Optimal. Leaf size=364 \[ \frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}-\frac{b^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b^2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{b^2 \sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b^2 \sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d} \]
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Rubi [A] time = 0.150337, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ \frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}-\frac{b^2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b^2 \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right ) \sqrt{b \tan ^3(c+d x)}}{\sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{b^2 \sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b^2 \sqrt{b \tan ^3(c+d x)} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d} \]
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 \left (b \tan ^3(c+d x)\right )^{5/2} \, dx &=\frac{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \int \tan ^{\frac{15}{2}}(c+d x) \, dx}{\tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}-\frac{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \int \tan ^{\frac{11}{2}}(c+d x) \, dx}{\tan ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}+\frac{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \int \tan ^{\frac{7}{2}}(c+d x) \, dx}{\tan ^{\frac{3}{2}}(c+d x)}\\ &=\frac{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}-\frac{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \int \tan ^{\frac{3}{2}}(c+d x) \, dx}{\tan ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b^2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}+\frac{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}+\frac{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \int \frac{1}{\sqrt{\tan (c+d x)}} \, dx}{\tan ^{\frac{3}{2}}(c+d x)}\\ &=-\frac{2 b^2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}+\frac{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}+\frac{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \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 b^2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}+\frac{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}+\frac{\left (2 b^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 b^2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}+\frac{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}+\frac{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \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{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \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 b^2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}+\frac{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}+\frac{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \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{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \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{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \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{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \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 b^2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}-\frac{b^2 \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{b^2 \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{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}+\frac{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \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{\left (b^2 \sqrt{b \tan ^3(c+d x)}\right ) \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 b^2 \cot (c+d x) \sqrt{b \tan ^3(c+d x)}}{d}-\frac{b^2 \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{b^2 \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{b^2 \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{b^2 \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{2 b^2 \tan (c+d x) \sqrt{b \tan ^3(c+d x)}}{5 d}-\frac{2 b^2 \tan ^3(c+d x) \sqrt{b \tan ^3(c+d x)}}{9 d}+\frac{2 b^2 \tan ^5(c+d x) \sqrt{b \tan ^3(c+d x)}}{13 d}\\ \end{align*}
Mathematica [A] time = 0.810332, size = 199, normalized size = 0.55 \[ \frac{b \left (b \tan ^3(c+d x)\right )^{3/2} \left (360 \tan ^{\frac{13}{2}}(c+d x)-520 \tan ^{\frac{9}{2}}(c+d x)+936 \tan ^{\frac{5}{2}}(c+d x)-1170 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )+1170 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-4680 \sqrt{\tan (c+d x)}-585 \sqrt{2} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )+585 \sqrt{2} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{2340 d \tan ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 265, normalized size = 0.7 \begin{align*}{\frac{1}{2340\,d \left ( \tan \left ( dx+c \right ) \right ) ^{5}{b}^{4}} \left ( b \left ( \tan \left ( dx+c \right ) \right ) ^{3} \right ) ^{{\frac{5}{2}}} \left ( 360\, \left ( b\tan \left ( dx+c \right ) \right ) ^{13/2}-520\,{b}^{2} \left ( b\tan \left ( dx+c \right ) \right ) ^{9/2}+585\,{b}^{6}\sqrt [4]{{b}^{2}}\sqrt{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 ) +1170\,{b}^{6}\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 ) -1170\,{b}^{6}\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 ) +936\,{b}^{4} \left ( b\tan \left ( dx+c \right ) \right ) ^{5/2}-4680\,{b}^{6}\sqrt{b\tan \left ( dx+c \right ) } \right ) \left ( b\tan \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5432, size = 240, normalized size = 0.66 \begin{align*} \frac{360 \, b^{\frac{5}{2}} \tan \left (d x + c\right )^{\frac{13}{2}} - 520 \, b^{\frac{5}{2}} \tan \left (d x + c\right )^{\frac{9}{2}} + 936 \, b^{\frac{5}{2}} \tan \left (d x + c\right )^{\frac{5}{2}} + 585 \,{\left (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 )\right )} b^{2} - 4680 \, b^{\frac{5}{2}} \sqrt{\tan \left (d x + c\right )}}{2340 \, 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 \left (b \tan ^{3}{\left (c + d x \right )}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.1667, size = 393, normalized size = 1.08 \begin{align*} \frac{1}{2340} \,{\left (\frac{1170 \, \sqrt{2} b \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{1170 \, \sqrt{2} b \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{585 \, \sqrt{2} b \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{585 \, \sqrt{2} b \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 \,{\left (45 \, \sqrt{b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{6} - 65 \, \sqrt{b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{4} + 117 \, \sqrt{b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{2} - 585 \, \sqrt{b \tan \left (d x + c\right )} b^{66} d^{12}\right )}}{b^{65} d^{13}}\right )} b \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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