Optimal. Leaf size=212 \[ \frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} b^{3/2} d}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right )}{\sqrt{2} b^{3/2} d}-\frac{\log \left (\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} b^{3/2} d}+\frac{\log \left (\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} b^{3/2} d}-\frac{2}{b d \sqrt{b \tan (c+d x)}} \]
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Rubi [A] time = 0.146247, antiderivative size = 212, 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 = {3474, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} b^{3/2} d}-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right )}{\sqrt{2} b^{3/2} d}-\frac{\log \left (\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} b^{3/2} d}+\frac{\log \left (\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} b^{3/2} d}-\frac{2}{b d \sqrt{b \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3474
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(b \tan (c+d x))^{3/2}} \, dx &=-\frac{2}{b d \sqrt{b \tan (c+d x)}}-\frac{\int \sqrt{b \tan (c+d x)} \, dx}{b^2}\\ &=-\frac{2}{b d \sqrt{b \tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac{2}{b d \sqrt{b \tan (c+d x)}}-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{b d}\\ &=-\frac{2}{b d \sqrt{b \tan (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{b-x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{b d}-\frac{\operatorname{Subst}\left (\int \frac{b+x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{b d}\\ &=-\frac{2}{b d \sqrt{b \tan (c+d x)}}-\frac{\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} b^{3/2} d}-\frac{\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} b^{3/2} d}-\frac{\operatorname{Subst}\left (\int \frac{1}{b-\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 b d}-\frac{\operatorname{Subst}\left (\int \frac{1}{b+\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 b d}\\ &=-\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} b^{3/2} d}+\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} b^{3/2} d}-\frac{2}{b d \sqrt{b \tan (c+d x)}}-\frac{\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} b^{3/2} d}+\frac{\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} b^{3/2} d}\\ &=\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} b^{3/2} d}-\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} b^{3/2} d}-\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} b^{3/2} d}+\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} b^{3/2} d}-\frac{2}{b d \sqrt{b \tan (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0646589, size = 38, normalized size = 0.18 \[ -\frac{2 \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\tan ^2(c+d x)\right )}{b d \sqrt{b \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 184, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}}{4\,bd}\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{1}{\sqrt [4]{{b}^{2}}}}}-{\frac{\sqrt{2}}{2\,bd}\arctan \left ({\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{b}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{b}^{2}}}}}+{\frac{\sqrt{2}}{2\,bd}\arctan \left ( -{\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{b}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{b}^{2}}}}}-2\,{\frac{1}{bd\sqrt{b\tan \left ( dx+c \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 = 1.7696, size = 1693, normalized size = 7.99 \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{1}{\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 [A] time = 1.57727, size = 275, normalized size = 1.3 \begin{align*} -\frac{1}{4} \, b{\left (\frac{2 \, \sqrt{2}{\left | b \right |}^{\frac{3}{2}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | b \right |}} + 2 \, \sqrt{b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt{{\left | b \right |}}}\right )}{b^{4} d} + \frac{2 \, \sqrt{2}{\left | b \right |}^{\frac{3}{2}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | b \right |}} - 2 \, \sqrt{b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt{{\left | b \right |}}}\right )}{b^{4} d} - \frac{\sqrt{2}{\left | b \right |}^{\frac{3}{2}} \log \left (b \tan \left (d x + c\right ) + \sqrt{2} \sqrt{b \tan \left (d x + c\right )} \sqrt{{\left | b \right |}} +{\left | b \right |}\right )}{b^{4} d} + \frac{\sqrt{2}{\left | b \right |}^{\frac{3}{2}} \log \left (b \tan \left (d x + c\right ) - \sqrt{2} \sqrt{b \tan \left (d x + c\right )} \sqrt{{\left | b \right |}} +{\left | b \right |}\right )}{b^{4} d} + \frac{8}{\sqrt{b \tan \left (d x + c\right )} b^{2} d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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