Optimal. Leaf size=67 \[ -\frac {\log (1+\sin (c+d x))}{a d}+\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{2 a d}+\frac {\sin ^3(c+d x)}{3 a d} \]
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Rubi [A]
time = 0.06, antiderivative size = 67, 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 = {2912, 12, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 2912
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {x^3}{a^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\text {Subst}\left (\int \frac {x^3}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\text {Subst}\left (\int \left (a^2-a x+x^2-\frac {a^3}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 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}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 50, normalized size = 0.75 \begin {gather*} \frac {-6 \log (1+\sin (c+d x))+6 \sin (c+d x)-3 \sin ^2(c+d x)+2 \sin ^3(c+d x)}{6 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 46, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )-\ln \left (1+\sin \left (d x +c \right )\right )}{d a}\) | \(46\) |
default | \(\frac {\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )-\ln \left (1+\sin \left (d x +c \right )\right )}{d a}\) | \(46\) |
risch | \(\frac {i x}{a}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )}}{8 a d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )}}{8 a d}+\frac {2 i c}{a d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}-\frac {\sin \left (3 d x +3 c \right )}{12 a d}+\frac {\cos \left (2 d x +2 c \right )}{4 a d}\) | \(110\) |
norman | \(\frac {-\frac {5}{3 a d}-\frac {20 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {20 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a d}+\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a d}-\frac {16 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}\) | \(212\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 53, normalized size = 0.79 \begin {gather*} \frac {\frac {2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right )}{a} - \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 48, normalized size = 0.72 \begin {gather*} \frac {3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{6 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.51, size = 66, normalized size = 0.99 \begin {gather*} \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 )}}{2 a d} + \frac {\sin {\left (c + d x \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\left (c \right )} \cos {\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 64, normalized size = 0.96 \begin {gather*} -\frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{3} - 3 \, a^{2} \sin \left (d x + c\right )^{2} + 6 \, a^{2} \sin \left (d x + c\right )}{a^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 56, normalized size = 0.84 \begin {gather*} -\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}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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