Optimal. Leaf size=65 \[ \frac {1}{4} \sqrt {\pi } e^{a-\frac {b^2}{4}} \text {erfi}\left (\frac {1}{2} (b+2 x)\right )-\frac {1}{4} \sqrt {\pi } e^{-a-\frac {b^2}{4}} \text {erfi}\left (\frac {1}{2} (2 x-b)\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5512, 2234, 2204} \[ \frac {1}{4} \sqrt {\pi } e^{a-\frac {b^2}{4}} \text {Erfi}\left (\frac {1}{2} (b+2 x)\right )-\frac {1}{4} \sqrt {\pi } e^{-a-\frac {b^2}{4}} \text {Erfi}\left (\frac {1}{2} (2 x-b)\right ) \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 5512
Rubi steps
\begin {align*} \int e^{x^2} \sinh (a+b x) \, dx &=\int \left (-\frac {1}{2} e^{-a-b x+x^2}+\frac {1}{2} e^{a+b x+x^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-a-b x+x^2} \, dx\right )+\frac {1}{2} \int e^{a+b x+x^2} \, dx\\ &=-\left (\frac {1}{2} e^{-a-\frac {b^2}{4}} \int e^{\frac {1}{4} (-b+2 x)^2} \, dx\right )+\frac {1}{2} e^{a-\frac {b^2}{4}} \int e^{\frac {1}{4} (b+2 x)^2} \, dx\\ &=-\frac {1}{4} e^{-a-\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-b+2 x)\right )+\frac {1}{4} e^{a-\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (b+2 x)\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 51, normalized size = 0.78 \[ \frac {1}{4} \sqrt {\pi } e^{-\frac {b^2}{4}} \left ((\cosh (a)-\sinh (a)) \text {erfi}\left (\frac {b}{2}-x\right )+(\sinh (a)+\cosh (a)) \text {erfi}\left (\frac {b}{2}+x\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 45, normalized size = 0.69 \[ \frac {1}{4} \, \sqrt {\pi } {\left (\operatorname {erfi}\left (\frac {1}{2} \, b + x\right ) e^{\left (\frac {1}{4} \, b^{2} + a\right )} - \operatorname {erfi}\left (-\frac {1}{2} \, b + x\right ) e^{\left (\frac {1}{4} \, b^{2} - a\right )}\right )} e^{\left (-\frac {1}{2} \, b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.12, size = 45, normalized size = 0.69 \[ \frac {1}{4} i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, b - i \, x\right ) e^{\left (-\frac {1}{4} \, b^{2} + a\right )} - \frac {1}{4} i \, \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, b - i \, x\right ) e^{\left (-\frac {1}{4} \, b^{2} - a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 52, normalized size = 0.80 \[ -\frac {i \sqrt {\pi }\, {\mathrm e}^{-a -\frac {b^{2}}{4}} \erf \left (-i x +\frac {1}{2} i b \right )}{4}-\frac {i \sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4}} \erf \left (i x +\frac {1}{2} i b \right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.31, size = 45, normalized size = 0.69 \[ -\frac {1}{4} i \, \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, b + i \, x\right ) e^{\left (-\frac {1}{4} \, b^{2} + a\right )} + \frac {1}{4} i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, b + i \, x\right ) e^{\left (-\frac {1}{4} \, b^{2} - a\right )} \]
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 \[ \int {\mathrm {e}}^{x^2}\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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 \[ \int e^{x^{2}} \sinh {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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