Optimal. Leaf size=35 \[ \frac {1}{2} \tan ^{-1}\left (\frac {\tan (x)}{\sqrt {-\sec ^2(x)}}\right )-\frac {1}{2} \tan (x) \sqrt {-\sec ^2(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3657, 4122, 195, 217, 203} \[ \frac {1}{2} \tan ^{-1}\left (\frac {\tan (x)}{\sqrt {-\sec ^2(x)}}\right )-\frac {1}{2} \tan (x) \sqrt {-\sec ^2(x)} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 3657
Rule 4122
Rubi steps
\begin {align*} \int \left (-1-\tan ^2(x)\right )^{3/2} \, dx &=\int \left (-\sec ^2(x)\right )^{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} \sqrt {-\sec ^2(x)} \tan (x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\tan (x)}{\sqrt {-\sec ^2(x)}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {\tan (x)}{\sqrt {-\sec ^2(x)}}\right )-\frac {1}{2} \sqrt {-\sec ^2(x)} \tan (x)\\ \end {align*}
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Mathematica [B] time = 0.07, size = 72, normalized size = 2.06 \[ \frac {1}{4} \cos (x) \sqrt {-\sec ^2(x)} \left (\frac {1}{\sin (x)-1}+\frac {1}{\left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2}+2 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-2 \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.45, size = 73, normalized size = 2.09 \[ \frac {{\left (-i \, e^{\left (4 i \, x\right )} - 2 i \, e^{\left (2 i \, x\right )} - i\right )} \log \left (e^{\left (i \, x\right )} + i\right ) + {\left (i \, e^{\left (4 i \, x\right )} + 2 i \, e^{\left (2 i \, x\right )} + i\right )} \log \left (e^{\left (i \, x\right )} - i\right ) - 2 \, e^{\left (3 i \, x\right )} + 2 \, e^{\left (i \, x\right )}}{2 \, {\left (e^{\left (4 i \, x\right )} + 2 \, e^{\left (2 i \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.30, size = 29, normalized size = 0.83 \[ -\frac {1}{2} i \, \sqrt {\tan \relax (x)^{2} + 1} \tan \relax (x) + \frac {1}{2} i \, \log \left (\sqrt {\tan \relax (x)^{2} + 1} - \tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 32, normalized size = 0.91 \[ -\frac {\tan \relax (x ) \sqrt {-1-\left (\tan ^{2}\relax (x )\right )}}{2}+\frac {\arctan \left (\frac {\tan \relax (x )}{\sqrt {-1-\left (\tan ^{2}\relax (x )\right )}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.58, size = 20, normalized size = 0.57 \[ -\frac {1}{2} \, \sqrt {-\tan \relax (x)^{2} - 1} \tan \relax (x) - \frac {1}{2} i \, \operatorname {arsinh}\left (\tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.59, size = 31, normalized size = 0.89 \[ \frac {\mathrm {atan}\left (\frac {\mathrm {tan}\relax (x)}{\sqrt {-{\mathrm {tan}\relax (x)}^2-1}}\right )}{2}-\frac {\mathrm {tan}\relax (x)\,\sqrt {-{\mathrm {tan}\relax (x)}^2-1}}{2} \]
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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