\(\int \frac {\text {erfi}(d (a+b \log (c x^n)))}{x} \, dx\) [249]

Optimal result

Integrand size = 17, antiderivative size = 64 \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {e^{\left (a d+b d \log \left (c x^n\right )\right )^2}}{b d n \sqrt {\pi }}+\frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

erfi(d*(a+b*ln(c*x^n)))*(a+b*ln(c*x^n))/b/n-exp((a*d+b*d*ln(c*x^n))^2)/b/d/n/Pi^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6486} \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {e^{\left (a d+b d \log \left (c x^n\right )\right )^2}}{\sqrt {\pi } b d n} \]

[In]

Int[Erfi[d*(a + b*Log[c*x^n])]/x,x]

[Out]

-(E^(a*d + b*d*Log[c*x^n])^2/(b*d*n*Sqrt[Pi])) + (Erfi[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n)

Rule 6486

Int[Erfi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(Erfi[a + b*x]/b), x] - Simp[E^(a + b*x)^2/(b*Sqrt[P
i]), x] /; FreeQ[{a, b}, x]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.30 \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {-e^{d^2 \left (a^2+b^2 \log ^2\left (c x^n\right )\right )} \left (c x^n\right )^{2 a b d^2}+d \sqrt {\pi } \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b d n \sqrt {\pi }} \]

[In]

Integrate[Erfi[d*(a + b*Log[c*x^n])]/x,x]

[Out]

(-(E^(d^2*(a^2 + b^2*Log[c*x^n]^2))*(c*x^n)^(2*a*b*d^2)) + d*Sqrt[Pi]*Erfi[d*(a + b*Log[c*x^n])]*(a + b*Log[c*
x^n]))/(b*d*n*Sqrt[Pi])

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \operatorname {erfi}\left (a d +b d \ln \left (c \,x^{n}\right )\right )-\frac {{\mathrm e}^{{\left (a d +b d \ln \left (c \,x^{n}\right )\right )}^{2}}}{\sqrt {\pi }}}{n d b}\)	\(61\)
default	\(\frac {\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \operatorname {erfi}\left (a d +b d \ln \left (c \,x^{n}\right )\right )-\frac {{\mathrm e}^{{\left (a d +b d \ln \left (c \,x^{n}\right )\right )}^{2}}}{\sqrt {\pi }}}{n d b}\)	\(61\)
parts	\(\ln \left (x \right ) \operatorname {erfi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\frac {2 d b n \left (\frac {{\mathrm e}^{\ln \left (x \right )^{2} b^{2} d^{2} n^{2}+2 d^{2} \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right ) n b \ln \left (x \right )+d^{2} {\left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )}^{2}}}{2 b^{2} d^{2} n^{2}}+\frac {i \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right ) \sqrt {\pi }\, \operatorname {erf}\left (i d b n \ln \left (x \right )+i d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{2 d \,n^{2} b^{2}}\right )}{\sqrt {\pi }}\)	\(162\)

[In]

int(erfi(d*(a+b*ln(c*x^n)))/x,x,method=_RETURNVERBOSE)

[Out]

1/n/d/b*((a*d+b*d*ln(c*x^n))*erfi(a*d+b*d*ln(c*x^n))-1/Pi^(1/2)*exp((a*d+b*d*ln(c*x^n))^2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.83 \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname {erfi}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \sqrt {\pi } e^{\left (b^{2} d^{2} n^{2} \log \left (x\right )^{2} + b^{2} d^{2} \log \left (c\right )^{2} + 2 \, a b d^{2} \log \left (c\right ) + a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n\right )} \log \left (x\right )\right )}}{\pi b d n} \]

[In]

integrate(erfi(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")

[Out]

((pi*b*d*n*log(x) + pi*b*d*log(c) + pi*a*d)*erfi(b*d*log(c*x^n) + a*d) - sqrt(pi)*e^(b^2*d^2*n^2*log(x)^2 + b^
2*d^2*log(c)^2 + 2*a*b*d^2*log(c) + a^2*d^2 + 2*(b^2*d^2*n*log(c) + a*b*d^2*n)*log(x)))/(pi*b*d*n)

Sympy [F]

\[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\operatorname {erfi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

[In]

integrate(erfi(d*(a+b*ln(c*x**n)))/x,x)

[Out]

Integral(erfi(a*d + b*d*log(c*x**n))/x, x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.91 \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) - \frac {e^{\left ({\left (b \log \left (c x^{n}\right ) + a\right )}^{2} d^{2}\right )}}{\sqrt {\pi }}}{b d n} \]

[In]

integrate(erfi(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")

[Out]

((b*log(c*x^n) + a)*d*erfi((b*log(c*x^n) + a)*d) - e^((b*log(c*x^n) + a)^2*d^2)/sqrt(pi))/(b*d*n)

Giac [F]

\[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {\operatorname {erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \]

[In]

integrate(erfi(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")

[Out]

integrate(erfi((b*log(c*x^n) + a)*d)/x, x)

Mupad [B] (verification not implemented)

Time = 5.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.75 \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\ln \left (c\,x^n\right )\,\mathrm {erfi}\left (a\,d+b\,d\,\ln \left (c\,x^n\right )\right )}{n}+\frac {a\,d\,\mathrm {erfi}\left (a\,\sqrt {d^2}+b\,\ln \left (c\,x^n\right )\,\sqrt {d^2}\right )}{b\,n\,\sqrt {d^2}}-\frac {{\mathrm {e}}^{b^2\,d^2\,{\ln \left (c\,x^n\right )}^2}\,{\mathrm {e}}^{a^2\,d^2}\,{\left (c\,x^n\right )}^{2\,a\,b\,d^2}}{b\,d\,n\,\sqrt {\pi }} \]

[In]

int(erfi(d*(a + b*log(c*x^n)))/x,x)

[Out]

(log(c*x^n)*erfi(a*d + b*d*log(c*x^n)))/n + (a*d*erfi(a*(d^2)^(1/2) + b*log(c*x^n)*(d^2)^(1/2)))/(b*n*(d^2)^(1
/2)) - (exp(b^2*d^2*log(c*x^n)^2)*exp(a^2*d^2)*(c*x^n)^(2*a*b*d^2))/(b*d*n*pi^(1/2))