Optimal. Leaf size=315 \[ -\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{2 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}-\frac{4 (\tan (e+f x)+1)^{5/2}}{35 f}-\frac{2 (\tan (e+f x)+1)^{3/2}}{3 f}-\frac{2 \sqrt{\tan (e+f x)+1}}{f}-\frac{\log \left (\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f}+\frac{\log \left (\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f} \]
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Rubi [A] time = 0.364896, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {3566, 3630, 12, 3528, 3485, 708, 1094, 634, 618, 204, 628} \[ -\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{2 \tan (e+f x) (\tan (e+f x)+1)^{5/2}}{7 f}-\frac{4 (\tan (e+f x)+1)^{5/2}}{35 f}-\frac{2 (\tan (e+f x)+1)^{3/2}}{3 f}-\frac{2 \sqrt{\tan (e+f x)+1}}{f}-\frac{\log \left (\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f}+\frac{\log \left (\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3630
Rule 12
Rule 3528
Rule 3485
Rule 708
Rule 1094
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \tan ^3(e+f x) (1+\tan (e+f x))^{3/2} \, dx &=\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}+\frac{2}{7} \int (1+\tan (e+f x))^{3/2} \left (-1-\frac{7}{2} \tan (e+f x)-\tan ^2(e+f x)\right ) \, dx\\ &=-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}+\frac{2}{7} \int -\frac{7}{2} \tan (e+f x) (1+\tan (e+f x))^{3/2} \, dx\\ &=-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}-\int \tan (e+f x) (1+\tan (e+f x))^{3/2} \, dx\\ &=-\frac{2 (1+\tan (e+f x))^{3/2}}{3 f}-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}-\int (-1+\tan (e+f x)) \sqrt{1+\tan (e+f x)} \, dx\\ &=-\frac{2 \sqrt{1+\tan (e+f x)}}{f}-\frac{2 (1+\tan (e+f x))^{3/2}}{3 f}-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}-\int -\frac{2}{\sqrt{1+\tan (e+f x)}} \, dx\\ &=-\frac{2 \sqrt{1+\tan (e+f x)}}{f}-\frac{2 (1+\tan (e+f x))^{3/2}}{3 f}-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}+2 \int \frac{1}{\sqrt{1+\tan (e+f x)}} \, dx\\ &=-\frac{2 \sqrt{1+\tan (e+f x)}}{f}-\frac{2 (1+\tan (e+f x))^{3/2}}{3 f}-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{2 \sqrt{1+\tan (e+f x)}}{f}-\frac{2 (1+\tan (e+f x))^{3/2}}{3 f}-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{2-2 x^2+x^4} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{f}\\ &=-\frac{2 \sqrt{1+\tan (e+f x)}}{f}-\frac{2 (1+\tan (e+f x))^{3/2}}{3 f}-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2 \left (1+\sqrt{2}\right )}-x}{\sqrt{2}-\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{\sqrt{1+\sqrt{2}} f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2 \left (1+\sqrt{2}\right )}+x}{\sqrt{2}+\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{\sqrt{1+\sqrt{2}} f}\\ &=-\frac{2 \sqrt{1+\tan (e+f x)}}{f}-\frac{2 (1+\tan (e+f x))^{3/2}}{3 f}-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{\sqrt{2} f}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{\sqrt{2} f}-\frac{\operatorname{Subst}\left (\int \frac{-\sqrt{2 \left (1+\sqrt{2}\right )}+2 x}{\sqrt{2}-\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{2 \sqrt{1+\sqrt{2}} f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2 \left (1+\sqrt{2}\right )}+2 x}{\sqrt{2}+\sqrt{2 \left (1+\sqrt{2}\right )} x+x^2} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{2 \sqrt{1+\sqrt{2}} f}\\ &=-\frac{\log \left (1+\sqrt{2}+\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+\tan (e+f x)}\right )}{2 \sqrt{1+\sqrt{2}} f}+\frac{\log \left (1+\sqrt{2}+\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+\tan (e+f x)}\right )}{2 \sqrt{1+\sqrt{2}} f}-\frac{2 \sqrt{1+\tan (e+f x)}}{f}-\frac{2 (1+\tan (e+f x))^{3/2}}{3 f}-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}-\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{2 \left (1-\sqrt{2}\right )-x^2} \, dx,x,-\sqrt{2 \left (1+\sqrt{2}\right )}+2 \sqrt{1+\tan (e+f x)}\right )}{f}-\frac{\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{2 \left (1-\sqrt{2}\right )-x^2} \, dx,x,\sqrt{2 \left (1+\sqrt{2}\right )}+2 \sqrt{1+\tan (e+f x)}\right )}{f}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{1+\tan (e+f x)}}{\sqrt{2 \left (-1+\sqrt{2}\right )}}\right )}{\sqrt{-1+\sqrt{2}} f}+\frac{\tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}+2 \sqrt{1+\tan (e+f x)}}{\sqrt{2 \left (-1+\sqrt{2}\right )}}\right )}{\sqrt{-1+\sqrt{2}} f}-\frac{\log \left (1+\sqrt{2}+\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+\tan (e+f x)}\right )}{2 \sqrt{1+\sqrt{2}} f}+\frac{\log \left (1+\sqrt{2}+\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+\tan (e+f x)}\right )}{2 \sqrt{1+\sqrt{2}} f}-\frac{2 \sqrt{1+\tan (e+f x)}}{f}-\frac{2 (1+\tan (e+f x))^{3/2}}{3 f}-\frac{4 (1+\tan (e+f x))^{5/2}}{35 f}+\frac{2 \tan (e+f x) (1+\tan (e+f x))^{5/2}}{7 f}\\ \end{align*}
Mathematica [C] time = 0.588238, size = 112, normalized size = 0.36 \[ \frac{2 \sqrt{\tan (e+f x)+1} \left (15 \tan ^3(e+f x)+24 \tan ^2(e+f x)-32 \tan (e+f x)-146\right )+105 (1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right )+105 (1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right )}{105 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 354, normalized size = 1.1 \begin{align*}{\frac{2}{7\,f} \left ( 1+\tan \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{2}{5\,f} \left ( 1+\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{2}{3\,f} \left ( 1+\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}-2\,{\frac{\sqrt{1+\tan \left ( fx+e \right ) }}{f}}+{\frac{\sqrt{2\,\sqrt{2}+2}\sqrt{2}}{4\,f}\ln \left ( 1+\sqrt{2}-\sqrt{2\,\sqrt{2}+2}\sqrt{1+\tan \left ( fx+e \right ) }+\tan \left ( fx+e \right ) \right ) }-{\frac{\sqrt{2\,\sqrt{2}+2}}{2\,f}\ln \left ( 1+\sqrt{2}-\sqrt{2\,\sqrt{2}+2}\sqrt{1+\tan \left ( fx+e \right ) }+\tan \left ( fx+e \right ) \right ) }+{\frac{\sqrt{2}}{f\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\tan \left ( fx+e \right ) }-\sqrt{2\,\sqrt{2}+2} \right ) } \right ) }-{\frac{\sqrt{2\,\sqrt{2}+2}\sqrt{2}}{4\,f}\ln \left ( 1+\sqrt{2}+\sqrt{2\,\sqrt{2}+2}\sqrt{1+\tan \left ( fx+e \right ) }+\tan \left ( fx+e \right ) \right ) }+{\frac{\sqrt{2\,\sqrt{2}+2}}{2\,f}\ln \left ( 1+\sqrt{2}+\sqrt{2\,\sqrt{2}+2}\sqrt{1+\tan \left ( fx+e \right ) }+\tan \left ( fx+e \right ) \right ) }+{\frac{\sqrt{2}}{f\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( \sqrt{2\,\sqrt{2}+2}+2\,\sqrt{1+\tan \left ( fx+e \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [B] time = 2.09714, size = 2635, normalized size = 8.37 \begin{align*} \text{result too large to display} \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 (\tan{\left (e + f x \right )} + 1\right )^{\frac{3}{2}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\tan \left (f x + e\right ) + 1\right )}^{\frac{3}{2}} \tan \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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