Optimal. Leaf size=53 \[ \frac {a^2 \sin ^5(e+f x)}{5 f}-\frac {2 a (a+b) \sin ^3(e+f x)}{3 f}+\frac {(a+b)^2 \sin (e+f x)}{f} \]
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Rubi [A] time = 0.07, 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 = {4147, 194} \[ \frac {a^2 \sin ^5(e+f x)}{5 f}-\frac {2 a (a+b) \sin ^3(e+f x)}{3 f}+\frac {(a+b)^2 \sin (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 194
Rule 4147
Rubi steps
\begin {align*} \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b-a x^2\right )^2 \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 \left (1+\frac {b (2 a+b)}{a^2}\right )-2 a^2 \left (1+\frac {b}{a}\right ) x^2+a^2 x^4\right ) \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {(a+b)^2 \sin (e+f x)}{f}-\frac {2 a (a+b) \sin ^3(e+f x)}{3 f}+\frac {a^2 \sin ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 106, normalized size = 2.00 \[ \frac {a^2 \sin ^5(e+f x)}{5 f}-\frac {2 a^2 \sin ^3(e+f x)}{3 f}+\frac {a^2 \sin (e+f x)}{f}-\frac {2 a b \sin ^3(e+f x)}{3 f}+\frac {2 a b \sin (e+f x)}{f}+\frac {b^2 \sin (e) \cos (f x)}{f}+\frac {b^2 \cos (e) \sin (f x)}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 59, normalized size = 1.11 \[ \frac {{\left (3 \, a^{2} \cos \left (f x + e\right )^{4} + 2 \, {\left (2 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} + 20 \, a b + 15 \, b^{2}\right )} \sin \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 82, normalized size = 1.55 \[ \frac {3 \, a^{2} \sin \left (f x + e\right )^{5} - 10 \, a^{2} \sin \left (f x + e\right )^{3} - 10 \, a b \sin \left (f x + e\right )^{3} + 15 \, a^{2} \sin \left (f x + e\right ) + 30 \, a b \sin \left (f x + e\right ) + 15 \, b^{2} \sin \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.60, size = 67, normalized size = 1.26 \[ \frac {\frac {a^{2} \left (\frac {8}{3}+\cos ^{4}\left (f x +e \right )+\frac {4 \left (\cos ^{2}\left (f x +e \right )\right )}{3}\right ) \sin \left (f x +e \right )}{5}+\frac {2 a b \left (2+\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )}{3}+b^{2} \sin \left (f x +e \right )}{f} \]
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 \[ \frac {3 \, a^{2} \sin \left (f x + e\right )^{5} - 10 \, {\left (a^{2} + a b\right )} \sin \left (f x + e\right )^{3} + 15 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sin \left (f x + e\right )}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.44, size = 44, normalized size = 0.83 \[ \frac {\sin \left (e+f\,x\right )\,{\left (a+b\right )}^2+\frac {a^2\,{\sin \left (e+f\,x\right )}^5}{5}-\frac {2\,a\,{\sin \left (e+f\,x\right )}^3\,\left (a+b\right )}{3}}{f} \]
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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