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 = 0.09, antiderivative size = 85, normalized size of antiderivative = 1.00, 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 12
Rule 43
Rule 2833
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*}
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Mathematica [A] time = 0.13, 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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fricas [A] time = 0.64, size = 58, normalized size = 0.68 \[ \frac {3 \, \cos \left (d x + c\right )^{4} - 12 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) + 12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{12 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 76, normalized size = 0.89 \[ \frac {\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4} - 4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \sin \left (d x + c\right )}{a^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 80, normalized size = 0.94 \[ \frac {\ln \left (1+\sin \left (d x +c \right )\right )}{a d}-\frac {\sin \left (d x +c \right )}{a d}+\frac {\sin ^{2}\left (d x +c \right )}{2 a d}-\frac {\sin ^{3}\left (d x +c \right )}{3 d a}+\frac {\sin ^{4}\left (d x +c \right )}{4 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 63, normalized size = 0.74 \[ \frac {\frac {3 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{3} + 6 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right )}{a} + \frac {12 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.46, size = 68, normalized size = 0.80 \[ \frac {\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{a}-\frac {\sin \left (c+d\,x\right )}{a}+\frac {{\sin \left (c+d\,x\right )}^2}{2\,a}-\frac {{\sin \left (c+d\,x\right )}^3}{3\,a}+\frac {{\sin \left (c+d\,x\right )}^4}{4\,a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.20, size = 80, normalized size = 0.94 \[ \begin {cases} \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} + \frac {\sin ^{4}{\left (c + d x \right )}}{4 a d} - \frac {\sin ^{3}{\left (c + d x \right )}}{3 a d} + \frac {\sin ^{2}{\left (c + d x \right )}}{2 a d} - \frac {\sin {\left (c + d x \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\relax (c )} \cos {\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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