Optimal. Leaf size=51 \[ \frac{(a+2 b) \cos ^3(c+d x)}{3 d}-\frac{(a+b) \cos (c+d x)}{d}-\frac{b \cos ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0451092, antiderivative size = 51, 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 = {3013, 373} \[ \frac{(a+2 b) \cos ^3(c+d x)}{3 d}-\frac{(a+b) \cos (c+d x)}{d}-\frac{b \cos ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3013
Rule 373
Rubi steps
\begin{align*} \int \sin ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a+b-b x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a \left (1+\frac{b}{a}\right )-(a+2 b) x^2+b x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{(a+b) \cos (c+d x)}{d}+\frac{(a+2 b) \cos ^3(c+d x)}{3 d}-\frac{b \cos ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.0285599, size = 77, normalized size = 1.51 \[ -\frac{3 a \cos (c+d x)}{4 d}+\frac{a \cos (3 (c+d x))}{12 d}-\frac{5 b \cos (c+d x)}{8 d}+\frac{5 b \cos (3 (c+d x))}{48 d}-\frac{b \cos (5 (c+d x))}{80 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 54, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( -{\frac{b\cos \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }-{\frac{a \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \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 = 0.963426, size = 58, normalized size = 1.14 \begin{align*} -\frac{3 \, b \cos \left (d x + c\right )^{5} - 5 \,{\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (a + b\right )} \cos \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92667, size = 115, normalized size = 2.25 \begin{align*} -\frac{3 \, b \cos \left (d x + c\right )^{5} - 5 \,{\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (a + b\right )} \cos \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.33809, size = 107, normalized size = 2.1 \begin{align*} \begin{cases} - \frac{a \sin ^{2}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} - \frac{2 a \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac{b \sin ^{4}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} - \frac{4 b \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac{8 b \cos ^{5}{\left (c + d x \right )}}{15 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin ^{2}{\left (c \right )}\right ) \sin ^{3}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10308, size = 90, normalized size = 1.76 \begin{align*} -\frac{b \cos \left (d x + c\right )^{5}}{5 \, d} + \frac{a \cos \left (d x + c\right )^{3}}{3 \, d} + \frac{2 \, b \cos \left (d x + c\right )^{3}}{3 \, d} - \frac{a \cos \left (d x + c\right )}{d} - \frac{b \cos \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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