Optimal. Leaf size=106 \[ \frac {\left (6 a^2+6 a b+b^2\right ) \tan ^5(e+f x)}{5 f}+\frac {a^2 \tan (e+f x)}{f}+\frac {(a+b)^2 \tan ^9(e+f x)}{9 f}+\frac {2 (a+b) (2 a+b) \tan ^7(e+f x)}{7 f}+\frac {2 a (2 a+b) \tan ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, 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 = {3191, 373} \[ \frac {\left (6 a^2+6 a b+b^2\right ) \tan ^5(e+f x)}{5 f}+\frac {a^2 \tan (e+f x)}{f}+\frac {(a+b)^2 \tan ^9(e+f x)}{9 f}+\frac {2 (a+b) (2 a+b) \tan ^7(e+f x)}{7 f}+\frac {2 a (2 a+b) \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3191
Rubi steps
\begin {align*} \int \sec ^{10}(e+f x) \left (a+b \sin ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (1+x^2\right )^2 \left (a+(a+b) x^2\right )^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2+2 a (2 a+b) x^2+\left (6 a^2+6 a b+b^2\right ) x^4+2 (a+b) (2 a+b) x^6+(a+b)^2 x^8\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a^2 \tan (e+f x)}{f}+\frac {2 a (2 a+b) \tan ^3(e+f x)}{3 f}+\frac {\left (6 a^2+6 a b+b^2\right ) \tan ^5(e+f x)}{5 f}+\frac {2 (a+b) (2 a+b) \tan ^7(e+f x)}{7 f}+\frac {(a+b)^2 \tan ^9(e+f x)}{9 f}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 107, normalized size = 1.01 \[ \frac {\sec ^9(e+f x) \left (252 \left (8 a^2+8 a b+3 b^2\right ) \sin (e+f x)+336 \left (4 a^2-a b-b^2\right ) \sin (3 (e+f x))+\left (16 a^2-4 a b+b^2\right ) (36 \sin (5 (e+f x))+9 \sin (7 (e+f x))+\sin (9 (e+f x)))\right )}{10080 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 128, normalized size = 1.21 \[ \frac {{\left (8 \, {\left (16 \, a^{2} - 4 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{8} + 4 \, {\left (16 \, a^{2} - 4 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{6} + 3 \, {\left (16 \, a^{2} - 4 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 10 \, {\left (4 \, a^{2} - a b - 5 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 35 \, a^{2} + 70 \, a b + 35 \, b^{2}\right )} \sin \left (f x + e\right )}{315 \, f \cos \left (f x + e\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 168, normalized size = 1.58 \[ \frac {35 \, a^{2} \tan \left (f x + e\right )^{9} + 70 \, a b \tan \left (f x + e\right )^{9} + 35 \, b^{2} \tan \left (f x + e\right )^{9} + 180 \, a^{2} \tan \left (f x + e\right )^{7} + 270 \, a b \tan \left (f x + e\right )^{7} + 90 \, b^{2} \tan \left (f x + e\right )^{7} + 378 \, a^{2} \tan \left (f x + e\right )^{5} + 378 \, a b \tan \left (f x + e\right )^{5} + 63 \, b^{2} \tan \left (f x + e\right )^{5} + 420 \, a^{2} \tan \left (f x + e\right )^{3} + 210 \, a b \tan \left (f x + e\right )^{3} + 315 \, a^{2} \tan \left (f x + e\right )}{315 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.61, size = 195, normalized size = 1.84 \[ \frac {-a^{2} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (f x +e \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (f x +e \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (f x +e \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (f x +e \right )\right )}{315}\right ) \tan \left (f x +e \right )+2 a b \left (\frac {\sin ^{3}\left (f x +e \right )}{9 \cos \left (f x +e \right )^{9}}+\frac {2 \left (\sin ^{3}\left (f x +e \right )\right )}{21 \cos \left (f x +e \right )^{7}}+\frac {8 \left (\sin ^{3}\left (f x +e \right )\right )}{105 \cos \left (f x +e \right )^{5}}+\frac {16 \left (\sin ^{3}\left (f x +e \right )\right )}{315 \cos \left (f x +e \right )^{3}}\right )+b^{2} \left (\frac {\sin ^{5}\left (f x +e \right )}{9 \cos \left (f x +e \right )^{9}}+\frac {4 \left (\sin ^{5}\left (f x +e \right )\right )}{63 \cos \left (f x +e \right )^{7}}+\frac {8 \left (\sin ^{5}\left (f x +e \right )\right )}{315 \cos \left (f x +e \right )^{5}}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 103, normalized size = 0.97 \[ \frac {35 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{9} + 90 \, {\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{7} + 63 \, {\left (6 \, a^{2} + 6 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{5} + 210 \, {\left (2 \, a^{2} + a b\right )} \tan \left (f x + e\right )^{3} + 315 \, a^{2} \tan \left (f x + e\right )}{315 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.22, size = 94, normalized size = 0.89 \[ \frac {a^2\,\mathrm {tan}\left (e+f\,x\right )+\frac {{\mathrm {tan}\left (e+f\,x\right )}^9\,{\left (a+b\right )}^2}{9}+{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {6\,a^2}{5}+\frac {6\,a\,b}{5}+\frac {b^2}{5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (\frac {4\,a^2}{7}+\frac {6\,a\,b}{7}+\frac {2\,b^2}{7}\right )+\frac {2\,a\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (2\,a+b\right )}{3}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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