+\frac{1}{2 (-1+x)}-\frac{x (1+x)^2}{2 \left (1+x^4\right )}\right ) \, dx,x,\sqrt{\tan (x)}\right )\\ &=\log \left (1-\sqrt{\tan (x)}\right )+\frac{1}{1-\sqrt{\tan (x)}}-\operatorname{Subst}\left (\int \frac{x (1+x)^2}{1+x^4} \, dx,x,\sqrt{\tan (x)}\right )\\ &=\log \left (1-\sqrt{\tan (x)}\right )+\frac{1}{1-\sqrt{\tan (x)}}-\operatorname{Subst}\left (\int \left (\frac{2 x^2}{1+x^4}+\frac{x \left (1+x^2\right )}{1+x^4}\right ) \, dx,x,\sqrt{\tan (x)}\right )\\ &=\log \left (1-\sqrt{\tan (x)}\right )+\frac{1}{1-\sqrt{\tan (x)}}-2 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{\tan (x)}\right )-\operatorname{Subst}\left (\int \frac{x \left (1+x^2\right )}{1+x^4} \, dx,x,\sqrt{\tan (x)}\right )\\ &=\log \left (1-\sqrt{\tan (x)}\right )+\frac{1}{1-\sqrt{\tan (x)}}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+x}{1+x^2} \, dx,x,\tan (x)\right )+\operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (x)}\right )-\operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (x)}\right )\\ &=\log \left (1-\sqrt{\tan (x)}\right )+\frac{1}{1-\sqrt{\tan (x)}}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (x)\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\tan (x)\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (x)}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (x)}\right )-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (x)}\right )}{2 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (x)}\right )}{2 \sqrt{2}}\\ &=-\frac{x}{2}+\frac{1}{2} \log (\cos (x))+\log \left (1-\sqrt{\tan (x)}\right )-\frac{\log \left (1-\sqrt{2} \sqrt{\tan (x)}+\tan (x)\right )}{2 \sqrt{2}}+\frac{\log \left (1+\sqrt{2} \sqrt{\tan (x)}+\tan (x)\right )}{2 \sqrt{2}}+\frac{1}{1-\sqrt{\tan (x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (x)}\right )}{\sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (x)}\right )}{\sqrt{2}}\\ &=-\frac{x}{2}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (x)}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (x)}\right )}{\sqrt{2}}+\frac{1}{2} \log (\cos (x))+\log \left (1-\sqrt{\tan (x)}\right )-\frac{\log \left (1-\sqrt{2} \sqrt{\tan (x)}+\tan (x)\right )}{2 \sqrt{2}}+\frac{\log \left (1+\sqrt{2} \sqrt{\tan (x)}+\tan (x)\right )}{2 \sqrt{2}}+\frac{1}{1-\sqrt{\tan (x)}}\\ \end{align*}
Mathematica [C] time = 0.292647, size = 62, normalized size = 0.74 \[ -\frac{2}{3} \tan ^{\frac{3}{2}}(x) \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\tan ^2(x)\right )-\frac{1}{2} \tan ^{-1}(\tan (x))+\frac{1}{1-\sqrt{\tan (x)}}+\log \left (1-\sqrt{\tan (x)}\right )+\frac{1}{2} \log (\cos (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 97, normalized size = 1.2 \begin{align*} - \left ( -1+\sqrt{\tan \left ( x \right ) } \right ) ^{-1}+\ln \left ( -1+\sqrt{\tan \left ( x \right ) } \right ) -{\frac{\sqrt{2}}{2}\arctan \left ( 1+\sqrt{2}\sqrt{\tan \left ( x \right ) } \right ) }-{\frac{\sqrt{2}}{2}\arctan \left ( -1+\sqrt{2}\sqrt{\tan \left ( x \right ) } \right ) }-{\frac{\sqrt{2}}{4}\ln \left ({ \left ( 1-\sqrt{2}\sqrt{\tan \left ( x \right ) }+\tan \left ( x \right ) \right ) \left ( 1+\sqrt{2}\sqrt{\tan \left ( x \right ) }+\tan \left ( x \right ) \right ) ^{-1}} \right ) }-{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }{4}}-{\frac{x}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42359, size = 158, normalized size = 1.88 \begin{align*} \frac{1}{4} \, \sqrt{2}{\left (\sqrt{2} - 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) - \frac{1}{4} \, \sqrt{2}{\left (\sqrt{2} + 2\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) - \frac{1}{8} \, \sqrt{2}{\left (\sqrt{2} - 2\right )} \log \left (\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \frac{1}{8} \, \sqrt{2}{\left (\sqrt{2} + 2\right )} \log \left (-\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \frac{1}{\sqrt{\tan \left (x\right )} - 1} + \log \left (\sqrt{\tan \left (x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 11.7255, size = 2503, normalized size = 29.8 \begin{align*} \text{result too large to display} \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 \frac{\tan{\left (x \right )}}{\left (\sqrt{\tan{\left (x \right )}} - 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21259, size = 150, normalized size = 1.79 \begin{align*} -\frac{1}{2} \,{\left (\sqrt{2} - 1\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) - \frac{1}{2} \,{\left (\sqrt{2} + 1\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (x\right )}\right )}\right ) + \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \frac{1}{\sqrt{\tan \left (x\right )} - 1} - \frac{1}{4} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \log \left ({\left | \sqrt{\tan \left (x\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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