Optimal. Leaf size=44 \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0315729, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3486, 2633} \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 2633
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac{b \cos ^3(c+d x)}{3 d}+a \int \cos ^3(c+d x) \, dx\\ &=-\frac{b \cos ^3(c+d x)}{3 d}-\frac{a \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac{b \cos ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{a \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.012072, size = 44, normalized size = 1. \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 36, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}b}{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 = 1.75519, size = 47, normalized size = 1.07 \begin{align*} -\frac{b \cos \left (d x + c\right )^{3} +{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7645, size = 90, normalized size = 2.05 \begin{align*} -\frac{b \cos \left (d x + c\right )^{3} -{\left (a \cos \left (d x + c\right )^{2} + 2 \, a\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{\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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