Optimal. Leaf size=106 \[ \frac{\left (a^2+6 a b+6 b^2\right ) \tan ^5(e+f x)}{5 f}+\frac{2 b (a+2 b) \tan ^7(e+f x)}{7 f}+\frac{2 (a+b) (a+2 b) \tan ^3(e+f x)}{3 f}+\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{b^2 \tan ^9(e+f x)}{9 f} \]
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Rubi [A] time = 0.089263, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4146, 373} \[ \frac{\left (a^2+6 a b+6 b^2\right ) \tan ^5(e+f x)}{5 f}+\frac{2 b (a+2 b) \tan ^7(e+f x)}{7 f}+\frac{2 (a+b) (a+2 b) \tan ^3(e+f x)}{3 f}+\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{b^2 \tan ^9(e+f x)}{9 f} \]
Antiderivative was successfully verified.
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Rule 4146
Rule 373
Rubi steps
\begin{align*} \int \sec ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (1+x^2\right )^2 \left (a+b+b x^2\right )^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left ((a+b)^2+2 (a+b) (a+2 b) x^2+\left (a^2+6 a b+6 b^2\right ) x^4+2 b (a+2 b) x^6+b^2 x^8\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a+b)^2 \tan (e+f x)}{f}+\frac{2 (a+b) (a+2 b) \tan ^3(e+f x)}{3 f}+\frac{\left (a^2+6 a b+6 b^2\right ) \tan ^5(e+f x)}{5 f}+\frac{2 b (a+2 b) \tan ^7(e+f x)}{7 f}+\frac{b^2 \tan ^9(e+f x)}{9 f}\\ \end{align*}
Mathematica [A] time = 0.416444, size = 96, normalized size = 0.91 \[ \frac{63 \left (a^2+6 a b+6 b^2\right ) \tan ^5(e+f x)+210 \left (a^2+3 a b+2 b^2\right ) \tan ^3(e+f x)+90 b (a+2 b) \tan ^7(e+f x)+315 (a+b)^2 \tan (e+f x)+35 b^2 \tan ^9(e+f x)}{315 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 134, normalized size = 1.3 \begin{align*}{\frac{1}{f} \left ( -{a}^{2} \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( fx+e \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{15}} \right ) \tan \left ( fx+e \right ) -2\,ab \left ( -{\frac{16}{35}}-1/7\, \left ( \sec \left ( fx+e \right ) \right ) ^{6}-{\frac{6\, \left ( \sec \left ( fx+e \right ) \right ) ^{4}}{35}}-{\frac{8\, \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{35}} \right ) \tan \left ( fx+e \right ) -{b}^{2} \left ( -{\frac{128}{315}}-{\frac{ \left ( \sec \left ( fx+e \right ) \right ) ^{8}}{9}}-{\frac{8\, \left ( \sec \left ( fx+e \right ) \right ) ^{6}}{63}}-{\frac{16\, \left ( \sec \left ( fx+e \right ) \right ) ^{4}}{105}}-{\frac{64\, \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{315}} \right ) \tan \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01043, size = 139, normalized size = 1.31 \begin{align*} \frac{35 \, b^{2} \tan \left (f x + e\right )^{9} + 90 \,{\left (a b + 2 \, b^{2}\right )} \tan \left (f x + e\right )^{7} + 63 \,{\left (a^{2} + 6 \, a b + 6 \, b^{2}\right )} \tan \left (f x + e\right )^{5} + 210 \,{\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \tan \left (f x + e\right )^{3} + 315 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )}{315 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.508377, size = 300, normalized size = 2.83 \begin{align*} \frac{{\left (8 \,{\left (21 \, a^{2} + 36 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{8} + 4 \,{\left (21 \, a^{2} + 36 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{6} + 3 \,{\left (21 \, a^{2} + 36 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 10 \,{\left (9 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 35 \, b^{2}\right )} \sin \left (f x + e\right )}{315 \, f \cos \left (f x + e\right )^{9}} \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 )^{2} \sec ^{6}{\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 = 1.28304, size = 221, normalized size = 2.08 \begin{align*} \frac{35 \, b^{2} \tan \left (f x + e\right )^{9} + 90 \, a b \tan \left (f x + e\right )^{7} + 180 \, b^{2} \tan \left (f x + e\right )^{7} + 63 \, a^{2} \tan \left (f x + e\right )^{5} + 378 \, a b \tan \left (f x + e\right )^{5} + 378 \, b^{2} \tan \left (f x + e\right )^{5} + 210 \, a^{2} \tan \left (f x + e\right )^{3} + 630 \, a b \tan \left (f x + e\right )^{3} + 420 \, b^{2} \tan \left (f x + e\right )^{3} + 315 \, a^{2} \tan \left (f x + e\right ) + 630 \, a b \tan \left (f x + e\right ) + 315 \, b^{2} \tan \left (f x + e\right )}{315 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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