Optimal. Leaf size=53 \[ \frac {a^2 \tan (e+f x)}{f}+\frac {(a+b)^2 \tan ^5(e+f x)}{5 f}+\frac {2 a (a+b) \tan ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.06, antiderivative size = 53, 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, 194} \[ \frac {a^2 \tan (e+f x)}{f}+\frac {(a+b)^2 \tan ^5(e+f x)}{5 f}+\frac {2 a (a+b) \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 194
Rule 3191
Rubi steps
\begin {align*} \int \sec ^6(e+f x) \left (a+b \sin ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \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 (a+b) x^2+(a+b)^2 x^4\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a^2 \tan (e+f x)}{f}+\frac {2 a (a+b) \tan ^3(e+f x)}{3 f}+\frac {(a+b)^2 \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 67, normalized size = 1.26 \[ \frac {\tan (e+f x) \left (\left (4 a^2-2 a b-6 b^2\right ) \sec ^2(e+f x)+8 a^2+3 (a+b)^2 \sec ^4(e+f x)-4 a b+3 b^2\right )}{15 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 83, normalized size = 1.57 \[ \frac {{\left ({\left (8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{2} - a b - 3 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}\right )} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 86, normalized size = 1.62 \[ \frac {3 \, a^{2} \tan \left (f x + e\right )^{5} + 6 \, a b \tan \left (f x + e\right )^{5} + 3 \, b^{2} \tan \left (f x + e\right )^{5} + 10 \, a^{2} \tan \left (f x + e\right )^{3} + 10 \, a b \tan \left (f x + e\right )^{3} + 15 \, a^{2} \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.60, size = 101, normalized size = 1.91 \[ \frac {-a^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )+2 a b \left (\frac {\sin ^{3}\left (f x +e \right )}{5 \cos \left (f x +e \right )^{5}}+\frac {2 \left (\sin ^{3}\left (f x +e \right )\right )}{15 \cos \left (f x +e \right )^{3}}\right )+\frac {b^{2} \left (\sin ^{5}\left (f x +e \right )\right )}{5 \cos \left (f x +e \right )^{5}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 55, normalized size = 1.04 \[ \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{5} + 10 \, {\left (a^{2} + a b\right )} \tan \left (f x + e\right )^{3} + 15 \, a^{2} \tan \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.83, size = 44, normalized size = 0.83 \[ \frac {a^2\,\mathrm {tan}\left (e+f\,x\right )+\frac {{\mathrm {tan}\left (e+f\,x\right )}^5\,{\left (a+b\right )}^2}{5}+\frac {2\,a\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (a+b\right )}{3}}{f} \]
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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