Optimal. Leaf size=28 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]
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Rubi [A] time = 0.08, 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 = {3230, 266, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 3230
Rubi steps
\begin {align*} \int \frac {\coth (x)}{\sqrt {a+b \sinh ^3(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^3}} \, dx,x,\sinh (x)\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sinh ^3(x)\right )\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^3(x)}\right )}{3 b}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 28, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth \relax (x)}{\sqrt {b \sinh \relax (x)^{3} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 21, normalized size = 0.75 \[ -\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\sinh ^{3}\relax (x )\right )}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth \relax (x)}{\sqrt {b \sinh \relax (x)^{3} + a}}\,{d x} \]
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 \[ \int \frac {\mathrm {coth}\relax (x)}{\sqrt {b\,{\mathrm {sinh}\relax (x)}^3+a}} \,d x \]
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 \[ \int \frac {\coth {\relax (x )}}{\sqrt {a + b \sinh ^{3}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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