Optimal. Leaf size=89 \[ \frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 b^2 f}-\frac{(a+2 b) \sqrt{a+b \sec ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{\sqrt{a} f} \]
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Rubi [A] time = 0.133072, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4139, 446, 88, 63, 208} \[ \frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 b^2 f}-\frac{(a+2 b) \sqrt{a+b \sec ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{\sqrt{a} f} \]
Antiderivative was successfully verified.
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Rule 4139
Rule 446
Rule 88
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^2}{x \sqrt{a+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-1+x)^2}{x \sqrt{a+b x}} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{-a-2 b}{b \sqrt{a+b x}}+\frac{1}{x \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=-\frac{(a+2 b) \sqrt{a+b \sec ^2(e+f x)}}{b^2 f}+\frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 b^2 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=-\frac{(a+2 b) \sqrt{a+b \sec ^2(e+f x)}}{b^2 f}+\frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 b^2 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sec ^2(e+f x)}\right )}{b f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{\sqrt{a} f}-\frac{(a+2 b) \sqrt{a+b \sec ^2(e+f x)}}{b^2 f}+\frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 b^2 f}\\ \end{align*}
Mathematica [F] time = 2.1216, size = 0, normalized size = 0. \[ \int \frac{\tan ^5(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [B] time = 0.44, size = 358, normalized size = 4. \begin{align*}{\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{3\,f{b}^{2} \left ( \cos \left ( fx+e \right ) \right ) ^{4} \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}-1 \right ) } \left ( 2\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{5/2}+3\,\sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2}}}}\ln \left ( 4\,\cos \left ( fx+e \right ) \sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2}}}}\sqrt{a}+4\,a\cos \left ( fx+e \right ) +4\,\sqrt{a}\sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2}}}} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}{b}^{2}+6\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{a}^{3/2}b+3\,\sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2}}}}\ln \left ( 4\,\cos \left ( fx+e \right ) \sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2}}}}\sqrt{a}+4\,a\cos \left ( fx+e \right ) +4\,\sqrt{a}\sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2}}}} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}{b}^{2}+ \left ( \cos \left ( fx+e \right ) \right ) ^{2}{a}^{{\frac{3}{2}}}b+6\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sqrt{a}{b}^{2}-\sqrt{a}{b}^{2} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}}}} \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 = 2.07901, size = 1006, normalized size = 11.3 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} \cos \left (f x + e\right )^{2} \log \left (128 \, a^{4} \cos \left (f x + e\right )^{8} + 256 \, a^{3} b \cos \left (f x + e\right )^{6} + 160 \, a^{2} b^{2} \cos \left (f x + e\right )^{4} + 32 \, a b^{3} \cos \left (f x + e\right )^{2} + b^{4} - 8 \,{\left (16 \, a^{3} \cos \left (f x + e\right )^{8} + 24 \, a^{2} b \cos \left (f x + e\right )^{6} + 10 \, a b^{2} \cos \left (f x + e\right )^{4} + b^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt{a} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}\right ) - 8 \,{\left (2 \,{\left (a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{2} - a b\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{24 \, a b^{2} f \cos \left (f x + e\right )^{2}}, \frac{3 \, \sqrt{-a} b^{2} \arctan \left (\frac{{\left (8 \, a^{2} \cos \left (f x + e\right )^{4} + 8 \, a b \cos \left (f x + e\right )^{2} + b^{2}\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \,{\left (2 \, a^{3} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b \cos \left (f x + e\right )^{2} + a b^{2}\right )}}\right ) \cos \left (f x + e\right )^{2} - 4 \,{\left (2 \,{\left (a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{2} - a b\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{12 \, a b^{2} f \cos \left (f x + e\right )^{2}}\right ] \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{\tan ^{5}{\left (e + f x \right )}}{\sqrt{a + b \sec ^{2}{\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 \frac{\tan \left (f x + e\right )^{5}}{\sqrt{b \sec \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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