Optimal. Leaf size=57 \[ \frac {2 x}{b \sqrt {\text {sech}(a+b x)}}+\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 57, 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 = {5444, 3771, 2639} \[ \frac {2 x}{b \sqrt {\text {sech}(a+b x)}}+\frac {4 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3771
Rule 5444
Rubi steps
\begin {align*} \int x \sqrt {\text {sech}(a+b x)} \sinh (a+b x) \, dx &=\frac {2 x}{b \sqrt {\text {sech}(a+b x)}}-\frac {2 \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx}{b}\\ &=\frac {2 x}{b \sqrt {\text {sech}(a+b x)}}-\frac {\left (2 \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \sqrt {\cosh (a+b x)} \, dx}{b}\\ &=\frac {2 x}{b \sqrt {\text {sech}(a+b x)}}+\frac {4 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{b^2}\\ \end {align*}
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Mathematica [C] time = 1.15, size = 100, normalized size = 1.75 \[ \frac {\sqrt {2} e^{-a-b x} \sqrt {\frac {e^{a+b x}}{e^{2 (a+b x)}+1}} \left (4 \sqrt {e^{2 (a+b x)}+1} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 (a+b x)}\right )+(b x-2) \left (e^{2 (a+b x)}+1\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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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 \[ \int x \sqrt {\operatorname {sech}\left (b x + a\right )} \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 250, normalized size = 4.39 \[ \frac {\left (b x -2\right ) \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}}\, {\mathrm e}^{-b x -a}}{b^{2}}-\frac {2 \left (-\frac {2 \left (1+{\mathrm e}^{2 b x +2 a}\right )}{\sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}}+\frac {i \sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{b x +a}-i\right )}\, \sqrt {i {\mathrm e}^{b x +a}}\, \left (-2 i \EllipticE \left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \EllipticF \left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}+{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}}\, \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}\, {\mathrm e}^{-b x -a}}{b^{2}} \]
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 \[ \int x \sqrt {\operatorname {sech}\left (b x + a\right )} \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\mathrm {sinh}\left (a+b\,x\right )\,\sqrt {\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}} \,d x \]
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 \[ \int x \sinh {\left (a + b x \right )} \sqrt {\operatorname {sech}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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