Optimal. Leaf size=88 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{a^{3/2} f}+\frac{(a+b)^2}{a b^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{a+b \sec ^2(e+f x)}}{b^2 f} \]
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Rubi [A] time = 0.150769, antiderivative size = 88, 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, 87, 63, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{a^{3/2} f}+\frac{(a+b)^2}{a b^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{a+b \sec ^2(e+f x)}}{b^2 f} \]
Antiderivative was successfully verified.
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Rule 4139
Rule 446
Rule 87
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^2}{x \left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-1+x)^2}{x (a+b x)^{3/2}} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{(a+b)^2}{a b (a+b x)^{3/2}}+\frac{1}{b \sqrt{a+b x}}+\frac{1}{a x \sqrt{a+b x}}\right ) \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{(a+b)^2}{a b^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{a+b \sec ^2(e+f x)}}{b^2 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sec ^2(e+f x)\right )}{2 a f}\\ &=\frac{(a+b)^2}{a b^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{a+b \sec ^2(e+f x)}}{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 )}{a b f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{a^{3/2} f}+\frac{(a+b)^2}{a b^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\sqrt{a+b \sec ^2(e+f x)}}{b^2 f}\\ \end{align*}
Mathematica [F] time = 5.26377, size = 0, normalized size = 0. \[ \int \frac{\tan ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [B] time = 0.529, size = 6593, normalized size = 74.9 \begin{align*} \text{output too large to display} \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 = 2.83191, size = 1095, normalized size = 12.44 \begin{align*} \left [\frac{{\left (a b^{2} \cos \left (f x + e\right )^{2} + b^{3}\right )} \sqrt{a} \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 (a^{2} b +{\left (2 \, a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{8 \,{\left (a^{3} b^{2} f \cos \left (f x + e\right )^{2} + a^{2} b^{3} f\right )}}, \frac{{\left (a b^{2} \cos \left (f x + e\right )^{2} + b^{3}\right )} \sqrt{-a} \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 ) + 4 \,{\left (a^{2} b +{\left (2 \, a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \,{\left (a^{3} b^{2} f \cos \left (f x + e\right )^{2} + a^{2} b^{3} f\right )}}\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 )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, 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}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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