\(\int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx\) [123]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx=-\frac {\text {erfc}(a+b x)}{d (c+d x)}-\frac {2 b \text {Int}\left (\frac {e^{-(a+b x)^2}}{c+d x},x\right )}{d \sqrt {\pi }} \]

[Out]

-erfc(b*x+a)/d/(d*x+c)-2*b*Unintegrable(1/exp((b*x+a)^2)/(d*x+c),x)/d/Pi^(1/2)

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx \]

[In]

Int[Erfc[a + b*x]/(c + d*x)^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.14 \[ \int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(-(erf(b*x + a) - 1)/(d^2*x^2 + 2*c*d*x + c^2), x)

Sympy [N/A]

Not integrable

Time = 10.72 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\operatorname {erfc}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {erfc}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathrm {erfc}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(erfc(a + b*x)/(c + d*x)^2,x)

[Out]

int(erfc(a + b*x)/(c + d*x)^2, x)