Optimal. Leaf size=16 \[ -\frac {\csc (x)}{a}-\frac {\log (\sin (x))}{a} \]
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Rubi [A]
time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3964, 45}
\begin {gather*} -\frac {\csc (x)}{a}-\frac {\log (\sin (x))}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3964
Rubi steps
\begin {align*} \int \frac {\cot ^3(x)}{a+a \csc (x)} \, dx &=\frac {\text {Subst}\left (\int \frac {a-a x}{x^2} \, dx,x,\sin (x)\right )}{a^2}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a}{x^2}-\frac {a}{x}\right ) \, dx,x,\sin (x)\right )}{a^2}\\ &=-\frac {\csc (x)}{a}-\frac {\log (\sin (x))}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 11, normalized size = 0.69 \begin {gather*} -\frac {\csc (x)+\log (\sin (x))}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 13, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {-\csc \left (x \right )+\ln \left (\csc \left (x \right )\right )}{a}\) | \(13\) |
default | \(\frac {-\csc \left (x \right )+\ln \left (\csc \left (x \right )\right )}{a}\) | \(13\) |
risch | \(\frac {i x}{a}-\frac {2 i {\mathrm e}^{i x}}{a \left ({\mathrm e}^{2 i x}-1\right )}-\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{a}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 18, normalized size = 1.12 \begin {gather*} -\frac {\log \left (\sin \left (x\right )\right )}{a} - \frac {1}{a \sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.07, size = 19, normalized size = 1.19 \begin {gather*} -\frac {\log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) + 1}{a \sin \left (x\right )} \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 {\cot ^{3}{\left (x \right )}}{\csc {\left (x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 14, normalized size = 0.88 \begin {gather*} -\frac {\frac {1}{\sin \left (x\right )} + \log \left ({\left | \sin \left (x\right ) \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 36, normalized size = 2.25 \begin {gather*} -\frac {\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2}-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )+\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )+\frac {1}{2\,\mathrm {tan}\left (\frac {x}{2}\right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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