Optimal. Leaf size=37 \[ \frac {2 x \sqrt {\cosh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5373, 2639} \[ \frac {2 x \sqrt {\cosh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 5373
Rubi steps
\begin {align*} \int \frac {x \sinh (a+b x)}{\sqrt {\cosh (a+b x)}} \, dx &=\frac {2 x \sqrt {\cosh (a+b x)}}{b}-\frac {2 \int \sqrt {\cosh (a+b x)} \, dx}{b}\\ &=\frac {2 x \sqrt {\cosh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2}\\ \end {align*}
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Mathematica [C] time = 1.55, size = 190, normalized size = 5.14 \[ \frac {\sqrt {2} e^{-a-b x} \sqrt {e^{2 (a+b x)}+1} \left (18 \, _3F_2\left (-\frac {1}{4},-\frac {1}{4},\frac {1}{2};\frac {3}{4},\frac {3}{4};-e^{2 (a+b x)}\right )-2 e^{2 (a+b x)} \, _3F_2\left (\frac {1}{2},\frac {3}{4},\frac {3}{4};\frac {7}{4},\frac {7}{4};-e^{2 (a+b x)}\right )+3 b x \left (3 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 (a+b x)}\right )+e^{2 (a+b x)} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 (a+b x)}\right )\right )\right )}{9 b^2 \sqrt {e^{-a-b x}+e^{a+b x}}} \]
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 \[ \int \frac {x \sinh \left (b x + a\right )}{\sqrt {\cosh \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 250, normalized size = 6.76 \[ \frac {\left (b x -2\right ) \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {2}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {{\mathrm e}^{-b x -a} \left (1+{\mathrm e}^{2 b x +2 a}\right )}}-\frac {2 \left (-\frac {2 \left (1+{\mathrm e}^{2 b x +2 a}\right )}{\sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}}+\frac {i \sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{b x +a}-i\right )}\, \sqrt {i {\mathrm e}^{b x +a}}\, \left (-2 i \EllipticE \left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}+{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {{\mathrm e}^{-b x -a} \left (1+{\mathrm e}^{2 b x +2 a}\right )}} \]
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 \[ \int \frac {x \sinh \left (b x + a\right )}{\sqrt {\cosh \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x\,\mathrm {sinh}\left (a+b\,x\right )}{\sqrt {\mathrm {cosh}\left (a+b\,x\right )}} \,d x \]
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 \[ \int \frac {x \sinh {\left (a + b x \right )}}{\sqrt {\cosh {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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