Optimal. Leaf size=271 \[ \frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac{2 \sqrt{\tan (e+f x)+1}}{f}+\frac{\log \left (\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f}-\frac{\log \left (\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f} \]
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Rubi [A] time = 0.231056, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {3528, 12, 3485, 708, 1094, 634, 618, 204, 628} \[ \frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}-\frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{2 (\tan (e+f x)+1)^{3/2}}{3 f}+\frac{2 \sqrt{\tan (e+f x)+1}}{f}+\frac{\log \left (\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f}-\frac{\log \left (\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{1+\sqrt{2}} f} \]
Antiderivative was successfully verified.
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Rule 3528
Rule 12
Rule 3485
Rule 708
Rule 1094
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \tan (e+f x) (1+\tan (e+f x))^{3/2} \, dx &=\frac{2 (1+\tan (e+f x))^{3/2}}{3 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}+\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}-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{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 \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{\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{\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{\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}\\ \end{align*}
Mathematica [C] time = 0.211466, size = 90, normalized size = 0.33 \[ \frac{2 \sqrt{\tan (e+f x)+1} (\tan (e+f x)+4)-3 (1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right )-3 (1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right )}{3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 326, normalized size = 1.2 \begin{align*}{\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] 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 )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02027, size = 2542, normalized size = 9.38 \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{\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 )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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