Optimal. Leaf size=210 \[ \frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{\sqrt{2} b}-\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right )}{\sqrt{2} b}+\frac{d^{3/2} \log \left (\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{2 \sqrt{2} b}-\frac{d^{3/2} \log \left (\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{2 \sqrt{2} b}+\frac{2 d \sqrt{d \tan (a+b x)}}{b} \]
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Rubi [A] time = 0.141573, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{\sqrt{2} b}-\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}+1\right )}{\sqrt{2} b}+\frac{d^{3/2} \log \left (\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{2 \sqrt{2} b}-\frac{d^{3/2} \log \left (\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}+\sqrt{d}\right )}{2 \sqrt{2} b}+\frac{2 d \sqrt{d \tan (a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int (d \tan (a+b x))^{3/2} \, dx &=\frac{2 d \sqrt{d \tan (a+b x)}}{b}-d^2 \int \frac{1}{\sqrt{d \tan (a+b x)}} \, dx\\ &=\frac{2 d \sqrt{d \tan (a+b x)}}{b}-\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (d^2+x^2\right )} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac{2 d \sqrt{d \tan (a+b x)}}{b}-\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{d^2+x^4} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{b}\\ &=\frac{2 d \sqrt{d \tan (a+b x)}}{b}-\frac{d^2 \operatorname{Subst}\left (\int \frac{d-x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{b}-\frac{d^2 \operatorname{Subst}\left (\int \frac{d+x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{b}\\ &=\frac{2 d \sqrt{d \tan (a+b x)}}{b}+\frac{d^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}+2 x}{-d-\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{2 \sqrt{2} b}+\frac{d^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}-2 x}{-d+\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{2 \sqrt{2} b}-\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{2 b}-\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (a+b x)}\right )}{2 b}\\ &=\frac{d^{3/2} \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{2 \sqrt{2} b}-\frac{d^{3/2} \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{2 \sqrt{2} b}+\frac{2 d \sqrt{d \tan (a+b x)}}{b}-\frac{d^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{\sqrt{2} b}+\frac{d^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{\sqrt{2} b}\\ &=\frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{\sqrt{2} b}-\frac{d^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \tan (a+b x)}}{\sqrt{d}}\right )}{\sqrt{2} b}+\frac{d^{3/2} \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)-\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{2 \sqrt{2} b}-\frac{d^{3/2} \log \left (\sqrt{d}+\sqrt{d} \tan (a+b x)+\sqrt{2} \sqrt{d \tan (a+b x)}\right )}{2 \sqrt{2} b}+\frac{2 d \sqrt{d \tan (a+b x)}}{b}\\ \end{align*}
Mathematica [A] time = 0.25994, size = 159, normalized size = 0.76 \[ \frac{(d \tan (a+b x))^{3/2} \left (2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (a+b x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (a+b x)}+1\right )+8 \sqrt{\tan (a+b x)}+\sqrt{2} \log \left (\tan (a+b x)-\sqrt{2} \sqrt{\tan (a+b x)}+1\right )-\sqrt{2} \log \left (\tan (a+b x)+\sqrt{2} \sqrt{\tan (a+b x)}+1\right )\right )}{4 b \tan ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 176, normalized size = 0.8 \begin{align*} 2\,{\frac{d\sqrt{d\tan \left ( bx+a \right ) }}{b}}-{\frac{d\sqrt{2}}{2\,b}\sqrt [4]{{d}^{2}}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( bx+a \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{d\sqrt{2}}{2\,b}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( bx+a \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }-{\frac{d\sqrt{2}}{4\,b}\sqrt [4]{{d}^{2}}\ln \left ({ \left ( d\tan \left ( bx+a \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( bx+a \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( bx+a \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( bx+a \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73654, size = 1328, normalized size = 6.32 \begin{align*} \frac{4 \, \sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{1}{4}} b \arctan \left (-\frac{d^{6} + \sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{3}{4}} b^{3} d \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} - \sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{3}{4}} b^{3} \sqrt{\frac{\sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{1}{4}} b d \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right ) + d^{3} \sin \left (b x + a\right ) + \sqrt{\frac{d^{6}}{b^{4}}} b^{2} \cos \left (b x + a\right )}{\cos \left (b x + a\right )}}}{d^{6}}\right ) + 4 \, \sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{1}{4}} b \arctan \left (\frac{d^{6} - \sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{3}{4}} b^{3} d \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} + \sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{3}{4}} b^{3} \sqrt{-\frac{\sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{1}{4}} b d \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right ) - d^{3} \sin \left (b x + a\right ) - \sqrt{\frac{d^{6}}{b^{4}}} b^{2} \cos \left (b x + a\right )}{\cos \left (b x + a\right )}}}{d^{6}}\right ) - \sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{1}{4}} b \log \left (\frac{\sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{1}{4}} b d \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right ) + d^{3} \sin \left (b x + a\right ) + \sqrt{\frac{d^{6}}{b^{4}}} b^{2} \cos \left (b x + a\right )}{\cos \left (b x + a\right )}\right ) + \sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{1}{4}} b \log \left (-\frac{\sqrt{2} \left (\frac{d^{6}}{b^{4}}\right )^{\frac{1}{4}} b d \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right ) - d^{3} \sin \left (b x + a\right ) - \sqrt{\frac{d^{6}}{b^{4}}} b^{2} \cos \left (b x + a\right )}{\cos \left (b x + a\right )}\right ) + 8 \, d \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{4 \, b} \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 (d \tan{\left (a + b x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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