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 = 0.0662634, antiderivative size = 53, 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 = {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 4147
Rule 194
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*}
Mathematica [A] time = 0.0261719, size = 106, normalized size = 2. \[ \frac{a^2 \sin ^5(e+f x)}{5 f}-\frac{2 a^2 \sin ^3(e+f x)}{3 f}+\frac{a^2 \sin (e+f x)}{f}-\frac{2 a b \sin ^3(e+f x)}{3 f}+\frac{2 a b \sin (e+f x)}{f}+\frac{b^2 \sin (e) \cos (f x)}{f}+\frac{b^2 \cos (e) \sin (f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 67, normalized size = 1.3 \begin{align*}{\frac{1}{f} \left ({\frac{{a}^{2}\sin \left ( fx+e \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( fx+e \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) }+{\frac{2\,ab \left ( 2+ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \sin \left ( fx+e \right ) }{3}}+{b}^{2}\sin \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973232, size = 74, normalized size = 1.4 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.486925, size = 142, normalized size = 2.68 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29164, size = 111, normalized size = 2.09 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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