Optimal. Leaf size=19 \[ \frac {\log (a+b \csc (x))}{a}+\frac {\log (\sin (x))}{a} \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3885, 36, 29, 31} \[ \frac {\log (a+b \csc (x))}{a}+\frac {\log (\sin (x))}{a} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 3885
Rubi steps
\begin {align*} \int \frac {\cot (x)}{a+b \csc (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,b \csc (x)\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,b \csc (x)\right )}{a}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \csc (x)\right )}{a}\\ &=\frac {\log (a+b \csc (x))}{a}+\frac {\log (\sin (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 0.58 \[ \frac {\log (a \sin (x)+b)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 11, normalized size = 0.58 \[ \frac {\log \left (a \sin \relax (x) + b\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 12, normalized size = 0.63 \[ \frac {\log \left ({\left | a \sin \relax (x) + b \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 21, normalized size = 1.11 \[ \frac {\ln \left (a +b \csc \relax (x )\right )}{a}-\frac {\ln \left (\csc \relax (x )\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 11, normalized size = 0.58 \[ \frac {\log \left (a \sin \relax (x) + b\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 55, normalized size = 2.89 \[ \frac {2\,\mathrm {atanh}\left (\frac {a\,\left (2\,b^3\,\sin \relax (x)+\frac {5\,a\,b^2}{2}-a^3-\frac {a\,b^2\,\cos \left (2\,x\right )}{2}\right )}{{\left (-a^2+\sin \relax (x)\,a\,b+2\,b^2\right )}^2}\right )}{a} \]
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 \[ \int \frac {\cot {\relax (x )}}{a + b \csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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