Optimal. Leaf size=32 \[ \frac {\log (\sin (c+d x))}{a d}-\frac {\log (1+\sin (c+d x))}{a d} \]
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Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2786, 36, 29,
31} \begin {gather*} \frac {\log (\sin (c+d x))}{a d}-\frac {\log (\sin (c+d x)+1)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2786
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,a \sin (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\log (\sin (c+d x))}{a d}-\frac {\log (1+\sin (c+d x))}{a d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 32, normalized size = 1.00 \begin {gather*} \frac {\log (\sin (c+d x))}{a d}-\frac {\log (1+\sin (c+d x))}{a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 27, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\ln \left (\sin \left (d x +c \right )\right )-\ln \left (1+\sin \left (d x +c \right )\right )}{a d}\) | \(27\) |
default | \(\frac {\ln \left (\sin \left (d x +c \right )\right )-\ln \left (1+\sin \left (d x +c \right )\right )}{a d}\) | \(27\) |
risch | \(-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a d}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 31, normalized size = 0.97 \begin {gather*} -\frac {\frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {\log \left (\sin \left (d x + c\right )\right )}{a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 28, normalized size = 0.88 \begin {gather*} \frac {\log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - \log \left (\sin \left (d x + c\right ) + 1\right )}{a d} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\cos {\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 33, normalized size = 1.03 \begin {gather*} -\frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.61, size = 32, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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