Optimal. Leaf size=97 \[ \frac {1}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {\log (\sin (c+d x))}{a^4 d}-\frac {\log (\sin (c+d x)+1)}{a^4 d}+\frac {1}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.06, antiderivative size = 97, 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 = {2707, 44} \[ \frac {1}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {1}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac {\log (\sin (c+d x))}{a^4 d}-\frac {\log (\sin (c+d x)+1)}{a^4 d}+\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^4 x}-\frac {1}{a (a+x)^4}-\frac {1}{a^2 (a+x)^3}-\frac {1}{a^3 (a+x)^2}-\frac {1}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\log (\sin (c+d x))}{a^4 d}-\frac {\log (1+\sin (c+d x))}{a^4 d}+\frac {1}{3 a d (a+a \sin (c+d x))^3}+\frac {1}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {1}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 62, normalized size = 0.64 \[ \frac {\frac {6 \sin ^2(c+d x)+15 \sin (c+d x)+11}{(\sin (c+d x)+1)^3}+6 \log (\sin (c+d x))-6 \log (\sin (c+d x)+1)}{6 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 152, normalized size = 1.57 \[ \frac {6 \, \cos \left (d x + c\right )^{2} + 6 \, {\left (3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 6 \, {\left (3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, \sin \left (d x + c\right ) - 17}{6 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 69, normalized size = 0.71 \[ -\frac {\frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} - \frac {6 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac {6 \, \sin \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) + 11}{a^{4} {\left (\sin \left (d x + c\right ) + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 86, normalized size = 0.89 \[ \frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{4} d}+\frac {1}{3 d \,a^{4} \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {1}{2 d \,a^{4} \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {1}{d \,a^{4} \left (1+\sin \left (d x +c \right )\right )}-\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 95, normalized size = 0.98 \[ \frac {\frac {6 \, \sin \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) + 11}{a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}} - \frac {6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {6 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.53, size = 206, normalized size = 2.12 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^4\,d}-\frac {6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+15\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+20\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+15\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a^4\right )} \]
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 \[ \frac {\int \frac {\cos {\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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