Optimal. Leaf size=68 \[ \frac {(a+b x) \text {erfi}(a+b x)^2}{b}-\frac {2 e^{(a+b x)^2} \text {erfi}(a+b x)}{\sqrt {\pi } b}+\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6354, 6384, 2204} \[ \frac {(a+b x) \text {Erfi}(a+b x)^2}{b}-\frac {2 e^{(a+b x)^2} \text {Erfi}(a+b x)}{\sqrt {\pi } b}+\frac {\sqrt {\frac {2}{\pi }} \text {Erfi}\left (\sqrt {2} (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2204
Rule 6354
Rule 6384
Rubi steps
\begin {align*} \int \text {erfi}(a+b x)^2 \, dx &=\frac {(a+b x) \text {erfi}(a+b x)^2}{b}-\frac {4 \int e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x) \, dx}{\sqrt {\pi }}\\ &=\frac {(a+b x) \text {erfi}(a+b x)^2}{b}-\frac {4 \operatorname {Subst}\left (\int e^{x^2} x \text {erfi}(x) \, dx,x,a+b x\right )}{b \sqrt {\pi }}\\ &=-\frac {2 e^{(a+b x)^2} \text {erfi}(a+b x)}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfi}(a+b x)^2}{b}+\frac {4 \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,a+b x\right )}{b \pi }\\ &=-\frac {2 e^{(a+b x)^2} \text {erfi}(a+b x)}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfi}(a+b x)^2}{b}+\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 64, normalized size = 0.94 \[ \frac {\sqrt {\pi } (a+b x) \text {erfi}(a+b x)^2-2 e^{(a+b x)^2} \text {erfi}(a+b x)+\sqrt {2} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{\sqrt {\pi } b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 90, normalized size = 1.32 \[ -\frac {2 \, \sqrt {\pi } b \operatorname {erfi}\left (b x + a\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - {\left (\pi b^{2} x + \pi a b\right )} \operatorname {erfi}\left (b x + a\right )^{2} - \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erfi}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right )}{\pi b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \erfi \left (b x +a \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {erfi}\left (a+b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________