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 = 0.37, antiderivative size = 133, normalized size of antiderivative = 1.58, number of steps used = 19, number of rules used = 13, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, 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 203
Rule 204
Rule 260
Rule 297
Rule 617
Rule 628
Rule 635
Rule 1162
Rule 1165
Rule 1248
Rule 1831
Rule 3670
Rule 6725
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*}
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Mathematica [C] time = 0.28, 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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan (x)}{\left (-1+\sqrt {\tan (x)}\right )^2} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 3.25, size = 603, normalized size = 7.18 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 111, normalized size = 1.32 \[ -\frac {1}{2} \, {\left (\sqrt {2} - 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \relax (x)}\right )}\right ) - \frac {1}{2} \, {\left (\sqrt {2} + 1\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \relax (x)}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right ) - \frac {1}{\sqrt {\tan \relax (x)} - 1} - \frac {1}{4} \, \log \left (\tan \relax (x)^{2} + 1\right ) + \log \left ({\left | \sqrt {\tan \relax (x)} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 94, normalized size = 1.12
method | result | size |
derivativedivides | \(-\frac {\arctan \left (\tan \relax (x )\right )}{2}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )+\tan \relax (x )}{1+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )+\tan \relax (x )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )\right )\right )}{4}-\frac {\ln \left (1+\tan ^{2}\relax (x )\right )}{4}-\frac {1}{-1+\sqrt {\tan }\relax (x )}+\ln \left (-1+\sqrt {\tan }\relax (x )\right )\) | \(94\) |
default | \(-\frac {\arctan \left (\tan \relax (x )\right )}{2}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )+\tan \relax (x )}{1+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )+\tan \relax (x )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\relax (x )\right )\right )\right )}{4}-\frac {\ln \left (1+\tan ^{2}\relax (x )\right )}{4}-\frac {1}{-1+\sqrt {\tan }\relax (x )}+\ln \left (-1+\sqrt {\tan }\relax (x )\right )\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 117, normalized size = 1.39 \[ \frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} - 2\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \relax (x)}\right )}\right ) - \frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} + 2\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \relax (x)}\right )}\right ) - \frac {1}{8} \, \sqrt {2} {\left (\sqrt {2} - 2\right )} \log \left (\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right ) - \frac {1}{8} \, \sqrt {2} {\left (\sqrt {2} + 2\right )} \log \left (-\sqrt {2} \sqrt {\tan \relax (x)} + \tan \relax (x) + 1\right ) - \frac {1}{\sqrt {\tan \relax (x)} - 1} + \log \left (\sqrt {\tan \relax (x)} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 228, normalized size = 2.71 \[ \ln \left (612\,\sqrt {\mathrm {tan}\relax (x)}-612\right )-\frac {1}{\sqrt {\mathrm {tan}\relax (x)}-1}+\left (\sum _{k=1}^4\ln \left (4\,\sqrt {\mathrm {tan}\relax (x)}+{\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}-\frac {z}{8}+\frac {1}{64},z,k\right )}^2\,\sqrt {\mathrm {tan}\relax (x)}\,80+{\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}-\frac {z}{8}+\frac {1}{64},z,k\right )}^3\,\sqrt {\mathrm {tan}\relax (x)}\,448+{\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}-\frac {z}{8}+\frac {1}{64},z,k\right )}^4\,\sqrt {\mathrm {tan}\relax (x)}\,128+32\,{\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}-\frac {z}{8}+\frac {1}{64},z,k\right )}^2-384\,{\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}-\frac {z}{8}+\frac {1}{64},z,k\right )}^3-256\,{\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}-\frac {z}{8}+\frac {1}{64},z,k\right )}^4-\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}-\frac {z}{8}+\frac {1}{64},z,k\right )\,\sqrt {\mathrm {tan}\relax (x)}\,48-4\right )\,\mathrm {root}\left (z^4+z^3+\frac {z^2}{2}-\frac {z}{8}+\frac {1}{64},z,k\right )\right ) \]
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 \[ \int \frac {\tan {\relax (x )}}{\left (\sqrt {\tan {\relax (x )}} - 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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