\(\int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx\) [302]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=-\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {2 a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfi}(a+b x)}{3 x}+\frac {2}{3} a b^2 \sqrt {b^2+d} e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )+\frac {4 a^2 b^3 \text {Int}\left (\frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4 b d \text {Int}\left (\frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {2 b \left (b^2+d\right ) \text {Int}\left (\frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4}{3} d^2 \text {Int}\left (e^{c+d x^2} \text {erfi}(a+b x),x\right ) \]

[Out]

-1/3*exp(d*x^2+c)*erfi(b*x+a)/x^3-2/3*d*exp(d*x^2+c)*erfi(b*x+a)/x+2/3*a*b^2*exp(c+a^2*d/(b^2+d))*erfi((a*b+(b
^2+d)*x)/(b^2+d)^(1/2))*(b^2+d)^(1/2)-1/3*b*exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/x^2/Pi^(1/2)-2/3*a*b^2*exp(a^2+c+2*
a*b*x+(b^2+d)*x^2)/x/Pi^(1/2)+4/3*a^2*b^3*Unintegrable(exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/x,x)/Pi^(1/2)+4/3*b*d*Un
integrable(exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/x,x)/Pi^(1/2)+2/3*b*(b^2+d)*Unintegrable(exp(a^2+c+2*a*b*x+(b^2+d)*x
^2)/x,x)/Pi^(1/2)+4/3*d^2*Unintegrable(exp(d*x^2+c)*erfi(b*x+a),x)

Rubi [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 19, 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 = {} \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx \]

[In]

Int[(E^(c + d*x^2)*Erfi[a + b*x])/x^4,x]

[Out]

-1/3*(b*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(Sqrt[Pi]*x^2) - (2*a*b^2*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)
)/(3*Sqrt[Pi]*x) - (E^(c + d*x^2)*Erfi[a + b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erfi[a + b*x])/(3*x) + (2*a*b^2*
Sqrt[b^2 + d]*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/3 + (4*a^2*b^3*Defer[Int][E^(
a^2 + c + 2*a*b*x + (b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) + (4*b*d*Defer[Int][E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^
2)/x, x])/(3*Sqrt[Pi]) + (2*b*(b^2 + d)*Defer[Int][E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) +
(4*d^2*Defer[Int][E^(c + d*x^2)*Erfi[a + b*x], x])/3

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}+\frac {1}{3} (2 d) \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^2} \, dx+\frac {(2 b) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x^3} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfi}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erfi}(a+b x) \, dx+\frac {\left (2 a b^2\right ) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x^2} \, dx}{3 \sqrt {\pi }}+\frac {(4 b d) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (2 b \left (b^2+d\right )\right ) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {2 a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfi}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erfi}(a+b x) \, dx+\frac {\left (4 a^2 b^3\right ) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {(4 b d) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (2 b \left (b^2+d\right )\right ) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (4 a b^2 \left (b^2+d\right )\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {2 a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfi}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erfi}(a+b x) \, dx+\frac {\left (4 a^2 b^3\right ) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {(4 b d) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (2 b \left (b^2+d\right )\right ) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (4 a b^2 \left (b^2+d\right ) e^{c+\frac {a^2 d}{b^2+d}}\right ) \int e^{\frac {\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{3 \sqrt {\pi }} \\ & = -\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {2 a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfi}(a+b x)}{3 x}+\frac {2}{3} a b^2 \sqrt {b^2+d} e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erfi}(a+b x) \, dx+\frac {\left (4 a^2 b^3\right ) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {(4 b d) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (2 b \left (b^2+d\right )\right ) \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx \]

[In]

Integrate[(E^(c + d*x^2)*Erfi[a + b*x])/x^4,x]

[Out]

Integrate[(E^(c + d*x^2)*Erfi[a + b*x])/x^4, x]

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erfi}\left (b x +a \right )}{x^{4}}d x\]

[In]

int(exp(d*x^2+c)*erfi(b*x+a)/x^4,x)

[Out]

int(exp(d*x^2+c)*erfi(b*x+a)/x^4,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erfi(b*x+a)/x^4,x, algorithm="fricas")

[Out]

integral(erfi(b*x + a)*e^(d*x^2 + c)/x^4, x)

Sympy [N/A]

Not integrable

Time = 65.54 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erfi}{\left (a + b x \right )}}{x^{4}}\, dx \]

[In]

integrate(exp(d*x**2+c)*erfi(b*x+a)/x**4,x)

[Out]

exp(c)*Integral(exp(d*x**2)*erfi(a + b*x)/x**4, x)

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erfi(b*x+a)/x^4,x, algorithm="maxima")

[Out]

integrate(erfi(b*x + a)*e^(d*x^2 + c)/x^4, x)

Giac [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erfi(b*x+a)/x^4,x, algorithm="giac")

[Out]

integrate(erfi(b*x + a)*e^(d*x^2 + c)/x^4, x)

Mupad [N/A]

Not integrable

Time = 5.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int \frac {\mathrm {erfi}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^4} \,d x \]

[In]

int((erfi(a + b*x)*exp(c + d*x^2))/x^4,x)

[Out]

int((erfi(a + b*x)*exp(c + d*x^2))/x^4, x)