Optimal. Leaf size=45 \[ \frac {\sin (c+d x)}{a^3 d}+\frac {\log (\sin (c+d x))}{a^3 d}-\frac {4 \log (\sin (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.09, antiderivative size = 45, 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, 72} \[ \frac {\sin (c+d x)}{a^3 d}+\frac {\log (\sin (c+d x))}{a^3 d}-\frac {4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 72
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a (a-x)^2}{x (a+x)} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2}{x (a+x)} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {a}{x}-\frac {4 a}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\log (\sin (c+d x))}{a^3 d}-\frac {4 \log (1+\sin (c+d x))}{a^3 d}+\frac {\sin (c+d x)}{a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 32, normalized size = 0.71 \[ \frac {\sin (c+d x)+\log (\sin (c+d x))-4 \log (\sin (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 34, normalized size = 0.76 \[ \frac {\log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + \sin \left (d x + c\right )}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 103, normalized size = 2.29 \[ \frac {\frac {3 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}{a^{3}} - \frac {8 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 46, normalized size = 1.02 \[ \frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{3} d}-\frac {4 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{3} d}+\frac {\sin \left (d x +c \right )}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 43, normalized size = 0.96 \[ -\frac {\frac {4 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac {\log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac {\sin \left (d x + c\right )}{a^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.92, size = 95, normalized size = 2.11 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {8\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^3\,d}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^3\,d} \]
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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