Optimal. Leaf size=119 \[ \frac{\left (5 a^2+12 a b+8 b^2\right ) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x \left (5 a^2+12 a b+8 b^2\right )+\frac{a (5 a+8 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{a \sin (e+f x) \cos ^5(e+f x) \left (a+b \tan ^2(e+f x)+b\right )}{6 f} \]
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Rubi [A] time = 0.145638, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4146, 413, 385, 199, 203} \[ \frac{\left (5 a^2+12 a b+8 b^2\right ) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{1}{16} x \left (5 a^2+12 a b+8 b^2\right )+\frac{a (5 a+8 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac{a \sin (e+f x) \cos ^5(e+f x) \left (a+b \tan ^2(e+f x)+b\right )}{6 f} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 413
Rule 385
Rule 199
Rule 203
Rubi steps
\begin{align*} \int \cos ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b+b x^2\right )^2}{\left (1+x^2\right )^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a \cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{6 f}+\frac{\operatorname{Subst}\left (\int \frac{(a+b) (5 a+6 b)+3 b (a+2 b) x^2}{\left (1+x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{6 f}\\ &=\frac{a (5 a+8 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a \cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{6 f}+\frac{\left (5 a^2+12 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac{\left (5 a^2+12 a b+8 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 f}+\frac{a (5 a+8 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a \cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{6 f}+\frac{\left (5 a^2+12 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{16 f}\\ &=\frac{1}{16} \left (5 a^2+12 a b+8 b^2\right ) x+\frac{\left (5 a^2+12 a b+8 b^2\right ) \cos (e+f x) \sin (e+f x)}{16 f}+\frac{a (5 a+8 b) \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac{a \cos ^5(e+f x) \sin (e+f x) \left (a+b+b \tan ^2(e+f x)\right )}{6 f}\\ \end{align*}
Mathematica [A] time = 0.196163, size = 99, normalized size = 0.83 \[ \frac{\left (45 a^2+96 a b+48 b^2\right ) \sin (2 (e+f x))+a^2 \sin (6 (e+f x))+60 a^2 e+60 a^2 f x+3 a (3 a+4 b) \sin (4 (e+f x))+144 a b e+144 a b f x+96 b^2 e+96 b^2 f x}{192 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 116, normalized size = 1. \begin{align*}{\frac{1}{f} \left ({a}^{2} \left ({\frac{\sin \left ( fx+e \right ) }{6} \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( fx+e \right ) }{8}} \right ) }+{\frac{5\,fx}{16}}+{\frac{5\,e}{16}} \right ) +2\,ab \left ( 1/4\, \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{3}+3/2\,\cos \left ( fx+e \right ) \right ) \sin \left ( fx+e \right ) +3/8\,fx+3/8\,e \right ) +{b}^{2} \left ({\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50327, size = 182, normalized size = 1.53 \begin{align*} \frac{3 \,{\left (5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )}{\left (f x + e\right )} + \frac{3 \,{\left (5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )^{5} + 8 \,{\left (5 \, a^{2} + 12 \, a b + 6 \, b^{2}\right )} \tan \left (f x + e\right )^{3} + 3 \,{\left (11 \, a^{2} + 20 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{6} + 3 \, \tan \left (f x + e\right )^{4} + 3 \, \tan \left (f x + e\right )^{2} + 1}}{48 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.503977, size = 212, normalized size = 1.78 \begin{align*} \frac{3 \,{\left (5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )} f x +{\left (8 \, a^{2} \cos \left (f x + e\right )^{5} + 2 \,{\left (5 \, a^{2} + 12 \, a b\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 1.31132, size = 217, normalized size = 1.82 \begin{align*} \frac{3 \,{\left (5 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )}{\left (f x + e\right )} + \frac{15 \, a^{2} \tan \left (f x + e\right )^{5} + 36 \, a b \tan \left (f x + e\right )^{5} + 24 \, b^{2} \tan \left (f x + e\right )^{5} + 40 \, a^{2} \tan \left (f x + e\right )^{3} + 96 \, a b \tan \left (f x + e\right )^{3} + 48 \, b^{2} \tan \left (f x + e\right )^{3} + 33 \, a^{2} \tan \left (f x + e\right ) + 60 \, a b \tan \left (f x + e\right ) + 24 \, b^{2} \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{3}}}{48 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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