Optimal. Leaf size=65 \[ \frac {\sin ^3(c+d x)}{3 a d}-\frac {\sin ^2(c+d x)}{2 a d}-\frac {\sin (c+d x)}{a d}+\frac {\log (\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.09, antiderivative size = 65, 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 = {2836, 12, 75} \[ \frac {\sin ^3(c+d x)}{3 a d}-\frac {\sin ^2(c+d x)}{2 a d}-\frac {\sin (c+d x)}{a d}+\frac {\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a (a-x)^2 (a+x)}{x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)}{x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^2+\frac {a^3}{x}-a x+x^2\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\log (\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 = 0.05, size = 49, normalized size = 0.75 \[ \frac {2 \sin ^3(c+d x)-3 \sin ^2(c+d x)-6 \sin (c+d x)+6 \log (\sin (c+d x))-2}{6 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 48, normalized size = 0.74 \[ \frac {3 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (\cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) + 6 \, \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right )}{6 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 61, normalized size = 0.94 \[ \frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{3} - 3 \, a^{2} \sin \left (d x + c\right )^{2} - 6 \, a^{2} \sin \left (d x + c\right )}{a^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 62, normalized size = 0.95 \[ \frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}-\frac {\sin \left (d x +c \right )}{a d}-\frac {\sin ^{2}\left (d x +c \right )}{2 a d}+\frac {\sin ^{3}\left (d x +c \right )}{3 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 51, normalized size = 0.78 \[ \frac {\frac {2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right )}{a} + \frac {6 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.05, size = 102, normalized size = 1.57 \[ \frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a\,d}-\frac {\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{a\,d}-\frac {2\,\sin \left (c+d\,x\right )}{3\,a\,d}+\frac {{\cos \left (c+d\,x\right )}^2}{2\,a\,d}-\frac {{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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