Optimal. Leaf size=83 \[ \frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{a^{5/2} f}+\frac{1}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.0883175, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4139, 266, 51, 63, 208} \[ \frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{a^{5/2} f}+\frac{1}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4139
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{5/2}} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{1}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,\sec ^2(e+f x)\right )}{2 a f}\\ &=\frac{1}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sec ^2(e+f x)\right )}{2 a^2 f}\\ &=\frac{1}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}+\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^2 b f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{a^{5/2} f}+\frac{1}{3 a f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{1}{a^2 f \sqrt{a+b \sec ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 7.6078, size = 613, normalized size = 7.39 \[ \frac{e^{i (e+f x)} \sec ^5(e+f x) \sqrt{4 b+a e^{-2 i (e+f x)} \left (1+e^{2 i (e+f x)}\right )^2} \left (\frac{-12 \log \left (\sqrt{a} \sqrt{a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b\right )-12 \log \left (\sqrt{a} \sqrt{a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}}+a e^{2 i (e+f x)}+a+2 b e^{2 i (e+f x)}\right )+24 i f x}{\sqrt{a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}}}-\frac{\sqrt{a} \left (1+e^{2 i (e+f x)}\right ) \left (-6 a^2 b \left (e^{2 i (e+f x)}+e^{4 i (e+f x)}+1\right )+a^3 \left (1+e^{2 i (e+f x)}\right )^2-32 a b^2 \left (1+e^{2 i (e+f x)}\right )^2-96 b^3 e^{2 i (e+f x)}\right )}{b^2 \left (a \left (1+e^{2 i (e+f x)}\right )^2+4 b e^{2 i (e+f x)}\right )^2}\right ) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{96 \sqrt{2} a^{5/2} f \left (a+b \sec ^2(e+f x)\right )^{5/2}}-\frac{\sec ^4(e+f x) (a \cos (2 (e+f x))+a+3 b) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{48 b^2 f (a \cos (2 (e+f x))+a+2 b)^{3/2} \left (a+b \sec ^2(e+f x)\right )^{5/2}}+\frac{\sec ^4(e+f x) ((a-2 b) \cos (2 (e+f x))+a+b) (a \cos (2 e+2 f x)+a+2 b)^{5/2}}{32 b^2 f (a \cos (2 (e+f x))+a+2 b)^{3/2} \left (a+b \sec ^2(e+f x)\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.056, size = 86, normalized size = 1. \begin{align*}{\frac{1}{3\,af} \left ( a+b \left ( \sec \left ( fx+e \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{{a}^{2}f}{\frac{1}{\sqrt{a+b \left ( \sec \left ( fx+e \right ) \right ) ^{2}}}}}-{\frac{1}{f}\ln \left ({\frac{1}{\sec \left ( fx+e \right ) } \left ( 2\,a+2\,\sqrt{a}\sqrt{a+b \left ( \sec \left ( fx+e \right ) \right ) ^{2}} \right ) } \right ){a}^{-{\frac{5}{2}}}} \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 \frac{\tan \left (f x + e\right )}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.36484, size = 1208, normalized size = 14.55 \begin{align*} \left [\frac{3 \,{\left (a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}\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 (4 \, a^{2} \cos \left (f x + e\right )^{4} + 3 \, a b \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}}}}{24 \,{\left (a^{5} f \cos \left (f x + e\right )^{4} + 2 \, a^{4} b f \cos \left (f x + e\right )^{2} + a^{3} b^{2} f\right )}}, \frac{3 \,{\left (a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}\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 (4 \, a^{2} \cos \left (f x + e\right )^{4} + 3 \, a b \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}}}}{12 \,{\left (a^{5} f \cos \left (f x + e\right )^{4} + 2 \, a^{4} b f \cos \left (f x + e\right )^{2} + a^{3} b^{2} f\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 66.105, size = 78, normalized size = 0.94 \begin{align*} \frac{1}{3 a f \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}} + \frac{1}{a^{2} f \sqrt{a + b \sec ^{2}{\left (e + f x \right )}}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b \sec ^{2}{\left (e + f x \right )}}}{\sqrt{- a}} \right )}}{a^{2} f \sqrt{- a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.04053, size = 613, normalized size = 7.39 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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