Optimal. Leaf size=364 \[ -\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}} \]
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Rubi [A] time = 0.147612, 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, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3658
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}{\left (b \tan ^3(e+f x)\right )^{5/2}} \, dx &=\frac{\tan ^{\frac{3}{2}}(e+f x) \int \frac{1}{\tan ^{\frac{15}{2}}(e+f x)} \, dx}{b^2 \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \int \frac{1}{\tan ^{\frac{11}{2}}(e+f x)} \, dx}{b^2 \sqrt{b \tan ^3(e+f x)}}\\ &=\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \int \frac{1}{\tan ^{\frac{7}{2}}(e+f x)} \, dx}{b^2 \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \int \frac{1}{\tan ^{\frac{3}{2}}(e+f x)} \, dx}{b^2 \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \int \sqrt{\tan (e+f x)} \, dx}{b^2 \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{b^2 f \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\left (2 \tan ^{\frac{3}{2}}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{\tan (e+f x)}\right )}{b^2 f \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (e+f x)}\right )}{b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (e+f x)}\right )}{b^2 f \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (e+f x)}\right )}{2 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (e+f x)}\right )}{2 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (e+f x)}\right )}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (e+f x)}\right )}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\log \left (1-\sqrt{2} \sqrt{\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac{3}{2}}(e+f x)}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\log \left (1+\sqrt{2} \sqrt{\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac{3}{2}}(e+f x)}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (e+f x)}\right )}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (e+f x)}\right )}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (e+f x)}\right ) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\log \left (1-\sqrt{2} \sqrt{\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac{3}{2}}(e+f x)}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\log \left (1+\sqrt{2} \sqrt{\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac{3}{2}}(e+f x)}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.0584859, size = 45, normalized size = 0.12 \[ -\frac{2 \tan (e+f x) \text{Hypergeometric2F1}\left (-\frac{13}{4},1,-\frac{9}{4},-\tan ^2(e+f x)\right )}{13 f \left (b \tan ^3(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 272, normalized size = 0.8 \begin{align*}{\frac{\tan \left ( fx+e \right ) }{2340\,f{b}^{6}} \left ( 585\,\sqrt{2} \left ( b\tan \left ( fx+e \right ) \right ) ^{13/2}\ln \left ( -{\frac{\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( fx+e \right ) }\sqrt{2}-b\tan \left ( fx+e \right ) -\sqrt{{b}^{2}}}{b\tan \left ( fx+e \right ) +\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{b}^{2}}}} \right ) +1170\,\sqrt{2} \left ( b\tan \left ( fx+e \right ) \right ) ^{13/2}\arctan \left ({\frac{\sqrt{2}\sqrt{b\tan \left ( fx+e \right ) }+\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) +1170\,\sqrt{2} \left ( b\tan \left ( fx+e \right ) \right ) ^{13/2}\arctan \left ({\frac{\sqrt{2}\sqrt{b\tan \left ( fx+e \right ) }-\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) +4680\,\sqrt [4]{{b}^{2}}{b}^{6} \left ( \tan \left ( fx+e \right ) \right ) ^{6}-936\,{b}^{6}\sqrt [4]{{b}^{2}} \left ( \tan \left ( fx+e \right ) \right ) ^{4}+520\,{b}^{6}\sqrt [4]{{b}^{2}} \left ( \tan \left ( fx+e \right ) \right ) ^{2}-360\,{b}^{6}\sqrt [4]{{b}^{2}} \right ) \left ( b \left ( \tan \left ( fx+e \right ) \right ) ^{3} \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt [4]{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66337, size = 232, normalized size = 0.64 \begin{align*} \frac{\frac{585 \,{\left (2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (f x + e\right )}\right )}\right ) + 2 \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (f x + e\right )}\right )}\right ) - \sqrt{2} \log \left (\sqrt{2} \sqrt{\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) + \sqrt{2} \log \left (-\sqrt{2} \sqrt{\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right )\right )}}{b^{\frac{5}{2}}} + \frac{8 \,{\left (\frac{585 \, \sqrt{b}}{\sqrt{\tan \left (f x + e\right )}} - \frac{117 \, \sqrt{b}}{\tan \left (f x + e\right )^{\frac{5}{2}}} + \frac{65 \, \sqrt{b}}{\tan \left (f x + e\right )^{\frac{9}{2}}} - \frac{45 \, \sqrt{b}}{\tan \left (f x + e\right )^{\frac{13}{2}}}\right )}}{b^{3}}}{2340 \, f} \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 (e + f x \right )}\right )^{\frac{5}{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 (f x + e\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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