Optimal. Leaf size=85 \[ \frac{\sin ^4(c+d x)}{4 a d}-\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x)+1)}{a d} \]
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Rubi [A] time = 0.0866874, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ \frac{\sin ^4(c+d x)}{4 a d}-\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin (c+d x)}{a d}+\frac{\log (\sin (c+d x)+1)}{a d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{a^4 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a^3+a^2 x-a x^2+x^3+\frac{a^4}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\log (1+\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{a d}+\frac{\sin ^2(c+d x)}{2 a d}-\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin ^4(c+d x)}{4 a d}\\ \end{align*}
Mathematica [A] time = 0.129097, size = 60, normalized size = 0.71 \[ \frac{3 \sin ^4(c+d x)-4 \sin ^3(c+d x)+6 \sin ^2(c+d x)-12 \sin (c+d x)+12 \log (\sin (c+d x)+1)}{12 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 80, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{da}}-{\frac{\sin \left ( dx+c \right ) }{da}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,da}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,da}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07133, size = 85, normalized size = 1. \begin{align*} \frac{\frac{3 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right )}{a} + \frac{12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49545, size = 157, normalized size = 1.85 \begin{align*} \frac{3 \, \cos \left (d x + c\right )^{4} - 12 \, \cos \left (d x + c\right )^{2} + 4 \,{\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) + 12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{12 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.59529, size = 102, normalized size = 1.2 \begin{align*} \begin{cases} \frac{\log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a d} - \frac{\sin ^{3}{\left (c + d x \right )}}{3 a d} - \frac{\sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 a d} - \frac{\sin{\left (c + d x \right )}}{a d} - \frac{\cos ^{4}{\left (c + d x \right )}}{4 a d} - \frac{\cos ^{2}{\left (c + d x \right )}}{2 a d} & \text{for}\: d \neq 0 \\\frac{x \sin ^{4}{\left (c \right )} \cos{\left (c \right )}}{a \sin{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29529, size = 103, normalized size = 1.21 \begin{align*} \frac{\frac{12 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{4} - 4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \sin \left (d x + c\right )}{a^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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