Optimal. Leaf size=54 \[ -\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } b}+x \text {erfi}(b x)^2+\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6354, 12, 6384, 2204} \[ -\frac {2 e^{b^2 x^2} \text {Erfi}(b x)}{\sqrt {\pi } b}+x \text {Erfi}(b x)^2+\frac {\sqrt {\frac {2}{\pi }} \text {Erfi}\left (\sqrt {2} b x\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2204
Rule 6354
Rule 6384
Rubi steps
\begin {align*} \int \text {erfi}(b x)^2 \, dx &=x \text {erfi}(b x)^2-\frac {4 \int b e^{b^2 x^2} x \text {erfi}(b x) \, dx}{\sqrt {\pi }}\\ &=x \text {erfi}(b x)^2-\frac {(4 b) \int e^{b^2 x^2} x \text {erfi}(b x) \, dx}{\sqrt {\pi }}\\ &=-\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{b \sqrt {\pi }}+x \text {erfi}(b x)^2+\frac {4 \int e^{2 b^2 x^2} \, dx}{\pi }\\ &=-\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{b \sqrt {\pi }}+x \text {erfi}(b x)^2+\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 54, normalized size = 1.00 \[ -\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } b}+x \text {erfi}(b x)^2+\frac {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 61, normalized size = 1.13 \[ \frac {\pi b^{2} x \operatorname {erfi}\left (b x\right )^{2} - 2 \, \sqrt {\pi } b \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erfi}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{\pi b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \erfi \left (b x \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {erfi}\left (b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erfi}^{2}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________