Optimal. Leaf size=210 \[ \frac{b^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right )}{\sqrt{2} d}+\frac{b^{3/2} \log \left (\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} d}-\frac{b^{3/2} \log \left (\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} d}+\frac{2 b \sqrt{b \tan (c+d x)}}{d} \]
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Rubi [A] time = 0.151164, 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} \[ \frac{b^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right )}{\sqrt{2} d}+\frac{b^{3/2} \log \left (\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} d}-\frac{b^{3/2} \log \left (\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} d}+\frac{2 b \sqrt{b \tan (c+d x)}}{d} \]
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 (b \tan (c+d x))^{3/2} \, dx &=\frac{2 b \sqrt{b \tan (c+d x)}}{d}-b^2 \int \frac{1}{\sqrt{b \tan (c+d x)}} \, dx\\ &=\frac{2 b \sqrt{b \tan (c+d x)}}{d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{2 b \sqrt{b \tan (c+d x)}}{d}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{d}\\ &=\frac{2 b \sqrt{b \tan (c+d x)}}{d}-\frac{b^2 \operatorname{Subst}\left (\int \frac{b-x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{d}-\frac{b^2 \operatorname{Subst}\left (\int \frac{b+x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{d}\\ &=\frac{2 b \sqrt{b \tan (c+d x)}}{d}+\frac{b^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{b}+2 x}{-b-\sqrt{2} \sqrt{b} x-x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}+\frac{b^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{b}-2 x}{-b+\sqrt{2} \sqrt{b} x-x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{b-\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 d}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{b+\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 d}\\ &=\frac{b^{3/2} \log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}-\frac{b^{3/2} \log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}+\frac{2 b \sqrt{b \tan (c+d x)}}{d}-\frac{b^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}+\frac{b^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}\\ &=\frac{b^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}-\frac{b^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} d}+\frac{b^{3/2} \log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}-\frac{b^{3/2} \log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} d}+\frac{2 b \sqrt{b \tan (c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.159505, size = 159, normalized size = 0.76 \[ \frac{(b \tan (c+d x))^{3/2} \left (2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )+8 \sqrt{\tan (c+d x)}+\sqrt{2} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-\sqrt{2} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{4 d \tan ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 176, normalized size = 0.8 \begin{align*} 2\,{\frac{b\sqrt{b\tan \left ( dx+c \right ) }}{d}}-{\frac{b\sqrt{2}}{4\,d}\sqrt [4]{{b}^{2}}\ln \left ({ \left ( b\tan \left ( dx+c \right ) +\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}+\sqrt{{b}^{2}} \right ) \left ( b\tan \left ( dx+c \right ) -\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}+\sqrt{{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{b\sqrt{2}}{2\,d}\sqrt [4]{{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{b}^{2}}}}}+1 \right ) }+{\frac{b\sqrt{2}}{2\,d}\sqrt [4]{{b}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{b}^{2}}}}}+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 = 1.696, size = 1328, normalized size = 6.32 \begin{align*} \frac{4 \, \sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{1}{4}} d \arctan \left (-\frac{b^{6} + \sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{3}{4}} b d^{3} \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} - \sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{3}{4}} d^{3} \sqrt{\frac{\sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{1}{4}} b d \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + b^{3} \sin \left (d x + c\right ) + \sqrt{\frac{b^{6}}{d^{4}}} d^{2} \cos \left (d x + c\right )}{\cos \left (d x + c\right )}}}{b^{6}}\right ) + 4 \, \sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{1}{4}} d \arctan \left (\frac{b^{6} - \sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{3}{4}} b d^{3} \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} + \sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{3}{4}} d^{3} \sqrt{-\frac{\sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{1}{4}} b d \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - b^{3} \sin \left (d x + c\right ) - \sqrt{\frac{b^{6}}{d^{4}}} d^{2} \cos \left (d x + c\right )}{\cos \left (d x + c\right )}}}{b^{6}}\right ) - \sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{1}{4}} d \log \left (\frac{\sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{1}{4}} b d \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + b^{3} \sin \left (d x + c\right ) + \sqrt{\frac{b^{6}}{d^{4}}} d^{2} \cos \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) + \sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{1}{4}} d \log \left (-\frac{\sqrt{2} \left (\frac{b^{6}}{d^{4}}\right )^{\frac{1}{4}} b d \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - b^{3} \sin \left (d x + c\right ) - \sqrt{\frac{b^{6}}{d^{4}}} d^{2} \cos \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) + 8 \, b \sqrt{\frac{b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}}}{4 \, d} \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 (b \tan{\left (c + d 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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