\(\int e^{c+d x^2} \text {erf}(b x) \, dx\) [61]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int e^{c+d x^2} \text {erf}(b x) \, dx=\text {Int}\left (e^{c+d x^2} \text {erf}(b x),x\right ) \]

[Out]

Unintegrable(exp(d*x^2+c)*erf(b*x),x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int e^{c+d x^2} \text {erf}(b x) \, dx=\int e^{c+d x^2} \text {erf}(b x) \, dx \]

[In]

Int[E^(c + d*x^2)*Erf[b*x],x]

[Out]

Defer[Int][E^(c + d*x^2)*Erf[b*x], x]

Rubi steps \begin{align*} \text {integral}& = \int e^{c+d x^2} \text {erf}(b x) \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int e^{c+d x^2} \text {erf}(b x) \, dx=\int e^{c+d x^2} \text {erf}(b x) \, dx \]

[In]

Integrate[E^(c + d*x^2)*Erf[b*x],x]

[Out]

Integrate[E^(c + d*x^2)*Erf[b*x], x]

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

\[\int {\mathrm e}^{d \,x^{2}+c} \operatorname {erf}\left (b x \right )d x\]

[In]

int(exp(d*x^2+c)*erf(b*x),x)

[Out]

int(exp(d*x^2+c)*erf(b*x),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int e^{c+d x^2} \text {erf}(b x) \, dx=\int { \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erf(b*x),x, algorithm="fricas")

[Out]

integral(erf(b*x)*e^(d*x^2 + c), x)

Sympy [N/A]

Not integrable

Time = 2.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int e^{c+d x^2} \text {erf}(b x) \, dx=e^{c} \int e^{d x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]

[In]

integrate(exp(d*x**2+c)*erf(b*x),x)

[Out]

exp(c)*Integral(exp(d*x**2)*erf(b*x), x)

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int e^{c+d x^2} \text {erf}(b x) \, dx=\int { \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erf(b*x),x, algorithm="maxima")

[Out]

integrate(erf(b*x)*e^(d*x^2 + c), x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int e^{c+d x^2} \text {erf}(b x) \, dx=\int { \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erf(b*x),x, algorithm="giac")

[Out]

integrate(erf(b*x)*e^(d*x^2 + c), x)

Mupad [N/A]

Not integrable

Time = 5.67 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int e^{c+d x^2} \text {erf}(b x) \, dx=\int {\mathrm {e}}^{d\,x^2+c}\,\mathrm {erf}\left (b\,x\right ) \,d x \]

[In]

int(exp(c + d*x^2)*erf(b*x),x)

[Out]

int(exp(c + d*x^2)*erf(b*x), x)