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

Optimal. Leaf size=102 \[ \frac{1}{3} x^3 \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{3} x^3 \left (c x^n\right )^{-3/n} e^{\frac{9-12 a b d^2 n}{4 b^2 d^2 n^2}} \text{Erf}\left (\frac{2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac{3}{n}}{2 b d}\right ) \]

[Out]

(x^3*Erf[d*(a + b*Log[c*x^n])])/3 - (E^((9 - 12*a*b*d^2*n)/(4*b^2*d^2*n^2))*x^3*Erf[(2*a*b*d^2 - 3/n + 2*b^2*d
^2*Log[c*x^n])/(2*b*d)])/(3*(c*x^n)^(3/n))

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Rubi [A]  time = 0.231717, antiderivative size = 102, 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} \[ \frac{1}{3} x^3 \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{3} x^3 \left (c x^n\right )^{-3/n} e^{\frac{9-12 a b d^2 n}{4 b^2 d^2 n^2}} \text{Erf}\left (\frac{2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac{3}{n}}{2 b d}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Erf[d*(a + b*Log[c*x^n])],x]

[Out]

(x^3*Erf[d*(a + b*Log[c*x^n])])/3 - (E^((9 - 12*a*b*d^2*n)/(4*b^2*d^2*n^2))*x^3*Erf[(2*a*b*d^2 - 3/n + 2*b^2*d
^2*Log[c*x^n])/(2*b*d)])/(3*(c*x^n)^(3/n))

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 x^2 \text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{1}{3} x^3 \text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{(2 b d n) \int e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2} x^2 \, dx}{3 \sqrt{\pi }}\\ &=\frac{1}{3} x^3 \text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{(2 b d n) \int \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}{3 \sqrt{\pi }}\\ &=\frac{1}{3} x^3 \text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{(2 b d n) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} x^2 \left (c x^n\right )^{-2 a b d^2} \, dx}{3 \sqrt{\pi }}\\ &=\frac{1}{3} x^3 \text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\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}{3 \sqrt{\pi }}\\ &=\frac{1}{3} x^3 \text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{\left (2 b d x^3 \left (c x^n\right )^{-2 a b d^2-\frac{3-2 a b d^2 n}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (-a^2 d^2+\frac{\left (3-2 a b d^2 n\right ) x}{n}-b^2 d^2 x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{3 \sqrt{\pi }}\\ &=\frac{1}{3} x^3 \text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{\left (2 b d e^{\frac{9-12 a b d^2 n}{4 b^2 d^2 n^2}} x^3 \left (c x^n\right )^{-2 a b d^2-\frac{3-2 a b d^2 n}{n}}\right ) \operatorname{Subst}\left (\int \exp \left (-\frac{\left (\frac{3-2 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 )}{3 \sqrt{\pi }}\\ &=\frac{1}{3} x^3 \text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac{1}{3} e^{\frac{9-12 a b d^2 n}{4 b^2 d^2 n^2}} x^3 \left (c x^n\right )^{-3/n} \text{erf}\left (\frac{2 a b d^2-\frac{3}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )\\ \end{align*}

Mathematica [A]  time = 0.317867, size = 88, normalized size = 0.86 \[ \frac{1}{3} \left (x^3 \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x^3 \text{Erf}\left (a d+b d \log \left (c x^n\right )-\frac{3}{2 b d n}\right ) \exp \left (-\frac{3 \left (\frac{4 a b n-\frac{3}{d^2}}{b^2}+4 n \log \left (c x^n\right )\right )}{4 n^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*Erf[d*(a + b*Log[c*x^n])] - (x^3*Erf[a*d - 3/(2*b*d*n) + b*d*Log[c*x^n]])/E^((3*((-3/d^2 + 4*a*b*n)/b^2 +
4*n*Log[c*x^n]))/(4*n^2)))/3

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Maple [F]  time = 0.158, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}{\it Erf} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Fricas [A]  time = 2.76454, size = 302, normalized size = 2.96 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \frac{1}{3} \, \sqrt{b^{2} d^{2} n^{2}} \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 - 3\right )} \sqrt{b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac{3 \,{\left (4 \, b^{2} d^{2} n \log \left (c\right ) + 4 \, a b d^{2} n - 3\right )}}{4 \, b^{2} d^{2} n^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*erf(b*d*log(c*x^n) + a*d) - 1/3*sqrt(b^2*d^2*n^2)*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 - 3)*sqrt(b^2*d^2*n^2)/(b^2*d^2*n^2))*e^(-3/4*(4*b^2*d^2*n*log(c) + 4*a*b*d^2*n - 3)/(b^2*d^2*n^2
))

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Sympy [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(x**2*erf(d*(a+b*ln(c*x**n))),x)

[Out]

Timed out

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Giac [A]  time = 1.51984, size = 115, normalized size = 1.13 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \frac{\operatorname{erf}\left (-b d n \log \left (x\right ) - b d \log \left (c\right ) - a d + \frac{3}{2 \, b d n}\right ) e^{\left (-\frac{3 \, a}{b n} + \frac{9}{4 \, b^{2} d^{2} n^{2}}\right )}}{3 \, c^{\frac{3}{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*erf(b*d*n*log(x) + b*d*log(c) + a*d) + 1/3*erf(-b*d*n*log(x) - b*d*log(c) - a*d + 3/2/(b*d*n))*e^(-3*a
/(b*n) + 9/4/(b^2*d^2*n^2))/c^(3/n)