Optimal. Leaf size=28 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]
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Rubi [A]
time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3309, 272, 65,
214} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 3309
Rubi steps
\begin {align*} \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx &=\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^3}} \, dx,x,\sin (x)\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^3(x)\right )\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^3(x)}\right )}{3 b}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.30, size = 21, normalized size = 0.75
method | result | size |
derivativedivides | \(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\sin ^{3}\left (x \right )\right )}}{\sqrt {a}}\right )}{3 \sqrt {a}}\) | \(21\) |
default | \(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\sin ^{3}\left (x \right )\right )}}{\sqrt {a}}\right )}{3 \sqrt {a}}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 39, normalized size = 1.39 \begin {gather*} \frac {\log \left (\frac {\sqrt {b \sin \left (x\right )^{3} + a} - \sqrt {a}}{\sqrt {b \sin \left (x\right )^{3} + a} + \sqrt {a}}\right )}{3 \, \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 (x \right )}}{\sqrt {a + b \sin ^{3}{\left (x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 24, normalized size = 0.86 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {b \sin \left (x\right )^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {cot}\left (x\right )}{\sqrt {b\,{\sin \left (x\right )}^3+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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