Integrand size = 12, antiderivative size = 77 \[ \int e^{x^2} \cos (a+b x) \, dx=\frac {1}{4} e^{-i a+\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i b+2 x)\right )+\frac {1}{4} e^{i a+\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i b+2 x)\right ) \]
[Out]
Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4561, 2266, 2235} \[ \int e^{x^2} \cos (a+b x) \, dx=\frac {1}{4} \sqrt {\pi } e^{\frac {b^2}{4}-i a} \text {erfi}\left (\frac {1}{2} (2 x-i b)\right )+\frac {1}{4} \sqrt {\pi } e^{\frac {b^2}{4}+i a} \text {erfi}\left (\frac {1}{2} (2 x+i b)\right ) \]
[In]
[Out]
Rule 2235
Rule 2266
Rule 4561
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} e^{-i a-i b x+x^2}+\frac {1}{2} e^{i a+i b x+x^2}\right ) \, dx \\ & = \frac {1}{2} \int e^{-i a-i b x+x^2} \, dx+\frac {1}{2} \int e^{i a+i b x+x^2} \, dx \\ & = \frac {1}{2} e^{-i a+\frac {b^2}{4}} \int e^{\frac {1}{4} (-i b+2 x)^2} \, dx+\frac {1}{2} e^{i a+\frac {b^2}{4}} \int e^{\frac {1}{4} (i b+2 x)^2} \, dx \\ & = \frac {1}{4} e^{-i a+\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-i b+2 x)\right )+\frac {1}{4} e^{i a+\frac {b^2}{4}} \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (i b+2 x)\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.06 \[ \int e^{x^2} \cos (a+b x) \, dx=\frac {1}{4} e^{\frac {b^2}{4}} \sqrt {\pi } \left (\cos (a) \text {erfi}\left (\frac {1}{2} (-i b+2 x)\right )+\cos (a) \text {erfi}\left (\frac {1}{2} (i b+2 x)\right )-\left (\text {erf}\left (\frac {b}{2}-i x\right )+\text {erf}\left (\frac {b}{2}+i x\right )\right ) \sin (a)\right ) \]
[In]
[Out]
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {i \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4}} {\mathrm e}^{-i a} \operatorname {erf}\left (i x +\frac {b}{2}\right )}{4}+\frac {i \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4}} {\mathrm e}^{i a} \operatorname {erf}\left (-i x +\frac {b}{2}\right )}{4}\) | \(54\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.60 \[ \int e^{x^2} \cos (a+b x) \, dx=\frac {1}{4} \, \sqrt {\pi } {\left (-i \, \operatorname {erf}\left (-\frac {1}{2} \, b + i \, x\right ) e^{\left (\frac {1}{4} \, b^{2} + i \, a\right )} - i \, \operatorname {erf}\left (\frac {1}{2} \, b + i \, x\right ) e^{\left (\frac {1}{4} \, b^{2} - i \, a\right )}\right )} \]
[In]
[Out]
\[ \int e^{x^2} \cos (a+b x) \, dx=\int e^{x^{2}} \cos {\left (a + b x \right )}\, dx \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.68 \[ \int e^{x^2} \cos (a+b x) \, dx=-\frac {1}{4} \, \sqrt {\pi } {\left ({\left (i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \operatorname {erf}\left (\frac {1}{2} \, b + i \, x\right ) e^{\left (\frac {1}{4} \, b^{2}\right )} + {\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \operatorname {erf}\left (-\frac {1}{2} \, b + i \, x\right ) e^{\left (\frac {1}{4} \, b^{2}\right )}\right )} \]
[In]
[Out]
\[ \int e^{x^2} \cos (a+b x) \, dx=\int { \cos \left (b x + a\right ) e^{\left (x^{2}\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int e^{x^2} \cos (a+b x) \, dx=\int \cos \left (a+b\,x\right )\,{\mathrm {e}}^{x^2} \,d x \]
[In]
[Out]