Optimal. Leaf size=85 \[ -\frac {b e^{-i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}-\frac {\sqrt {\pi } e^{i c} \text {erfc}(b x)^2}{8 b}+\frac {\sqrt {\pi } e^{-i c} \text {erfi}(b x)}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6408, 6377, 2204, 6376, 6374, 30} \[ -\frac {b e^{-i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}-\frac {\sqrt {\pi } e^{i c} \text {Erfc}(b x)^2}{8 b}+\frac {\sqrt {\pi } e^{-i c} \text {Erfi}(b x)}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2204
Rule 6374
Rule 6376
Rule 6377
Rule 6408
Rubi steps
\begin {align*} \int \cos \left (c+i b^2 x^2\right ) \text {erfc}(b x) \, dx &=\frac {1}{2} \int e^{i c-b^2 x^2} \text {erfc}(b x) \, dx+\frac {1}{2} \int e^{-i c+b^2 x^2} \text {erfc}(b x) \, dx\\ &=\frac {1}{2} \int e^{-i c+b^2 x^2} \, dx-\frac {1}{2} \int e^{-i c+b^2 x^2} \text {erf}(b x) \, dx-\frac {\left (e^{i c} \sqrt {\pi }\right ) \operatorname {Subst}(\int x \, dx,x,\text {erfc}(b x))}{4 b}\\ &=-\frac {e^{i c} \sqrt {\pi } \text {erfc}(b x)^2}{8 b}+\frac {e^{-i c} \sqrt {\pi } \text {erfi}(b x)}{4 b}-\frac {b e^{-i c} x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi }}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 1.74, size = 0, normalized size = 0.00 \[ \int \cos \left (c+i b^2 x^2\right ) \text {erfc}(b x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{2} \, {\left ({\left (\operatorname {erf}\left (b x\right ) - 1\right )} e^{\left (-2 \, b^{2} x^{2} + 2 i \, c\right )} + \operatorname {erf}\left (b x\right ) - 1\right )} e^{\left (b^{2} x^{2} - i \, c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (i \, b^{2} x^{2} + c\right ) \operatorname {erfc}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \cos \left (i b^{2} x^{2}+c \right ) \mathrm {erfc}\left (b x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\sqrt {\pi } \cos \relax (c) \operatorname {erfc}\left (b x\right )^{2}}{8 \, b} - \frac {i \, \sqrt {\pi } \operatorname {erfc}\left (b x\right )^{2} \sin \relax (c)}{8 \, b} + \frac {1}{2} \, \cos \relax (c) \int \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2}\right )}\,{d x} - \frac {1}{2} i \, \int \operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2}\right )}\,{d x} \sin \relax (c) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (b^2\,x^2\,1{}\mathrm {i}+c\right )\,\mathrm {erfc}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\left (i b^{2} x^{2} + c \right )} \operatorname {erfc}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________