Optimal. Leaf size=37 \[ -2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \sinh (x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2800, 52, 65,
213} \begin {gather*} 2 \sqrt {a+b \sinh (x)}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 213
Rule 2800
Rubi steps
\begin {align*} \int \coth (x) \sqrt {a+b \sinh (x)} \, dx &=\text {Subst}\left (\int \frac {\sqrt {a+x}}{x} \, dx,x,b \sinh (x)\right )\\ &=2 \sqrt {a+b \sinh (x)}+a \text {Subst}\left (\int \frac {1}{x \sqrt {a+x}} \, dx,x,b \sinh (x)\right )\\ &=2 \sqrt {a+b \sinh (x)}+(2 a) \text {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \sinh (x)}\right )\\ &=-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \sinh (x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 37, normalized size = 1.00 \begin {gather*} -2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sinh (x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \sinh (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 30, normalized size = 0.81
method | result | size |
derivativedivides | \(-2 \arctanh \left (\frac {\sqrt {a +b \sinh \left (x \right )}}{\sqrt {a}}\right ) \sqrt {a}+2 \sqrt {a +b \sinh \left (x \right )}\) | \(30\) |
default | \(-2 \arctanh \left (\frac {\sqrt {a +b \sinh \left (x \right )}}{\sqrt {a}}\right ) \sqrt {a}+2 \sqrt {a +b \sinh \left (x \right )}\) | \(30\) |
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 \begin {gather*} \text {Failed to integrate} \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. 73 vs.
\(2 (29) = 58\).
time = 0.58, size = 356, normalized size = 9.62 \begin {gather*} \left [\frac {1}{2} \, \sqrt {a} \log \left (-\frac {b^{2} \cosh \left (x\right )^{4} + b^{2} \sinh \left (x\right )^{4} + 16 \, a b \cosh \left (x\right )^{3} + 4 \, {\left (b^{2} \cosh \left (x\right ) + 4 \, a b\right )} \sinh \left (x\right )^{3} - 16 \, a b \cosh \left (x\right ) + 2 \, {\left (16 \, a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} + 24 \, a b \cosh \left (x\right ) + 16 \, a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} - 8 \, {\left (b \cosh \left (x\right )^{3} + b \sinh \left (x\right )^{3} + 4 \, a \cosh \left (x\right )^{2} + {\left (3 \, b \cosh \left (x\right ) + 4 \, a\right )} \sinh \left (x\right )^{2} - b \cosh \left (x\right ) + {\left (3 \, b \cosh \left (x\right )^{2} + 8 \, a \cosh \left (x\right ) - b\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a} \sqrt {a} + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} + 12 \, a b \cosh \left (x\right )^{2} - 4 \, a b + {\left (16 \, a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right ) + 2 \, \sqrt {b \sinh \left (x\right ) + a}, \sqrt {-a} \arctan \left (\frac {4 \, \sqrt {b \sinh \left (x\right ) + a} \sqrt {-a} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right ) - b}\right ) + 2 \, \sqrt {b \sinh \left (x\right ) + a}\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*} \int \sqrt {a + b \sinh {\left (x \right )}} \coth {\left (x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.03 \begin {gather*} \int \mathrm {coth}\left (x\right )\,\sqrt {a+b\,\mathrm {sinh}\left (x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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