Optimal. Leaf size=71 \[ \frac {2 x}{b \sqrt {\text {csch}(a+b x)}}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}} \]
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Rubi [A] time = 0.04, 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 = {5445, 3771, 2639} \[ \frac {2 x}{b \sqrt {\text {csch}(a+b x)}}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b^2 \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3771
Rule 5445
Rubi steps
\begin {align*} \int x \cosh (a+b x) \sqrt {\text {csch}(a+b x)} \, dx &=\frac {2 x}{b \sqrt {\text {csch}(a+b x)}}-\frac {2 \int \frac {1}{\sqrt {\text {csch}(a+b x)}} \, dx}{b}\\ &=\frac {2 x}{b \sqrt {\text {csch}(a+b x)}}-\frac {2 \int \sqrt {i \sinh (a+b x)} \, dx}{b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}}\\ &=\frac {2 x}{b \sqrt {\text {csch}(a+b x)}}+\frac {4 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{b^2 \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}}\\ \end {align*}
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Mathematica [C] time = 1.10, size = 183, normalized size = 2.58 \[ \frac {e^{-a-b x} \sqrt {2-2 e^{2 (a+b x)}} \sqrt {\frac {e^{a+b x}}{e^{2 (a+b x)}-1}} \left (-18 \, _3F_2\left (-\frac {1}{4},-\frac {1}{4},\frac {1}{2};\frac {3}{4},\frac {3}{4};e^{2 (a+b x)}\right )-2 e^{2 (a+b x)} \, _3F_2\left (\frac {1}{2},\frac {3}{4},\frac {3}{4};\frac {7}{4},\frac {7}{4};e^{2 (a+b x)}\right )-3 b x \left (3 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};e^{2 (a+b x)}\right )-e^{2 (a+b x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 (a+b x)}\right )\right )\right )}{9 b^2} \]
Warning: Unable to verify antiderivative.
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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 \[ \int x \cosh \left (b x + a\right ) \sqrt {\operatorname {csch}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 229, normalized size = 3.23 \[ \frac {\left (b x -2\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{{\mathrm e}^{2 b x +2 a}-1}}\, {\mathrm e}^{-b x -a}}{b^{2}}+\frac {2 \left (\frac {2 \,{\mathrm e}^{2 b x +2 a}-2}{\sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{b x +a}}}-\frac {\sqrt {1+{\mathrm e}^{b x +a}}\, \sqrt {2-2 \,{\mathrm e}^{b x +a}}\, \sqrt {-{\mathrm e}^{b x +a}}\, \left (-2 \EllipticE \left (\sqrt {1+{\mathrm e}^{b x +a}}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {1+{\mathrm e}^{b x +a}}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}-{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{{\mathrm e}^{2 b x +2 a}-1}}\, \sqrt {\left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{b x +a}}\, {\mathrm e}^{-b x -a}}{b^{2}} \]
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 \[ \int x \cosh \left (b x + a\right ) \sqrt {\operatorname {csch}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {cosh}\left (a+b\,x\right )\,\sqrt {\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}} \,d x \]
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 \[ \int x \cosh {\left (a + b x \right )} \sqrt {\operatorname {csch}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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