Optimal. Leaf size=298 \[ -\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right ) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{2}{3 b d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{\frac{3}{2}}(c+d x) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}} \]
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Rubi [A] time = 0.130004, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {3658, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right ) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{2}{3 b d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{\frac{3}{2}}(c+d x) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3474
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\left (b \tan ^3(c+d x)\right )^{3/2}} \, dx &=\frac{\tan ^{\frac{3}{2}}(c+d x) \int \frac{1}{\tan ^{\frac{9}{2}}(c+d x)} \, dx}{b \sqrt{b \tan ^3(c+d x)}}\\ &=-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{\frac{3}{2}}(c+d x) \int \frac{1}{\tan ^{\frac{5}{2}}(c+d x)} \, dx}{b \sqrt{b \tan ^3(c+d x)}}\\ &=\frac{2}{3 b d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \int \frac{1}{\sqrt{\tan (c+d x)}} \, dx}{b \sqrt{b \tan ^3(c+d x)}}\\ &=\frac{2}{3 b d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b d \sqrt{b \tan ^3(c+d x)}}\\ &=\frac{2}{3 b d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}}+\frac{\left (2 \tan ^{\frac{3}{2}}(c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{b d \sqrt{b \tan ^3(c+d x)}}\\ &=\frac{2}{3 b d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{b d \sqrt{b \tan ^3(c+d x)}}\\ &=\frac{2}{3 b d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 b d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{\frac{3}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{\frac{3}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}\\ &=\frac{2}{3 b d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}}-\frac{\log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac{3}{2}}(c+d x)}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{\log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac{3}{2}}(c+d x)}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{\frac{3}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{\frac{3}{2}}(c+d x) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}\\ &=\frac{2}{3 b d \sqrt{b \tan ^3(c+d x)}}-\frac{2 \cot ^2(c+d x)}{7 b d \sqrt{b \tan ^3(c+d x)}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right ) \tan ^{\frac{3}{2}}(c+d x)}{\sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}-\frac{\log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac{3}{2}}(c+d x)}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}+\frac{\log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac{3}{2}}(c+d x)}{2 \sqrt{2} b d \sqrt{b \tan ^3(c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0667278, size = 45, normalized size = 0.15 \[ -\frac{2 \tan (c+d x) \, _2F_1\left (-\frac{7}{4},1;-\frac{3}{4};-\tan ^2(c+d x)\right )}{7 d \left (b \tan ^3(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 235, normalized size = 0.8 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{84\,d{b}^{4}} \left ( 21\,\sqrt [4]{{b}^{2}}\sqrt{2} \left ( b\tan \left ( dx+c \right ) \right ) ^{7/2}\ln \left ( -{\frac{b\tan \left ( dx+c \right ) +\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}+\sqrt{{b}^{2}}}{\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}-b\tan \left ( dx+c \right ) -\sqrt{{b}^{2}}}} \right ) +42\,\sqrt [4]{{b}^{2}}\sqrt{2} \left ( b\tan \left ( dx+c \right ) \right ) ^{7/2}\arctan \left ({\frac{\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }+\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) -42\,\sqrt [4]{{b}^{2}}\sqrt{2} \left ( b\tan \left ( dx+c \right ) \right ) ^{7/2}\arctan \left ({\frac{-\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }+\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) +56\,{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}-24\,{b}^{4} \right ) \left ( b \left ( \tan \left ( dx+c \right ) \right ) ^{3} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4358, size = 220, normalized size = 0.74 \begin{align*} \frac{\frac{21 \,{\left (2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + \sqrt{2} \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt{2} \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{b^{\frac{3}{2}}} + \frac{8 \,{\left (21 \, \sqrt{\tan \left (d x + c\right )} + \frac{7}{\tan \left (d x + c\right )^{\frac{3}{2}}} - \frac{3}{\tan \left (d x + c\right )^{\frac{7}{2}}}\right )}}{b^{\frac{3}{2}}} - \frac{168 \, \sqrt{\tan \left (d x + c\right )}}{b^{\frac{3}{2}}}}{84 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \tan ^{3}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \tan \left (d x + c\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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