Optimal. Leaf size=50 \[ -\frac{(2 a+b) \sin ^3(e+f x)}{3 f}+\frac{(a+b) \sin (e+f x)}{f}+\frac{a \sin ^5(e+f x)}{5 f} \]
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Rubi [A] time = 0.0662125, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4044, 3013, 373} \[ -\frac{(2 a+b) \sin ^3(e+f x)}{3 f}+\frac{(a+b) \sin (e+f x)}{f}+\frac{a \sin ^5(e+f x)}{5 f} \]
Antiderivative was successfully verified.
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Rule 4044
Rule 3013
Rule 373
Rubi steps
\begin{align*} \int \cos ^5(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\int \cos ^3(e+f x) \left (b+a \cos ^2(e+f x)\right ) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a+b-a x^2\right ) \, dx,x,-\sin (e+f x)\right )}{f}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a \left (1+\frac{b}{a}\right )-(2 a+b) x^2+a x^4\right ) \, dx,x,-\sin (e+f x)\right )}{f}\\ &=\frac{(a+b) \sin (e+f x)}{f}-\frac{(2 a+b) \sin ^3(e+f x)}{3 f}+\frac{a \sin ^5(e+f x)}{5 f}\\ \end{align*}
Mathematica [A] time = 0.0238516, size = 71, normalized size = 1.42 \[ \frac{a \sin ^5(e+f x)}{5 f}-\frac{2 a \sin ^3(e+f x)}{3 f}+\frac{a \sin (e+f x)}{f}-\frac{b \sin ^3(e+f x)}{3 f}+\frac{b \sin (e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 54, normalized size = 1.1 \begin{align*}{\frac{1}{f} \left ({\frac{\sin \left ( fx+e \right ) a}{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{b \left ( 2+ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \sin \left ( fx+e \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00095, size = 58, normalized size = 1.16 \begin{align*} \frac{3 \, a \sin \left (f x + e\right )^{5} - 5 \,{\left (2 \, a + b\right )} \sin \left (f x + e\right )^{3} + 15 \,{\left (a + b\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 = 0.477763, size = 113, normalized size = 2.26 \begin{align*} \frac{{\left (3 \, a \cos \left (f x + e\right )^{4} +{\left (4 \, a + 5 \, b\right )} \cos \left (f x + e\right )^{2} + 8 \, a + 10 \, b\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 = 1.22154, size = 84, normalized size = 1.68 \begin{align*} \frac{3 \, a \sin \left (f x + e\right )^{5} - 10 \, a \sin \left (f x + e\right )^{3} - 5 \, b \sin \left (f x + e\right )^{3} + 15 \, a \sin \left (f x + e\right ) + 15 \, b \sin \left (f x + e\right )}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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