Optimal. Leaf size=67 \[ \frac {a^3 \log (\sin (c+d x))}{d}+\frac {3 a^2 b \sin (c+d x)}{d}+\frac {3 a b^2 \sin ^2(c+d x)}{2 d}+\frac {b^3 \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2721, 43} \[ \frac {3 a^2 b \sin (c+d x)}{d}+\frac {a^3 \log (\sin (c+d x))}{d}+\frac {3 a b^2 \sin ^2(c+d x)}{2 d}+\frac {b^3 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2721
Rubi steps
\begin {align*} \int \cot (c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+x)^3}{x} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (3 a^2+\frac {a^3}{x}+3 a x+x^2\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \log (\sin (c+d x))}{d}+\frac {3 a^2 b \sin (c+d x)}{d}+\frac {3 a b^2 \sin ^2(c+d x)}{2 d}+\frac {b^3 \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 1.00 \[ \frac {a^3 \log (\sin (c+d x))}{d}+\frac {3 a^2 b \sin (c+d x)}{d}+\frac {3 a b^2 \sin ^2(c+d x)}{2 d}+\frac {b^3 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 66, normalized size = 0.99 \[ -\frac {9 \, a b^{2} \cos \left (d x + c\right )^{2} - 6 \, a^{3} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 2 \, {\left (b^{3} \cos \left (d x + c\right )^{2} - 9 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 58, normalized size = 0.87 \[ \frac {2 \, b^{3} \sin \left (d x + c\right )^{3} + 9 \, a b^{2} \sin \left (d x + c\right )^{2} + 6 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 18 \, a^{2} b \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 64, normalized size = 0.96 \[ \frac {a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {3 a^{2} b \sin \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 57, normalized size = 0.85 \[ \frac {2 \, b^{3} \sin \left (d x + c\right )^{3} + 9 \, a b^{2} \sin \left (d x + c\right )^{2} + 6 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 18 \, a^{2} b \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.68, size = 118, normalized size = 1.76 \[ \frac {b^3\,\sin \left (c+d\,x\right )}{3\,d}-\frac {a^3\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {3\,a\,b^2\,{\cos \left (c+d\,x\right )}^2}{2\,d}-\frac {b^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {3\,a^2\,b\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (c + d x \right )}\right )^{3} \cot {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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