Optimal. Leaf size=71 \[ \frac {2 x \sqrt {\sinh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{b^2 \sqrt {i \sinh (a+b x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5480, 2721,
2719} \begin {gather*} \frac {2 x \sqrt {\sinh (a+b x)}}{b}+\frac {4 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b^2 \sqrt {i \sinh (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2721
Rule 5480
Rubi steps
\begin {align*} \int \frac {x \cosh (a+b x)}{\sqrt {\sinh (a+b x)}} \, dx &=\frac {2 x \sqrt {\sinh (a+b x)}}{b}-\frac {2 \int \sqrt {\sinh (a+b x)} \, dx}{b}\\ &=\frac {2 x \sqrt {\sinh (a+b x)}}{b}-\frac {\left (2 \sqrt {\sinh (a+b x)}\right ) \int \sqrt {i \sinh (a+b x)} \, dx}{b \sqrt {i \sinh (a+b x)}}\\ &=\frac {2 x \sqrt {\sinh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{b^2 \sqrt {i \sinh (a+b x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 1.07, size = 182, normalized size = 2.56 \begin {gather*} \frac {e^{-a-b x} \sqrt {2-2 e^{2 (a+b x)}} \left (-3 b x \left (3 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};e^{2 (a+b x)}\right )-e^{2 (a+b x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 (a+b x)}\right )\right )-18 \, _3F_2\left (-\frac {1}{4},-\frac {1}{4},\frac {1}{2};\frac {3}{4},\frac {3}{4};e^{2 (a+b x)}\right )-2 e^{2 (a+b x)} \, _3F_2\left (\frac {1}{2},\frac {3}{4},\frac {3}{4};\frac {7}{4},\frac {7}{4};e^{2 (a+b x)}\right )\right )}{9 b^2 \sqrt {-e^{-a-b x}+e^{a+b 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. \(228\) vs.
\(2(92)=184\).
time = 0.52, size = 229, normalized size = 3.23
method | result | size |
risch | \(\frac {\left (b x -2\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right ) \sqrt {2}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{-b x -a}}}+\frac {2 \left (\frac {2 \,{\mathrm e}^{2 b x +2 a}-2}{\sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{b x +a}}}-\frac {\sqrt {{\mathrm e}^{b x +a}+1}\, \sqrt {-2 \,{\mathrm e}^{b x +a}+2}\, \sqrt {-{\mathrm e}^{b x +a}}\, \left (-2 \EllipticE \left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {{\mathrm e}^{b x +a}+1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}-{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{b x +a}}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{-b x -a}}}\) | \(229\) |
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 [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 {x \cosh {\left (a + b x \right )}}{\sqrt {\sinh {\left (a + b 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 = -0.01 \begin {gather*} \int \frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{\sqrt {\mathrm {sinh}\left (a+b\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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