Optimal. Leaf size=56 \[ -\frac {2 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{d \sqrt {i \sinh (c+d x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2721, 2719}
\begin {gather*} -\frac {2 i E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{d \sqrt {i \sinh (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2721
Rubi steps
\begin {align*} \int \sqrt {b \sinh (c+d x)} \, dx &=\frac {\sqrt {b \sinh (c+d x)} \int \sqrt {i \sinh (c+d x)} \, dx}{\sqrt {i \sinh (c+d x)}}\\ &=-\frac {2 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right ) \sqrt {b \sinh (c+d x)}}{d \sqrt {i \sinh (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 52, normalized size = 0.93 \begin {gather*} \frac {2 i E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right ) \sqrt {b \sinh (c+d x)}}{d \sqrt {i \sinh (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 111, normalized size = 1.98
method | result | size |
default | \(\frac {b \sqrt {-i \left (\sinh \left (d x +c \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (i-\sinh \left (d x +c \right )\right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \left (2 \EllipticE \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d}\) | \(111\) |
risch | \(\frac {\sqrt {2}\, \sqrt {b \left ({\mathrm e}^{2 d x +2 c}-1\right ) {\mathrm e}^{-d x -c}}}{d}-\frac {\left (\frac {2 b \,{\mathrm e}^{2 d x +2 c}-2 b}{b \sqrt {{\mathrm e}^{d x +c} \left (b \,{\mathrm e}^{2 d x +2 c}-b \right )}}-\frac {\sqrt {{\mathrm e}^{d x +c}+1}\, \sqrt {-2 \,{\mathrm e}^{d x +c}+2}\, \sqrt {-{\mathrm e}^{d x +c}}\, \left (-2 \EllipticE \left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {{\mathrm e}^{d x +c}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {b \,{\mathrm e}^{3 d x +3 c}-b \,{\mathrm e}^{d x +c}}}\right ) \sqrt {2}\, \sqrt {b \left ({\mathrm e}^{2 d x +2 c}-1\right ) {\mathrm e}^{-d x -c}}\, \sqrt {b \left ({\mathrm e}^{2 d x +2 c}-1\right ) {\mathrm e}^{d x +c}}}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}\) | \(227\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 42, normalized size = 0.75 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + \sqrt {b \sinh \left (d x + c\right )}\right )}}{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 \sqrt {b \sinh {\left (c + d 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 = -0.02 \begin {gather*} \int \sqrt {b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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