Optimal. Leaf size=83 \[ \frac{(a+b)^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a b^{5/2} f}-\frac{(a+2 b) \tan (e+f x)}{b^2 f}-\frac{x}{a}+\frac{\tan ^3(e+f x)}{3 b f} \]
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Rubi [A] time = 0.272673, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4141, 1975, 479, 582, 522, 203, 205} \[ \frac{(a+b)^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a b^{5/2} f}-\frac{(a+2 b) \tan (e+f x)}{b^2 f}-\frac{x}{a}+\frac{\tan ^3(e+f x)}{3 b f} \]
Antiderivative was successfully verified.
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Rule 4141
Rule 1975
Rule 479
Rule 582
Rule 522
Rule 203
Rule 205
Rubi steps
\begin{align*} \int \frac{\tan ^6(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right ) \left (a+b \left (1+x^2\right )\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\tan ^3(e+f x)}{3 b f}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 (a+b)+3 (a+2 b) x^2\right )}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 b f}\\ &=-\frac{(a+2 b) \tan (e+f x)}{b^2 f}+\frac{\tan ^3(e+f x)}{3 b f}+\frac{\operatorname{Subst}\left (\int \frac{3 (a+b) (a+2 b)+3 \left (a^2+3 a b+3 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{3 b^2 f}\\ &=-\frac{(a+2 b) \tan (e+f x)}{b^2 f}+\frac{\tan ^3(e+f x)}{3 b f}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{a f}+\frac{(a+b)^3 \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{a b^2 f}\\ &=-\frac{x}{a}+\frac{(a+b)^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{a b^{5/2} f}-\frac{(a+2 b) \tan (e+f x)}{b^2 f}+\frac{\tan ^3(e+f x)}{3 b f}\\ \end{align*}
Mathematica [C] time = 3.00328, size = 229, normalized size = 2.76 \[ \frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (-\frac{(3 a+7 b) \sec (e) \sin (f x) \sec (e+f x)}{b^2 f}-\frac{3 (a+b)^{5/2} (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left (\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right )}{a b^2 f \sqrt{b (\cos (e)-i \sin (e))^4}}-\frac{3 x}{a}+\frac{\sec (e) \sin (f x) \sec ^3(e+f x)}{b f}+\frac{\tan (e) \sec ^2(e+f x)}{b f}\right )}{6 \left (a+b \sec ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.085, size = 186, normalized size = 2.2 \begin{align*}{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{3\,fb}}-{\frac{\tan \left ( fx+e \right ) a}{f{b}^{2}}}-2\,{\frac{\tan \left ( fx+e \right ) }{fb}}-{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) }{fa}}+{\frac{{a}^{2}}{f{b}^{2}}\arctan \left ({\tan \left ( fx+e \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}}+3\,{\frac{a}{fb\sqrt{ \left ( a+b \right ) b}}\arctan \left ({\frac{\tan \left ( fx+e \right ) b}{\sqrt{ \left ( a+b \right ) b}}} \right ) }+3\,{\frac{1}{f\sqrt{ \left ( a+b \right ) b}}\arctan \left ({\frac{\tan \left ( fx+e \right ) b}{\sqrt{ \left ( a+b \right ) b}}} \right ) }+{\frac{b}{fa}\arctan \left ({\tan \left ( fx+e \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.595782, size = 890, normalized size = 10.72 \begin{align*} \left [-\frac{12 \, b^{2} f x \cos \left (f x + e\right )^{3} - 3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{-\frac{a + b}{b}} \cos \left (f x + e\right )^{3} \log \left (\frac{{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \,{\left ({\left (a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - b^{2} \cos \left (f x + e\right )\right )} \sqrt{-\frac{a + b}{b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) + 4 \,{\left ({\left (3 \, a^{2} + 7 \, a b\right )} \cos \left (f x + e\right )^{2} - a b\right )} \sin \left (f x + e\right )}{12 \, a b^{2} f \cos \left (f x + e\right )^{3}}, -\frac{6 \, b^{2} f x \cos \left (f x + e\right )^{3} + 3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{\frac{a + b}{b}} \arctan \left (\frac{{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt{\frac{a + b}{b}}}{2 \,{\left (a + b\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{3} + 2 \,{\left ({\left (3 \, a^{2} + 7 \, a b\right )} \cos \left (f x + e\right )^{2} - a b\right )} \sin \left (f x + e\right )}{6 \, a b^{2} f \cos \left (f x + e\right )^{3}}\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 ^{6}{\left (e + f x \right )}}{a + b \sec ^{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 = 4.81404, size = 178, normalized size = 2.14 \begin{align*} -\frac{\frac{3 \,{\left (f x + e\right )}}{a} - \frac{3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (f x + e\right )}{\sqrt{a b + b^{2}}}\right )\right )}}{\sqrt{a b + b^{2}} a b^{2}} - \frac{b^{2} \tan \left (f x + e\right )^{3} - 3 \, a b \tan \left (f x + e\right ) - 6 \, b^{2} \tan \left (f x + e\right )}{b^{3}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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