Integrand size = 14, antiderivative size = 65 \[ \int e^{x^2} \cosh \left (a+c x^2\right ) \, dx=\frac {e^{-a} \sqrt {\pi } \text {erfi}\left (\sqrt {1-c} x\right )}{4 \sqrt {1-c}}+\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {1+c} x\right )}{4 \sqrt {1+c}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5624, 2235} \[ \int e^{x^2} \cosh \left (a+c x^2\right ) \, dx=\frac {\sqrt {\pi } e^{-a} \text {erfi}\left (\sqrt {1-c} x\right )}{4 \sqrt {1-c}}+\frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {c+1} x\right )}{4 \sqrt {c+1}} \]
[In]
[Out]
Rule 2235
Rule 5624
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} e^{-a+(1-c) x^2}+\frac {1}{2} e^{a+(1+c) x^2}\right ) \, dx \\ & = \frac {1}{2} \int e^{-a+(1-c) x^2} \, dx+\frac {1}{2} \int e^{a+(1+c) x^2} \, dx \\ & = \frac {e^{-a} \sqrt {\pi } \text {erfi}\left (\sqrt {1-c} x\right )}{4 \sqrt {1-c}}+\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {1+c} x\right )}{4 \sqrt {1+c}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.09 \[ \int e^{x^2} \cosh \left (a+c x^2\right ) \, dx=\frac {\sqrt {\pi } \left (\sqrt {-1+c} (1+c) \text {erf}\left (\sqrt {-1+c} x\right ) (\cosh (a)-\sinh (a))+(-1+c) \sqrt {1+c} \text {erfi}\left (\sqrt {1+c} x\right ) (\cosh (a)+\sinh (a))\right )}{4 \left (-1+c^2\right )} \]
[In]
[Out]
Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {\sqrt {\pi }\, {\mathrm e}^{-a} \operatorname {erf}\left (\sqrt {c -1}\, x \right )}{4 \sqrt {c -1}}+\frac {\sqrt {\pi }\, {\mathrm e}^{a} \operatorname {erf}\left (\sqrt {-c -1}\, x \right )}{4 \sqrt {-c -1}}\) | \(48\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.17 \[ \int e^{x^2} \cosh \left (a+c x^2\right ) \, dx=\frac {\sqrt {\pi } {\left ({\left (c + 1\right )} \cosh \left (a\right ) - {\left (c + 1\right )} \sinh \left (a\right )\right )} \sqrt {c - 1} \operatorname {erf}\left (\sqrt {c - 1} x\right ) - \sqrt {\pi } {\left ({\left (c - 1\right )} \cosh \left (a\right ) + {\left (c - 1\right )} \sinh \left (a\right )\right )} \sqrt {-c - 1} \operatorname {erf}\left (\sqrt {-c - 1} x\right )}{4 \, {\left (c^{2} - 1\right )}} \]
[In]
[Out]
\[ \int e^{x^2} \cosh \left (a+c x^2\right ) \, dx=\int e^{x^{2}} \cosh {\left (a + c x^{2} \right )}\, dx \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int e^{x^2} \cosh \left (a+c x^2\right ) \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {c - 1} x\right ) e^{\left (-a\right )}}{4 \, \sqrt {c - 1}} + \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-c - 1} x\right ) e^{a}}{4 \, \sqrt {-c - 1}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int e^{x^2} \cosh \left (a+c x^2\right ) \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {c - 1} x\right ) e^{\left (-a\right )}}{4 \, \sqrt {c - 1}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c - 1} x\right ) e^{a}}{4 \, \sqrt {-c - 1}} \]
[In]
[Out]
Timed out. \[ \int e^{x^2} \cosh \left (a+c x^2\right ) \, dx=\int {\mathrm {e}}^{x^2}\,\mathrm {cosh}\left (c\,x^2+a\right ) \,d x \]
[In]
[Out]