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

Optimal. Leaf size=94 \[ \frac {\left (c x^n\right )^{\frac {1}{n}} e^{\frac {a}{b n}-\frac {1}{4 b^2 d^2 n^2}} \text {erfi}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac {1}{n}}{2 b d}\right )}{x}-\frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

[Out]

-erfi(d*(a+b*ln(c*x^n)))/x+exp(-1/4/b^2/d^2/n^2+a/b/n)*(c*x^n)^(1/n)*erfi(1/2*(2*a*b*d^2-1/n+2*b^2*d^2*ln(c*x^
n))/b/d)/x

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Rubi [A]  time = 0.22, antiderivative size = 94, 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 = {6403, 2278, 2274, 15, 2276, 2234, 2204} \[ \frac {\left (c x^n\right )^{\frac {1}{n}} e^{\frac {a}{b n}-\frac {1}{4 b^2 d^2 n^2}} \text {Erfi}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac {1}{n}}{2 b d}\right )}{x}-\frac {\text {Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

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

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.31, size = 82, normalized size = 0.87 \[ \frac {\left (c x^n\right )^{\frac {1}{n}} e^{\frac {4 a b d^2 n-1}{4 b^2 d^2 n^2}} \text {erfi}\left (a d+b d \log \left (c x^n\right )-\frac {1}{2 b d n}\right )-\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.56, size = 126, normalized size = 1.34 \[ \frac {\sqrt {b^{2} d^{2} n^{2}} x \operatorname {erfi}\left (\frac {{\left (2 \, b^{2} d^{2} n^{2} \log \relax (x) + 2 \, b^{2} d^{2} n \log \relax (c) + 2 \, a b d^{2} n - 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (\frac {4 \, b^{2} d^{2} n \log \relax (c) + 4 \, a b d^{2} n - 1}{4 \, b^{2} d^{2} n^{2}}\right )} - \operatorname {erfi}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\erfi \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {erfi}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(d*(a + b*log(c*x^n)))/x^2,x)

[Out]

int(erfi(d*(a + b*log(c*x^n)))/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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