Optimal. Leaf size=76 \[ -\frac {a^3 \log (a+b \sin (c+d x))}{b^4 d}+\frac {a^2 \sin (c+d x)}{b^3 d}-\frac {a \sin ^2(c+d x)}{2 b^2 d}+\frac {\sin ^3(c+d x)}{3 b d} \]
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Rubi [A] time = 0.09, antiderivative size = 76, 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} \[ \frac {a^2 \sin (c+d x)}{b^3 d}-\frac {a^3 \log (a+b \sin (c+d x))}{b^4 d}-\frac {a \sin ^2(c+d x)}{2 b^2 d}+\frac {\sin ^3(c+d x)}{3 b d} \]
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 ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{b^3 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2-a x+x^2-\frac {a^3}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=-\frac {a^3 \log (a+b \sin (c+d x))}{b^4 d}+\frac {a^2 \sin (c+d x)}{b^3 d}-\frac {a \sin ^2(c+d x)}{2 b^2 d}+\frac {\sin ^3(c+d x)}{3 b d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 66, normalized size = 0.87 \[ \frac {-6 a^3 \log (a+b \sin (c+d x))+6 a^2 b \sin (c+d x)-3 a b^2 \sin ^2(c+d x)+2 b^3 \sin ^3(c+d x)}{6 b^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 71, normalized size = 0.93 \[ \frac {3 \, a b^{2} \cos \left (d x + c\right )^{2} - 6 \, a^{3} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \, {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}{6 \, b^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 68, normalized size = 0.89 \[ -\frac {\frac {6 \, a^{3} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{4}} - \frac {2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \, a^{2} \sin \left (d x + c\right )}{b^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 73, normalized size = 0.96 \[ -\frac {a^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{4} d}+\frac {a^{2} \sin \left (d x +c \right )}{b^{3} d}-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{2 b^{2} d}+\frac {\sin ^{3}\left (d x +c \right )}{3 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 67, normalized size = 0.88 \[ -\frac {\frac {6 \, a^{3} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{4}} - \frac {2 \, b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 6 \, a^{2} \sin \left (d x + c\right )}{b^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 64, normalized size = 0.84 \[ \frac {\frac {{\sin \left (c+d\,x\right )}^3}{3\,b}-\frac {a^3\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{b^4}-\frac {a\,{\sin \left (c+d\,x\right )}^2}{2\,b^2}+\frac {a^2\,\sin \left (c+d\,x\right )}{b^3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.71, size = 105, normalized size = 1.38 \[ \begin {cases} \frac {x \sin ^{3}{\relax (c )} \cos {\relax (c )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sin ^{4}{\left (c + d x \right )}}{4 a d} & \text {for}\: b = 0 \\\frac {x \sin ^{3}{\relax (c )} \cos {\relax (c )}}{a + b \sin {\relax (c )}} & \text {for}\: d = 0 \\- \frac {a^{3} \log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )}}{b^{4} d} + \frac {a^{2} \sin {\left (c + d x \right )}}{b^{3} d} + \frac {a \cos ^{2}{\left (c + d x \right )}}{2 b^{2} d} + \frac {\sin ^{3}{\left (c + d x \right )}}{3 b d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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