Optimal. Leaf size=255 \[ \frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d} \]
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Rubi [A] time = 0.113415, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {3658, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \sqrt{b \tan ^3(c+d x)} \, dx &=\frac{\sqrt{b \tan ^3(c+d x)} \int \tan ^{\frac{3}{2}}(c+d x) \, dx}{\tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}-\frac{\sqrt{b \tan ^3(c+d x)} \int \frac{1}{\sqrt{\tan (c+d x)}} \, dx}{\tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}-\frac{\sqrt{b \tan ^3(c+d x)} \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}-\frac{\left (2 \sqrt{b \tan ^3(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}-\frac{\sqrt{b \tan ^3(c+d x)} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\sqrt{b \tan ^3(c+d x)} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}-\frac{\sqrt{b \tan ^3(c+d x)} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\sqrt{b \tan ^3(c+d x)} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\sqrt{b \tan ^3(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} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\sqrt{b \tan ^3(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} d \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}+\frac{\log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \sqrt{b \tan ^3(c+d x)}}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \sqrt{b \tan ^3(c+d x)}}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\sqrt{b \tan ^3(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} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\sqrt{b \tan ^3(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} d \tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{\log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \sqrt{b \tan ^3(c+d x)}}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{\log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right ) \sqrt{b \tan ^3(c+d x)}}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.242687, size = 161, normalized size = 0.63 \[ \frac{\sqrt{b \tan ^3(c+d x)} \left (2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )+8 \sqrt{\tan (c+d x)}+\sqrt{2} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-\sqrt{2} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{4 d \tan ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 207, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,d\tan \left ( dx+c \right ) }\sqrt{b \left ( \tan \left ( dx+c \right ) \right ) ^{3}} \left ( \sqrt [4]{{b}^{2}}\sqrt{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 ( \sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}-b\tan \left ( dx+c \right ) -\sqrt{{b}^{2}} \right ) ^{-1}} \right ) +2\,\sqrt [4]{{b}^{2}}\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }+\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) -2\,\sqrt [4]{{b}^{2}}\sqrt{2}\arctan \left ({\frac{-\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }+\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) -8\,\sqrt{b\tan \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{b\tan \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46182, size = 180, normalized size = 0.71 \begin{align*} -\frac{2 \, \sqrt{2} \sqrt{b} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt{2} \sqrt{b} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + \sqrt{2} \sqrt{b} \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt{2} \sqrt{b} \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 8 \, \sqrt{b} \sqrt{\tan \left (d x + c\right )}}{4 \, 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 \sqrt{b \tan ^{3}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3278, size = 263, normalized size = 1.03 \begin{align*} -\frac{1}{4} \,{\left (\frac{2 \, \sqrt{2} \sqrt{{\left | b \right |}} \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 )}{d} + \frac{2 \, \sqrt{2} \sqrt{{\left | b \right |}} \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 )}{d} + \frac{\sqrt{2} \sqrt{{\left | b \right |}} \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 )}{d} - \frac{\sqrt{2} \sqrt{{\left | b \right |}} \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 )}{d} - \frac{8 \, \sqrt{b \tan \left (d x + c\right )}}{d}\right )} \mathrm{sgn}\left (\tan \left (d x + c\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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