Optimal. Leaf size=85 \[ \frac {e^{-a+\frac {1}{4 c}} \sqrt {\pi } \text {Erf}\left (\frac {1-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e^{a-\frac {1}{4 c}} \sqrt {\pi } \text {Erfi}\left (\frac {1+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \]
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Rubi [A]
time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5623, 2266,
2236, 2235} \begin {gather*} \frac {\sqrt {\pi } e^{\frac {1}{4 c}-a} \text {Erf}\left (\frac {1-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {\sqrt {\pi } e^{a-\frac {1}{4 c}} \text {Erfi}\left (\frac {2 c x+1}{2 \sqrt {c}}\right )}{4 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2236
Rule 2266
Rule 5623
Rubi steps
\begin {align*} \int e^x \sinh \left (a+c x^2\right ) \, dx &=\int \left (-\frac {1}{2} e^{-a+x-c x^2}+\frac {1}{2} e^{a+x+c x^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-a+x-c x^2} \, dx\right )+\frac {1}{2} \int e^{a+x+c x^2} \, dx\\ &=\frac {1}{2} e^{a-\frac {1}{4 c}} \int e^{\frac {(1+2 c x)^2}{4 c}} \, dx-\frac {1}{2} e^{-a+\frac {1}{4 c}} \int e^{-\frac {(1-2 c x)^2}{4 c}} \, dx\\ &=\frac {e^{-a+\frac {1}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {1-2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e^{a-\frac {1}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {1+2 c x}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 80, normalized size = 0.94 \begin {gather*} \frac {e^{\left .-\frac {1}{4}\right /c} \sqrt {\pi } \left (-e^{\left .\frac {1}{2}\right /c} \text {Erf}\left (\frac {-1+2 c x}{2 \sqrt {c}}\right ) (\cosh (a)-\sinh (a))+\text {Erfi}\left (\frac {1+2 c x}{2 \sqrt {c}}\right ) (\cosh (a)+\sinh (a))\right )}{4 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.20, size = 72, normalized size = 0.85
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -1}{4 c}} \erf \left (\sqrt {c}\, x -\frac {1}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -1}{4 c}} \erf \left (x \sqrt {-c}-\frac {1}{2 \sqrt {-c}}\right )}{4 \sqrt {-c}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 65, normalized size = 0.76 \begin {gather*} \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-c} x - \frac {1}{2 \, \sqrt {-c}}\right ) e^{\left (a - \frac {1}{4 \, c}\right )}}{4 \, \sqrt {-c}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {c} x - \frac {1}{2 \, \sqrt {c}}\right ) e^{\left (-a + \frac {1}{4 \, c}\right )}}{4 \, \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 103, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-c} {\left (\cosh \left (\frac {4 \, a c - 1}{4 \, c}\right ) + \sinh \left (\frac {4 \, a c - 1}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + 1\right )} \sqrt {-c}}{2 \, c}\right ) + \sqrt {\pi } \sqrt {c} {\left (\cosh \left (\frac {4 \, a c - 1}{4 \, c}\right ) - \sinh \left (\frac {4 \, a c - 1}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 \, c x - 1}{2 \, \sqrt {c}}\right )}{4 \, c} \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} \sinh {\left (a + c x^{2} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 73, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x + \frac {1}{c}\right )}\right ) e^{\left (\frac {4 \, a c - 1}{4 \, c}\right )}}{4 \, \sqrt {-c}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x - \frac {1}{c}\right )}\right ) e^{\left (-\frac {4 \, a c - 1}{4 \, c}\right )}}{4 \, \sqrt {c}} \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 = -0.01 \begin {gather*} \int {\mathrm {e}}^x\,\mathrm {sinh}\left (c\,x^2+a\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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