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

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

[Out]

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

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Rubi [A]  time = 0.205497, antiderivative size = 95, 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 = {6403, 2278, 2274, 15, 2276, 2234, 2204} \[ \frac{\left (c x^n\right )^{2/n} e^{-\frac{1-2 a b d^2 n}{b^2 d^2 n^2}} \text{Erfi}\left (\frac{a b d^2+b^2 d^2 \log \left (c x^n\right )-\frac{1}{n}}{b d}\right )}{2 x^2}-\frac{\text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[d*(a + b*Log[c*x^n])]/x^3,x]

[Out]

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

Rule 6403

Int[Erfi[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((e*x)^(m + 1)*Erf
i[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 2204

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

Rubi steps

\begin{align*} \int \frac{\text{erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=-\frac{\text{erfi}\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{erfi}\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{erfi}\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{erfi}\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{erfi}\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{erfi}\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{erfi}\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{erfi}\left (\frac{a b d^2-\frac{1}{n}+b^2 d^2 \log \left (c x^n\right )}{b d}\right )}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.265812, size = 80, normalized size = 0.84 \[ \frac{e^{\frac{\frac{2 a b n-\frac{1}{d^2}}{b^2}+2 n \log \left (c x^n\right )}{n^2}} \text{Erfi}\left (a d+b d \log \left (c x^n\right )-\frac{1}{b d n}\right )-\text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.217, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfi} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.00626, size = 288, normalized size = 3.03 \begin{align*} \frac{\sqrt{b^{2} d^{2} n^{2}} x^{2} \operatorname{erfi}\left (\frac{{\left (b^{2} d^{2} n^{2} \log \left (x\right ) + b^{2} d^{2} n \log \left (c\right ) + 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 \left (c\right ) + 2 \, a b d^{2} n - 1}{b^{2} d^{2} n^{2}}\right )} - \operatorname{erfi}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(b^2*d^2*n^2)*x^2*erfi((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)) - erfi(b*d*log(c*x^n) + a*d))/x^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(erfi(d*(a+b*ln(c*x**n)))/x**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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