Optimal. Leaf size=78 \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{f}+\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f} \]
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Rubi [A] time = 0.0824348, antiderivative size = 78, 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, 50, 63, 208} \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{f}+\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 4139
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan (e+f x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{x} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f}+\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f}+\frac{a^2 \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{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{f}+\frac{a \sqrt{a+b \sec ^2(e+f x)}}{f}+\frac{\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f}\\ \end{align*}
Mathematica [C] time = 0.278951, size = 84, normalized size = 1.08 \[ \frac{2 b \left (a+b \sec ^2(e+f x)\right )^{3/2} \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3}{2},-\frac{1}{2},-\frac{a \cos ^2(e+f x)}{b}\right )}{3 f \sqrt{\frac{a \cos ^2(e+f x)}{b}+1} (a \cos (2 (e+f x))+a+2 b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 81, normalized size = 1. \begin{align*}{\frac{1}{3\,f} \left ( a+b \left ( \sec \left ( fx+e \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{f}{a}^{{\frac{3}{2}}}\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 ) }+{\frac{a}{f}\sqrt{a+b \left ( \sec \left ( fx+e \right ) \right ) ^{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{\left (b \sec \left (f x + e\right )^{2} + a\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 = 1.85961, size = 944, normalized size = 12.1 \begin{align*} \left [\frac{3 \, a^{\frac{3}{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 (4 \, a \cos \left (f x + e\right )^{2} + b\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{24 \, f \cos \left (f x + e\right )^{2}}, \frac{3 \, \sqrt{-a} 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 ) \cos \left (f x + e\right )^{2} + 4 \,{\left (4 \, a \cos \left (f x + e\right )^{2} + b\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{12 \, 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 \left (a + b \sec ^{2}{\left (e + f x \right )}\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 (b \sec \left (f x + e\right )^{2} + a\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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