Optimal. Leaf size=65 \[ -\frac {1}{4} e^{-a-\frac {b^2}{4}} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (-b+2 x)\right )+\frac {1}{4} e^{a-\frac {b^2}{4}} \sqrt {\pi } \text {Erfi}\left (\frac {1}{2} (b+2 x)\right ) \]
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Rubi [A]
time = 0.05, 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 = {5623, 2266,
2235} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 5623
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 = 0.05, size = 51, normalized size = 0.78 \begin {gather*} \frac {1}{4} e^{-\frac {b^2}{4}} \sqrt {\pi } \left (\text {Erfi}\left (\frac {b}{2}-x\right ) (\cosh (a)-\sinh (a))+\text {Erfi}\left (\frac {b}{2}+x\right ) (\cosh (a)+\sinh (a))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.15, size = 52, normalized size = 0.80
method | result | size |
risch | \(-\frac {i \sqrt {\pi }\, {\mathrm e}^{-a -\frac {b^{2}}{4}} \erf \left (-i x +\frac {1}{2} i b \right )}{4}-\frac {i \sqrt {\pi }\, {\mathrm e}^{a -\frac {b^{2}}{4}} \erf \left (i x +\frac {1}{2} i b \right )}{4}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.27, size = 45, normalized size = 0.69 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 45, normalized size = 0.69 \begin {gather*} \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 )} \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 e^{x^{2}} \sinh {\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.41, size = 45, normalized size = 0.69 \begin {gather*} \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 )} \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 = -0.02 \begin {gather*} \int {\mathrm {e}}^{x^2}\,\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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