Optimal. Leaf size=318 \[ \frac{2 (\tan (e+f x)+1)^{3/2} \tan ^2(e+f x)}{7 f}-\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}-\frac{8 (\tan (e+f x)+1)^{3/2} \tan (e+f x)}{35 f}-\frac{18 (\tan (e+f x)+1)^{3/2}}{35 f}+\frac{\log \left (\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )} f}-\frac{\log \left (\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )} f} \]
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Rubi [A] time = 0.433513, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {3566, 3647, 3631, 3485, 700, 1127, 1161, 618, 204, 1164, 628} \[ \frac{2 (\tan (e+f x)+1)^{3/2} \tan ^2(e+f x)}{7 f}-\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\tan (e+f x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}+\frac{\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 \sqrt{\tan (e+f x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )}{f}-\frac{8 (\tan (e+f x)+1)^{3/2} \tan (e+f x)}{35 f}-\frac{18 (\tan (e+f x)+1)^{3/2}}{35 f}+\frac{\log \left (\tan (e+f x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )} f}-\frac{\log \left (\tan (e+f x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\tan (e+f x)+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )} f} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3647
Rule 3631
Rule 3485
Rule 700
Rule 1127
Rule 1161
Rule 618
Rule 204
Rule 1164
Rule 628
Rubi steps
\begin{align*} \int \tan ^4(e+f x) \sqrt{1+\tan (e+f x)} \, dx &=\frac{2 \tan ^2(e+f x) (1+\tan (e+f x))^{3/2}}{7 f}+\frac{2}{7} \int \tan (e+f x) \sqrt{1+\tan (e+f x)} \left (-2-\frac{7}{2} \tan (e+f x)-2 \tan ^2(e+f x)\right ) \, dx\\ &=-\frac{8 \tan (e+f x) (1+\tan (e+f x))^{3/2}}{35 f}+\frac{2 \tan ^2(e+f x) (1+\tan (e+f x))^{3/2}}{7 f}+\frac{4}{35} \int \sqrt{1+\tan (e+f x)} \left (2-\frac{27}{4} \tan ^2(e+f x)\right ) \, dx\\ &=-\frac{18 (1+\tan (e+f x))^{3/2}}{35 f}-\frac{8 \tan (e+f x) (1+\tan (e+f x))^{3/2}}{35 f}+\frac{2 \tan ^2(e+f x) (1+\tan (e+f x))^{3/2}}{7 f}+\int \sqrt{1+\tan (e+f x)} \, dx\\ &=-\frac{18 (1+\tan (e+f x))^{3/2}}{35 f}-\frac{8 \tan (e+f x) (1+\tan (e+f x))^{3/2}}{35 f}+\frac{2 \tan ^2(e+f x) (1+\tan (e+f x))^{3/2}}{7 f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{18 (1+\tan (e+f x))^{3/2}}{35 f}-\frac{8 \tan (e+f x) (1+\tan (e+f x))^{3/2}}{35 f}+\frac{2 \tan ^2(e+f x) (1+\tan (e+f x))^{3/2}}{7 f}+\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{2-2 x^2+x^4} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{f}\\ &=-\frac{18 (1+\tan (e+f x))^{3/2}}{35 f}-\frac{8 \tan (e+f x) (1+\tan (e+f x))^{3/2}}{35 f}+\frac{2 \tan ^2(e+f x) (1+\tan (e+f x))^{3/2}}{7 f}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-x^2}{2-2 x^2+x^4} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{f}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+x^2}{2-2 x^2+x^4} \, dx,x,\sqrt{1+\tan (e+f x)}\right )}{f}\\ &=-\frac{18 (1+\tan (e+f x))^{3/2}}{35 f}-\frac{8 \tan (e+f x) (1+\tan (e+f x))^{3/2}}{35 f}+\frac{2 \tan ^2(e+f x) (1+\tan (e+f x))^{3/2}}{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 )}{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 )}{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{2 \left (1+\sqrt{2}\right )} 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{2 \left (1+\sqrt{2}\right )} 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{2 \left (1+\sqrt{2}\right )} 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{2 \left (1+\sqrt{2}\right )} f}-\frac{18 (1+\tan (e+f x))^{3/2}}{35 f}-\frac{8 \tan (e+f x) (1+\tan (e+f x))^{3/2}}{35 f}+\frac{2 \tan ^2(e+f x) (1+\tan (e+f x))^{3/2}}{7 f}-\frac{\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{\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{2 \left (-1+\sqrt{2}\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{2 \left (-1+\sqrt{2}\right )} 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{2 \left (1+\sqrt{2}\right )} 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{2 \left (1+\sqrt{2}\right )} f}-\frac{18 (1+\tan (e+f x))^{3/2}}{35 f}-\frac{8 \tan (e+f x) (1+\tan (e+f x))^{3/2}}{35 f}+\frac{2 \tan ^2(e+f x) (1+\tan (e+f x))^{3/2}}{7 f}\\ \end{align*}
Mathematica [C] time = 0.713089, size = 118, normalized size = 0.37 \[ \frac{-35 i \sqrt{1-i} \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1-i}}\right )+35 i \sqrt{1+i} \tanh ^{-1}\left (\frac{\sqrt{\tan (e+f x)+1}}{\sqrt{1+i}}\right )+2 \sqrt{\tan (e+f x)+1} \left ((5 \tan (e+f x)+1) \sec ^2(e+f x)-2 (9 \tan (e+f x)+5)\right )}{35 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 318, normalized size = 1. \begin{align*}{\frac{2}{7\,f} \left ( 1+\tan \left ( fx+e \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{4}{5\,f} \left ( 1+\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}}+{\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{1}{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}}{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}\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{1}{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 ) }+{\frac{\sqrt{2\,\sqrt{2}+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 ) } \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 \sqrt{\tan \left (f x + e\right ) + 1} \tan \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04833, size = 2626, normalized size = 8.26 \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 \sqrt{\tan{\left (e + f x \right )} + 1} \tan ^{4}{\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 \sqrt{\tan \left (f x + e\right ) + 1} \tan \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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