Optimal. Leaf size=13 \[ 2 \sqrt {\cosh (x) \coth (x)} \tanh (x) \]
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Rubi [A]
time = 0.05, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4482, 4483,
4485, 2669} \begin {gather*} 2 \tanh (x) \sqrt {\cosh (x) \coth (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2669
Rule 4482
Rule 4483
Rule 4485
Rubi steps
\begin {align*} \int \sqrt {\text {csch}(x)+\sinh (x)} \, dx &=\int \sqrt {\cosh (x) \coth (x)} \, dx\\ &=\frac {\sqrt {\cosh (x) \coth (x)} \int \sqrt {-i \cosh (x) \coth (x)} \, dx}{\sqrt {-i \cosh (x) \coth (x)}}\\ &=\frac {\sqrt {\cosh (x) \coth (x)} \int \sqrt {\cosh (x)} \sqrt {-i \coth (x)} \, dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\\ &=2 \sqrt {\cosh (x) \coth (x)} \tanh (x)\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(35\) vs. \(2(13)=26\).
time = 0.05, size = 35, normalized size = 2.69 \begin {gather*} \frac {2 \sqrt {\cosh (x) \coth (x)} \left (-1+\sqrt [4]{-\sinh ^2(x)}\right ) \tanh (x)}{\sqrt [4]{-\sinh ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs.
\(2(11)=22\).
time = 3.43, size = 42, normalized size = 3.23
method | result | size |
risch | \(\frac {\sqrt {2}\, \sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-x}}{{\mathrm e}^{2 x}-1}}\, \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\) | \(42\) |
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. 54 vs.
\(2 (11) = 22\).
time = 0.48, size = 54, normalized size = 4.15 \begin {gather*} \frac {\sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {e^{\left (-x\right )} + 1} \sqrt {-e^{\left (-x\right )} + 1}} - \frac {\sqrt {2} e^{\left (-\frac {3}{2} \, x\right )}}{\sqrt {e^{\left (-x\right )} + 1} \sqrt {-e^{\left (-x\right )} + 1}} \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. 55 vs.
\(2 (11) = 22\).
time = 0.46, size = 55, normalized size = 4.23 \begin {gather*} \frac {2 \, \sqrt {\frac {1}{2}} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}}{\sqrt {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \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*} \int \sqrt {\sinh {\left (x \right )} + \operatorname {csch}{\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 [B]
time = 1.54, size = 13, normalized size = 1.00 \begin {gather*} 2\,\mathrm {tanh}\left (x\right )\,\sqrt {\mathrm {sinh}\left (x\right )+\frac {1}{\mathrm {sinh}\left (x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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