Optimal. Leaf size=30 \[ \frac{(a+b) \sin (e+f x)}{f}-\frac{a \sin ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.045325, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4044, 3013} \[ \frac{(a+b) \sin (e+f x)}{f}-\frac{a \sin ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 4044
Rule 3013
Rubi steps
\begin{align*} \int \cos ^3(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\int \cos (e+f x) \left (b+a \cos ^2(e+f x)\right ) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \left (a+b-a x^2\right ) \, dx,x,-\sin (e+f x)\right )}{f}\\ &=\frac{(a+b) \sin (e+f x)}{f}-\frac{a \sin ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.0226969, size = 50, normalized size = 1.67 \[ -\frac{a \sin ^3(e+f x)}{3 f}+\frac{a \sin (e+f x)}{f}+\frac{b \sin (e) \cos (f x)}{f}+\frac{b \cos (e) \sin (f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 33, normalized size = 1.1 \begin{align*}{\frac{1}{f} \left ({\frac{a \left ( 2+ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \sin \left ( fx+e \right ) }{3}}+\sin \left ( fx+e \right ) b \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991559, size = 36, normalized size = 1.2 \begin{align*} -\frac{a \sin \left (f x + e\right )^{3} - 3 \,{\left (a + b\right )} \sin \left (f x + e\right )}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.47222, size = 69, normalized size = 2.3 \begin{align*} \frac{{\left (a \cos \left (f x + e\right )^{2} + 2 \, a + 3 \, b\right )} \sin \left (f x + e\right )}{3 \, 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 = 1.30849, size = 50, normalized size = 1.67 \begin{align*} -\frac{a \sin \left (f x + e\right )^{3} - 3 \, a \sin \left (f x + e\right ) - 3 \, b \sin \left (f x + e\right )}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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