Optimal. Leaf size=33 \[ -\frac{a \cos ^4(c+d x)}{4 d}-\frac{b \cos ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0619245, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4377, 12, 2565, 30} \[ -\frac{a \cos ^4(c+d x)}{4 d}-\frac{b \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4377
Rule 12
Rule 2565
Rule 30
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx &=a \int \cos ^3(c+d x) \sin (c+d x) \, dx+\int b \cos ^2(c+d x) \sin (c+d x) \, dx\\ &=b \int \cos ^2(c+d x) \sin (c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int x^3 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a \cos ^4(c+d x)}{4 d}-\frac{b \operatorname{Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{b \cos ^3(c+d x)}{3 d}-\frac{a \cos ^4(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0120894, size = 33, normalized size = 1. \[ -\frac{a \cos ^4(c+d x)}{4 d}-\frac{b \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 29, normalized size = 0.9 \begin{align*} -{\frac{1}{d} \left ({\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}a}{4}}+{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10022, size = 38, normalized size = 1.15 \begin{align*} -\frac{3 \, a \cos \left (d x + c\right )^{4} + 4 \, b \cos \left (d x + c\right )^{3}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.487187, size = 69, normalized size = 2.09 \begin{align*} -\frac{3 \, a \cos \left (d x + c\right )^{4} + 4 \, b \cos \left (d x + c\right )^{3}}{12 \, 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 \sin{\left (c + d x \right )} + 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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