3.44 \(\int \frac{\text{Erf}(d (a+b \log (c x^n)))}{x^2} \, dx\)

Optimal. Leaf size=92 \[ \frac{\left (c x^n\right )^{\frac{1}{n}} e^{\frac{a}{b n}+\frac{1}{4 b^2 d^2 n^2}} \text{Erf}\left (\frac{2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}}{2 b d}\right )}{x}-\frac{\text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

[Out]

-(Erf[d*(a + b*Log[c*x^n])]/x) + (E^(1/(4*b^2*d^2*n^2) + a/(b*n))*(c*x^n)^n^(-1)*Erf[(2*a*b*d^2 + n^(-1) + 2*b
^2*d^2*Log[c*x^n])/(2*b*d)])/x

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Rubi [A]  time = 0.213331, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {6401, 2278, 2274, 15, 2276, 2234, 2205} \[ \frac{\left (c x^n\right )^{\frac{1}{n}} e^{\frac{a}{b n}+\frac{1}{4 b^2 d^2 n^2}} \text{Erf}\left (\frac{2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )+\frac{1}{n}}{2 b d}\right )}{x}-\frac{\text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[Erf[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

-(Erf[d*(a + b*Log[c*x^n])]/x) + (E^(1/(4*b^2*d^2*n^2) + a/(b*n))*(c*x^n)^n^(-1)*Erf[(2*a*b*d^2 + n^(-1) + 2*b
^2*d^2*Log[c*x^n])/(2*b*d)])/x

Rule 6401

Int[Erf[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((e*x)^(m + 1)*Erf[
d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(2*b*d*n)/(Sqrt[Pi]*(m + 1)), Int[(e*x)^m/E^(d*(a + b*Log[c*x^n]
))^2, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]

Rule 2278

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*F^(a^2*d
+ 2*a*b*d*Log[c*x^n] + b^2*d*Log[c*x^n]^2), x] /; FreeQ[{F, a, b, c, d, e, m, n}, x]

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2276

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]^2*(b_.))*(d_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)/(
e*n*(c*x^n)^((m + 1)/n)), Subst[Int[E^(a*d*Log[F] + ((m + 1)*x)/n + b*d*Log[F]*x^2), x], x, Log[c*x^n]], x] /;
FreeQ[{F, a, b, c, d, e, m, n}, x]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=-\frac{\text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{(2 b d n) \int \frac{e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{\text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{(2 b d n) \int \frac{\exp \left (-a^2 d^2-2 a b d^2 \log \left (c x^n\right )-b^2 d^2 \log ^2\left (c x^n\right )\right )}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{\text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{(2 b d n) \int \frac{e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} \left (c x^n\right )^{-2 a b d^2}}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{\text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{\left (2 b d n x^{2 a b d^2 n} \left (c x^n\right )^{-2 a b d^2}\right ) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} x^{-2-2 a b d^2 n} \, dx}{\sqrt{\pi }}\\ &=-\frac{\text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{\left (2 b d \left (c x^n\right )^{-2 a b d^2-\frac{-1-2 a b d^2 n}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (-a^2 d^2+\frac{\left (-1-2 a b d^2 n\right ) x}{n}-b^2 d^2 x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{\sqrt{\pi } x}\\ &=-\frac{\text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{\left (2 b d e^{\frac{1}{4 b^2 d^2 n^2}+\frac{a}{b n}} \left (c x^n\right )^{-2 a b d^2-\frac{-1-2 a b d^2 n}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (-\frac{\left (\frac{-1-2 a b d^2 n}{n}-2 b^2 d^2 x\right )^2}{4 b^2 d^2}\right ) \, dx,x,\log \left (c x^n\right )\right )}{\sqrt{\pi } x}\\ &=-\frac{\text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac{e^{\frac{1}{4 b^2 d^2 n^2}+\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \text{erf}\left (\frac{2 a b d^2+\frac{1}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )}{x}\\ \end{align*}

Mathematica [A]  time = 0.248685, size = 80, normalized size = 0.87 \[ \frac{e^{\frac{\frac{4 a b n+\frac{1}{d^2}}{b^2}+4 n \log \left (c x^n\right )}{4 n^2}} \text{Erf}\left (a d+b d \log \left (c x^n\right )+\frac{1}{2 b d n}\right )-\text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-Erf[d*(a + b*Log[c*x^n])] + E^(((d^(-2) + 4*a*b*n)/b^2 + 4*n*Log[c*x^n])/(4*n^2))*Erf[a*d + 1/(2*b*d*n) + b*
d*Log[c*x^n]])/x

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Maple [F]  time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Erf} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 2.70359, size = 293, normalized size = 3.18 \begin{align*} \frac{\sqrt{b^{2} d^{2} n^{2}} x \operatorname{erf}\left (\frac{{\left (2 \, b^{2} d^{2} n^{2} \log \left (x\right ) + 2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n + 1\right )} \sqrt{b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (\frac{4 \, b^{2} d^{2} n \log \left (c\right ) + 4 \, a b d^{2} n + 1}{4 \, b^{2} d^{2} n^{2}}\right )} - \operatorname{erf}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(b^2*d^2*n^2)*x*erf(1/2*(2*b^2*d^2*n^2*log(x) + 2*b^2*d^2*n*log(c) + 2*a*b*d^2*n + 1)*sqrt(b^2*d^2*n^2)/(
b^2*d^2*n^2))*e^(1/4*(4*b^2*d^2*n*log(c) + 4*a*b*d^2*n + 1)/(b^2*d^2*n^2)) - erf(b*d*log(c*x^n) + a*d))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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