\(\int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx\) [39]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx=\text {Int}\left (\frac {\text {erf}(a+b x)^2}{(c+d x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx=\int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx \]

[In]

Int[Erf[a + b*x]^2/(c + d*x)^2,x]

[Out]

Defer[Int][Erf[a + b*x]^2/(c + d*x)^2, x]

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

Mathematica [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx=\int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {erf}\left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(erf(b*x + a)^2/(d^2*x^2 + 2*c*d*x + c^2), x)

Sympy [N/A]

Not integrable

Time = 11.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx=\int \frac {\operatorname {erf}^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

Integral(erf(a + b*x)**2/(c + d*x)**2, x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 4.62 \[ \int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

4*b*integrate(erf(b*x + a)*e^(-b^2*x^2 - 2*a*b*x)/(sqrt(pi)*d^2*x*e^(a^2) + sqrt(pi)*c*d*e^(a^2)), x) - erf(b*
x + a)^2/(d^2*x + c*d)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate(erf(b*x + a)^2/(d*x + c)^2, x)

Mupad [N/A]

Not integrable

Time = 5.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx=\int \frac {{\mathrm {erf}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(erf(a + b*x)^2/(c + d*x)^2,x)

[Out]

int(erf(a + b*x)^2/(c + d*x)^2, x)