Optimal. Leaf size=32 \[ \frac{a \sin (c+d x)}{d}-\frac{(a-b) \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0345442, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3676} \[ \frac{a \sin (c+d x)}{d}-\frac{(a-b) \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3676
Rubi steps
\begin{align*} \int \cos ^3(c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (a-(a-b) x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{a \sin (c+d x)}{d}-\frac{(a-b) \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0168688, size = 44, normalized size = 1.38 \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 36, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{a \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05688, size = 39, normalized size = 1.22 \begin{align*} -\frac{{\left (a - b\right )} \sin \left (d x + c\right )^{3} - 3 \, a \sin \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42889, size = 74, normalized size = 2.31 \begin{align*} \frac{{\left ({\left (a - b\right )} \cos \left (d x + c\right )^{2} + 2 \, a + b\right )} \sin \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.55848, size = 49, normalized size = 1.53 \begin{align*} -\frac{a \sin \left (d x + c\right )^{3} - b \sin \left (d x + c\right )^{3} - 3 \, a \sin \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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