Optimal. Leaf size=95 \[ \frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b^2 f}-\frac{(a+b) \sqrt{a+b \tan ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{f \sqrt{a-b}} \]
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Rubi [A] time = 0.140269, antiderivative size = 95, 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 = {3670, 446, 88, 63, 208} \[ \frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b^2 f}-\frac{(a+b) \sqrt{a+b \tan ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{f \sqrt{a-b}} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 88
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan ^5(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(1+x) \sqrt{a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{-a-b}{b \sqrt{a+b x}}+\frac{1}{(1+x) \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac{(a+b) \sqrt{a+b \tan ^2(e+f x)}}{b^2 f}+\frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b^2 f}+\frac{\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)}}{b^2 f}+\frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b^2 f}+\frac{\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{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b} f}-\frac{(a+b) \sqrt{a+b \tan ^2(e+f x)}}{b^2 f}+\frac{\left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 b^2 f}\\ \end{align*}
Mathematica [A] time = 2.34322, size = 87, normalized size = 0.92 \[ -\frac{\frac{2 \left (2 a-b \tan ^2(e+f x)+3 b\right ) \sqrt{a+b \tan ^2(e+f x)}}{3 b^2}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b}}}{2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 111, normalized size = 1.2 \begin{align*}{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{2}}{3\,fb}\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}-{\frac{2\,a}{3\,f{b}^{2}}\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}-{\frac{1}{fb}\sqrt{a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2}}}+{\frac{1}{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(-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 [A] time = 2.16573, size = 717, normalized size = 7.55 \begin{align*} \left [\frac{3 \, \sqrt{a - b} b^{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 ({\left (a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2} - a b + 3 \, b^{2}\right )} \sqrt{b \tan \left (f x + e\right )^{2} + a}}{12 \,{\left (a b^{2} - b^{3}\right )} f}, \frac{3 \, \sqrt{-a + b} b^{2} \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 ({\left (a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - 2 \, a^{2} - a b + 3 \, b^{2}\right )} \sqrt{b \tan \left (f x + e\right )^{2} + a}}{6 \,{\left (a b^{2} - b^{3}\right )} 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 \frac{\tan ^{5}{\left (e + f x \right )}}{\sqrt{a + b \tan ^{2}{\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 = 1.63544, size = 154, normalized size = 1.62 \begin{align*} \frac{\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}} b^{4} f^{2} - 3 \, \sqrt{b \tan \left (f x + e\right )^{2} + a} a b^{4} f^{2} - 3 \, \sqrt{b \tan \left (f x + e\right )^{2} + a} b^{5} f^{2}}{3 \, b^{6} f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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