Optimal. Leaf size=255 \[ \frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \tan (e+f x)}{f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} f \sqrt{b \tan ^3(e+f x)}} \]
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Rubi [A] time = 0.114648, 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, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \tan (e+f x)}{f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )}{2 \sqrt{2} f \sqrt{b \tan ^3(e+f x)}} \]
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}{\sqrt{b \tan ^3(e+f x)}} \, dx &=\frac{\tan ^{\frac{3}{2}}(e+f x) \int \frac{1}{\tan ^{\frac{3}{2}}(e+f x)} \, dx}{\sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \tan (e+f x)}{f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \int \sqrt{\tan (e+f x)} \, dx}{\sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \tan (e+f x)}{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 )}{f \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \tan (e+f x)}{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 )}{f \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \tan (e+f x)}{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 )}{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 )}{f \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \tan (e+f x)}{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 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 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} 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} f \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \tan (e+f x)}{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} 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} 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} 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} f \sqrt{b \tan ^3(e+f x)}}\\ &=-\frac{2 \tan (e+f x)}{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} 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} 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} 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} f \sqrt{b \tan ^3(e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.0340779, size = 43, normalized size = 0.17 \[ -\frac{2 \tan (e+f x) \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},-\tan ^2(e+f x)\right )}{f \sqrt{b \tan ^3(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 211, normalized size = 0.8 \begin{align*} -{\frac{\tan \left ( fx+e \right ) }{4\,f} \left ( \sqrt{2}\sqrt{b\tan \left ( fx+e \right ) }\ln \left ( -{ \left ( \sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( fx+e \right ) }\sqrt{2}-b\tan \left ( fx+e \right ) -\sqrt{{b}^{2}} \right ) \left ( b\tan \left ( fx+e \right ) +\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{b}^{2}} \right ) ^{-1}} \right ) +2\,\sqrt{2}\sqrt{b\tan \left ( fx+e \right ) }\arctan \left ({\frac{\sqrt{2}\sqrt{b\tan \left ( fx+e \right ) }+\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) +2\,\sqrt{2}\sqrt{b\tan \left ( fx+e \right ) }\arctan \left ({\frac{\sqrt{2}\sqrt{b\tan \left ( fx+e \right ) }-\sqrt [4]{{b}^{2}}}{\sqrt [4]{{b}^{2}}}} \right ) +8\,\sqrt [4]{{b}^{2}} \right ){\frac{1}{\sqrt{b \left ( \tan \left ( fx+e \right ) \right ) ^{3}}}}{\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 = 1.6107, size = 170, normalized size = 0.67 \begin{align*} -\frac{\frac{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 )}{\sqrt{b}} + \frac{8}{\sqrt{b} \sqrt{\tan \left (f x + e\right )}}}{4 \, 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}{\sqrt{b \tan ^{3}{\left (e + f x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32153, size = 355, normalized size = 1.39 \begin{align*} -\frac{1}{4} \, b^{2}{\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 (f x + e\right )}\right )}}{2 \, \sqrt{{\left | b \right |}}}\right )}{b^{4} f \mathrm{sgn}\left (\tan \left (f x + e\right )\right )} + \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 (f x + e\right )}\right )}}{2 \, \sqrt{{\left | b \right |}}}\right )}{b^{4} f \mathrm{sgn}\left (\tan \left (f x + e\right )\right )} - \frac{\sqrt{2}{\left | b \right |}^{\frac{3}{2}} \log \left (b \tan \left (f x + e\right ) + \sqrt{2} \sqrt{b \tan \left (f x + e\right )} \sqrt{{\left | b \right |}} +{\left | b \right |}\right )}{b^{4} f \mathrm{sgn}\left (\tan \left (f x + e\right )\right )} + \frac{\sqrt{2}{\left | b \right |}^{\frac{3}{2}} \log \left (b \tan \left (f x + e\right ) - \sqrt{2} \sqrt{b \tan \left (f x + e\right )} \sqrt{{\left | b \right |}} +{\left | b \right |}\right )}{b^{4} f \mathrm{sgn}\left (\tan \left (f x + e\right )\right )} + \frac{8}{\sqrt{b \tan \left (f x + e\right )} b^{2} f \mathrm{sgn}\left (\tan \left (f x + e\right )\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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