Optimal. Leaf size=90 \[ \frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{f} \]
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Rubi [A] time = 0.0968939, antiderivative size = 90, 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 = {3670, 444, 50, 63, 208} \[ \frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 444
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x \left (a+b x^2\right )^{3/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}+\frac{(a-b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{1+x} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}+\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}+\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{1-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \tan ^2(e+f x)}\right )}{b f}\\ &=-\frac{(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{f}+\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 f}\\ \end{align*}
Mathematica [A] time = 0.382881, size = 80, normalized size = 0.89 \[ \frac{\sqrt{a+b \tan ^2(e+f x)} \left (4 a+b \tan ^2(e+f x)-3 b\right )-3 (a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{3 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 181, normalized size = 2. \begin{align*}{\frac{b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}{3\,f}\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}+{\frac{4\,a}{3\,f}\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}-{\frac{b}{f}\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}+{\frac{{b}^{2}}{f}\arctan \left ({\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}}-2\,{\frac{ab}{f\sqrt{-a+b}}\arctan \left ({\frac{\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}{\sqrt{-a+b}}} \right ) }+{\frac{{a}^{2}}{f}\arctan \left ({\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}} \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 \tan \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 [A] time = 2.23965, size = 624, normalized size = 6.93 \begin{align*} \left [-\frac{3 \,{\left (a - b\right )}^{\frac{3}{2}} \log \left (-\frac{b^{2} \tan \left (f x + e\right )^{4} + 2 \,{\left (4 \, a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + 4 \,{\left (b \tan \left (f x + e\right )^{2} + 2 \, a - b\right )} \sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{a - b} + 8 \, a^{2} - 8 \, a b + b^{2}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) - 4 \,{\left (b \tan \left (f x + e\right )^{2} + 4 \, a - 3 \, b\right )} \sqrt{b \tan \left (f x + e\right )^{2} + a}}{12 \, f}, \frac{3 \,{\left (a - b\right )} \sqrt{-a + b} \arctan \left (\frac{2 \, \sqrt{b \tan \left (f x + e\right )^{2} + a} \sqrt{-a + b}}{b \tan \left (f x + e\right )^{2} + 2 \, a - b}\right ) + 2 \,{\left (b \tan \left (f x + e\right )^{2} + 4 \, a - 3 \, b\right )} \sqrt{b \tan \left (f x + e\right )^{2} + a}}{6 \, f}\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 \tan ^{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 [A] time = 1.20663, size = 154, normalized size = 1.71 \begin{align*} \frac{{\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (\frac{\sqrt{b \tan \left (f x + e\right )^{2} + a}}{\sqrt{-a + b}}\right )}{\sqrt{-a + b} f} + \frac{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} f^{2} + 3 \, \sqrt{b \tan \left (f x + e\right )^{2} + a} a f^{2} - 3 \, \sqrt{b \tan \left (f x + e\right )^{2} + a} b f^{2}}{3 \, f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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