3.45 \(\int \frac {\text {erf}(d (a+b \log (c x^n)))}{x^3} \, dx\)

Optimal. Leaf size=95 \[ \frac {\left (c x^n\right )^{2/n} e^{\frac {2 a b d^2 n+1}{b^2 d^2 n^2}} \text {erf}\left (\frac {a b d^2 n+b^2 d^2 n \log \left (c x^n\right )+1}{b d n}\right )}{2 x^2}-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 95, normalized size of antiderivative = 1.00, 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} \[ \frac {\left (c x^n\right )^{2/n} e^{\frac {2 a b d^2 n+1}{b^2 d^2 n^2}} \text {Erf}\left (\frac {a b d^2 n+b^2 d^2 n \log \left (c x^n\right )+1}{b d n}\right )}{2 x^2}-\frac {\text {Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Erf[d*(a + b*Log[c*x^n])]/x^3,x]

[Out]

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

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 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]

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 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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {(b d n) \int \frac {e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{x^3} \, dx}{\sqrt {\pi }}\\ &=-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {(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^3} \, dx}{\sqrt {\pi }}\\ &=-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {(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^3} \, dx}{\sqrt {\pi }}\\ &=-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (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^{-3-2 a b d^2 n} \, dx}{\sqrt {\pi }}\\ &=-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (b d \left (c x^n\right )^{-2 a b d^2-\frac {-2-2 a b d^2 n}{n}}\right ) \operatorname {Subst}\left (\int \exp \left (-a^2 d^2+\frac {\left (-2-2 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^2}\\ &=-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {\left (b d e^{\frac {1+2 a b d^2 n}{b^2 d^2 n^2}} \left (c x^n\right )^{-2 a b d^2-\frac {-2-2 a b d^2 n}{n}}\right ) \operatorname {Subst}\left (\int \exp \left (-\frac {\left (\frac {-2-2 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^2}\\ &=-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {1+2 a b d^2 n}{b^2 d^2 n^2}} \left (c x^n\right )^{2/n} \text {erf}\left (\frac {1+a b d^2 n+b^2 d^2 n \log \left (c x^n\right )}{b d n}\right )}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 77, normalized size = 0.81 \[ \frac {e^{\frac {\frac {2 a b n+\frac {1}{d^2}}{b^2}+2 n \log \left (c x^n\right )}{n^2}} \text {erf}\left (a d+b d \log \left (c x^n\right )+\frac {1}{b d n}\right )-\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[d*(a + b*Log[c*x^n])]/x^3,x]

[Out]

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

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fricas [A]  time = 0.56, size = 124, normalized size = 1.31 \[ \frac {\sqrt {b^{2} d^{2} n^{2}} x^{2} \operatorname {erf}\left (\frac {{\left (b^{2} d^{2} n^{2} \log \relax (x) + b^{2} d^{2} n \log \relax (c) + a b d^{2} n + 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{b^{2} d^{2} n^{2}}\right ) e^{\left (\frac {2 \, b^{2} d^{2} n \log \relax (c) + 2 \, a b d^{2} n + 1}{b^{2} d^{2} n^{2}}\right )} - \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(d*(a+b*log(c*x^n)))/x^3,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(d*(a+b*log(c*x^n)))/x^3,x, algorithm="giac")

[Out]

integrate(erf((b*log(c*x^n) + a)*d)/x^3, x)

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maple [F]  time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {\erf \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(d*(a+b*ln(c*x^n)))/x^3,x)

[Out]

int(erf(d*(a+b*ln(c*x^n)))/x^3,x)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(d*(a+b*log(c*x^n)))/x^3,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {erf}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(d*(a + b*log(c*x^n)))/x^3,x)

[Out]

int(erf(d*(a + b*log(c*x^n)))/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(d*(a+b*ln(c*x**n)))/x**3,x)

[Out]

Integral(erf(a*d + b*d*log(c*x**n))/x**3, x)

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