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

Optimal. Leaf size=94 \[ \frac {1}{2} x^2 \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} x^2 \left (c x^n\right )^{-2/n} e^{\frac {1-2 a b d^2 n}{b^2 d^2 n^2}} \text {erf}\left (\frac {a b d^2+b^2 d^2 \log \left (c x^n\right )-\frac {1}{n}}{b d}\right ) \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {6401, 2278, 2274, 15, 2276, 2234, 2205} \[ \frac {1}{2} x^2 \text {Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} x^2 \left (c x^n\right )^{-2/n} e^{\frac {1-2 a b d^2 n}{b^2 d^2 n^2}} \text {Erf}\left (\frac {a b d^2+b^2 d^2 \log \left (c x^n\right )-\frac {1}{n}}{b d}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*Erf[d*(a + b*Log[c*x^n])],x]

[Out]

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

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

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Mathematica [A]  time = 0.30, size = 84, normalized size = 0.89 \[ \frac {1}{2} \left (x^2 \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x^2 e^{-\frac {\frac {2 a b n-\frac {1}{d^2}}{b^2}+2 n \log \left (c x^n\right )}{n^2}} \text {erf}\left (a d+b d \log \left (c x^n\right )-\frac {1}{b d n}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 121, normalized size = 1.29 \[ \frac {1}{2} \, x^{2} \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \frac {1}{2} \, \sqrt {b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (b^{2} d^{2} n^{2} \log \relax (x) + b^{2} d^{2} n \log \relax (c) + a b d^{2} n - 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac {2 \, b^{2} d^{2} n \log \relax (c) + 2 \, a b d^{2} n - 1}{b^{2} d^{2} n^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.61, size = 83, normalized size = 0.88 \[ \frac {1}{2} \, x^{2} \operatorname {erf}\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) + \frac {\operatorname {erf}\left (-b d n \log \relax (x) - b d \log \relax (c) - a d + \frac {1}{b d n}\right ) e^{\left (-\frac {2 \, a}{b n} + \frac {1}{b^{2} d^{2} n^{2}}\right )}}{2 \, c^{\frac {2}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int x \erf \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \operatorname {erf}\left (b d \log \left (x^{n}\right ) + {\left (b \log \relax (c) + a\right )} d\right ) - \frac {b d n e^{\left (-b^{2} d^{2} \log \relax (c)^{2} - a^{2} d^{2}\right )} \int \frac {x e^{\left (-2 \, b^{2} d^{2} \log \relax (c) \log \left (x^{n}\right ) - b^{2} d^{2} \log \left (x^{n}\right )^{2}\right )}}{{\left (x^{n}\right )}^{2 \, a b d^{2}}}\,{d x}}{\sqrt {\pi } c^{2 \, a b d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*erf(d*(a + b*log(c*x^n))),x)

[Out]

int(x*erf(d*(a + b*log(c*x^n))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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