Optimal. Leaf size=75 \[ -\frac {\log (a+b \sin (c+d x))}{a^3 d}+\frac {\log (\sin (c+d x))}{a^3 d}+\frac {1}{a^2 d (a+b \sin (c+d x))}+\frac {1}{2 a d (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 0.06, antiderivative size = 75, 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, 44} \[ \frac {1}{a^2 d (a+b \sin (c+d x))}-\frac {\log (a+b \sin (c+d x))}{a^3 d}+\frac {\log (\sin (c+d x))}{a^3 d}+\frac {1}{2 a d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2721
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+x)^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^3 x}-\frac {1}{a (a+x)^3}-\frac {1}{a^2 (a+x)^2}-\frac {1}{a^3 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\log (\sin (c+d x))}{a^3 d}-\frac {\log (a+b \sin (c+d x))}{a^3 d}+\frac {1}{2 a d (a+b \sin (c+d x))^2}+\frac {1}{a^2 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 60, normalized size = 0.80 \[ \frac {\frac {a (3 a+2 b \sin (c+d x))}{(a+b \sin (c+d x))^2}-2 \log (a+b \sin (c+d x))+2 \log (\sin (c+d x))}{2 a^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 154, normalized size = 2.05 \[ -\frac {2 \, a b \sin \left (d x + c\right ) + 3 \, a^{2} + 2 \, {\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \, {\left (b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right )}{2 \, {\left (a^{3} b^{2} d \cos \left (d x + c\right )^{2} - 2 \, a^{4} b d \sin \left (d x + c\right ) - {\left (a^{5} + a^{3} b^{2}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 69, normalized size = 0.92 \[ -\frac {\frac {2 \, \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{3}} - \frac {2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {2 \, a b \sin \left (d x + c\right ) + 3 \, a^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2} a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 74, normalized size = 0.99 \[ \frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{3} d}-\frac {\ln \left (a +b \sin \left (d x +c \right )\right )}{a^{3} d}+\frac {1}{2 a d \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {1}{a^{2} d \left (a +b \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 81, normalized size = 1.08 \[ \frac {\frac {2 \, b \sin \left (d x + c\right ) + 3 \, a}{a^{2} b^{2} \sin \left (d x + c\right )^{2} + 2 \, a^{3} b \sin \left (d x + c\right ) + a^{4}} - \frac {2 \, \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{3}} + \frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.54, size = 369, normalized size = 4.92 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}{a^3\,d}-\frac {6\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d\,\left (a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^5+4\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4\,a^3\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4+4\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4\,a^2\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4+4\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4\,a^2\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )} \]
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 \[ \int \frac {\cot {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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