Optimal. Leaf size=11 \[ \frac {\tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A]
time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4423, 212}
\begin {gather*} \frac {\tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 4423
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\tanh ^{-1}(\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}(\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 12, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {\arctanh \left (\sin \left (d x +c \right )\right )}{d}\) | \(12\) |
default | \(\frac {\arctanh \left (\sin \left (d x +c \right )\right )}{d}\) | \(12\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (11) = 22\).
time = 0.28, size = 26, normalized size = 2.36 \begin {gather*} \frac {\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (11) = 22\).
time = 3.55, size = 28, normalized size = 2.55 \begin {gather*} \frac {\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, 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*} \int \frac {\cot {\left (c + d x \right )}}{- \sin {\left (c + d x \right )} + \csc {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (11) = 22\).
time = 0.42, size = 28, normalized size = 2.55 \begin {gather*} \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.64, size = 15, normalized size = 1.36 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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