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

Optimal. Leaf size=19 \[ \text {Int}\left (\frac {\text {erf}(a+b x)^2}{(c+d x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {Erf}(a+b x)^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\text {erf}(a+b x)^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erf}\left (b x + a\right )^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}\left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\erf \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ 4 \, b \int \frac {\operatorname {erf}\left (b x + a\right ) e^{\left (-b^{2} x^{2} - 2 \, a b x\right )}}{\sqrt {\pi } d^{2} x e^{\left (a^{2}\right )} + \sqrt {\pi } c d e^{\left (a^{2}\right )}}\,{d x} - \frac {\operatorname {erf}\left (b x + a\right )^{2}}{d^{2} x + c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\mathrm {erf}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(a + b*x)^2/(c + d*x)^2,x)

[Out]

int(erf(a + b*x)^2/(c + d*x)^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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