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 = 0.09, 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 = {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 373
Rule 4146
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*}
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Mathematica [A] time = 0.40, 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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fricas [A] time = 0.76, size = 120, normalized size = 1.13 \[ \frac {{\left (8 \, {\left (21 \, a^{2} + 36 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{8} + 4 \, {\left (21 \, a^{2} + 36 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{6} + 3 \, {\left (21 \, a^{2} + 36 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 10 \, {\left (9 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 35 \, b^{2}\right )} \sin \left (f x + e\right )}{315 \, f \cos \left (f x + e\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 164, normalized size = 1.55 \[ \frac {35 \, b^{2} \tan \left (f x + e\right )^{9} + 90 \, a b \tan \left (f x + e\right )^{7} + 180 \, b^{2} \tan \left (f x + e\right )^{7} + 63 \, a^{2} \tan \left (f x + e\right )^{5} + 378 \, a b \tan \left (f x + e\right )^{5} + 378 \, b^{2} \tan \left (f x + e\right )^{5} + 210 \, a^{2} \tan \left (f x + e\right )^{3} + 630 \, a b \tan \left (f x + e\right )^{3} + 420 \, b^{2} \tan \left (f x + e\right )^{3} + 315 \, a^{2} \tan \left (f x + e\right ) + 630 \, a b \tan \left (f x + e\right ) + 315 \, b^{2} \tan \left (f x + e\right )}{315 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.26, size = 134, normalized size = 1.26 \[ \frac {-a^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-2 a b \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (f x +e \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (f x +e \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (f x +e \right )\right )}{35}\right ) \tan \left (f x +e \right )-b^{2} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (f x +e \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (f x +e \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (f x +e \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (f x +e \right )\right )}{315}\right ) \tan \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 103, normalized size = 0.97 \[ \frac {35 \, b^{2} \tan \left (f x + e\right )^{9} + 90 \, {\left (a b + 2 \, b^{2}\right )} \tan \left (f x + e\right )^{7} + 63 \, {\left (a^{2} + 6 \, a b + 6 \, b^{2}\right )} \tan \left (f x + e\right )^{5} + 210 \, {\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \tan \left (f x + e\right )^{3} + 315 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )}{315 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.54, size = 94, normalized size = 0.89 \[ \frac {\mathrm {tan}\left (e+f\,x\right )\,{\left (a+b\right )}^2+\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^9}{9}+{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {2\,a^2}{3}+2\,a\,b+\frac {4\,b^2}{3}\right )+{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {a^2}{5}+\frac {6\,a\,b}{5}+\frac {6\,b^2}{5}\right )+\frac {2\,b\,{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (a+2\,b\right )}{7}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \sec ^{6}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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