Optimal. Leaf size=52 \[ \frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {\log (\sin (c+d x))}{a^2 d}-\frac {\log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 52, 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} \[ \frac {1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {\log (\sin (c+d x))}{a^2 d}-\frac {\log (\sin (c+d x)+1)}{a^2 d} \]
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))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {1}{a (a+x)^2}-\frac {1}{a^2 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\log (\sin (c+d x))}{a^2 d}-\frac {\log (1+\sin (c+d x))}{a^2 d}+\frac {1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 36, normalized size = 0.69 \[ \frac {\frac {1}{\sin (c+d x)+1}+\log (\sin (c+d x))-\log (\sin (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 59, normalized size = 1.13 \[ \frac {{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - {\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 1}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 45, normalized size = 0.87 \[ \frac {a {\left (\frac {\log \left ({\left | -\frac {a}{a \sin \left (d x + c\right ) + a} + 1 \right |}\right )}{a^{3}} + \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )} a^{2}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 50, normalized size = 0.96 \[ \frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{2} d}+\frac {1}{d \,a^{2} \left (1+\sin \left (d x +c \right )\right )}-\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 46, normalized size = 0.88 \[ \frac {\frac {1}{a^{2} \sin \left (d x + c\right ) + a^{2}} - \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {\log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.58, size = 87, normalized size = 1.67 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^2\,d}-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a^2\right )} \]
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 \[ \frac {\int \frac {\cos {\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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