3.252 \(\int (e x)^m \text{Erfi}(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=126 \[ \frac{(e x)^{m+1} \text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac{x (e x)^m \left (c x^n\right )^{-\frac{m+1}{n}} \exp \left (-\frac{(m+1) \left (4 a b d^2 n+m+1\right )}{4 b^2 d^2 n^2}\right ) \text{Erfi}\left (\frac{2 a b d^2 n+2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right )}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.322351, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6403, 2278, 2274, 15, 20, 2276, 2234, 2204} \[ \frac{(e x)^{m+1} \text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac{x (e x)^m \left (c x^n\right )^{-\frac{m+1}{n}} \exp \left (-\frac{(m+1) \left (4 a b d^2 n+m+1\right )}{4 b^2 d^2 n^2}\right ) \text{Erfi}\left (\frac{2 a b d^2 n+2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*Erfi[d*(a + b*Log[c*x^n])],x]

[Out]

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

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 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

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

Mathematica [A]  time = 0.441594, size = 126, normalized size = 1. \[ \frac{(e x)^m \left (x \text{Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x^{-m} \text{Erfi}\left (\frac{2 a b d^2 n+m+1}{2 b d n}+b d \log \left (c x^n\right )\right ) \exp \left (-\frac{(m+1) \left (4 a b d^2 n+4 b^2 d^2 n \log \left (c x^n\right )-4 b^2 d^2 n^2 \log (x)+m+1\right )}{4 b^2 d^2 n^2}\right )\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}{\it erfi} \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((e*x)^m*erfi(d*(a+b*ln(c*x^n))),x)

[Out]

int((e*x)^m*erfi(d*(a+b*ln(c*x^n))),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*erfi((b*log(c*x^n) + a)*d), x)

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Fricas [A]  time = 3.10303, size = 429, normalized size = 3.4 \begin{align*} \frac{x \operatorname{erfi}\left (b d \log \left (c x^{n}\right ) + a d\right ) e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} - \sqrt{b^{2} d^{2} n^{2}} \operatorname{erfi}\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 + m + 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} m n^{2} \log \left (e\right ) - 4 \,{\left (b^{2} d^{2} m + b^{2} d^{2}\right )} n \log \left (c\right ) - m^{2} - 4 \,{\left (a b d^{2} m + a b d^{2}\right )} n - 2 \, m - 1}{4 \, b^{2} d^{2} n^{2}}\right )}}{m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*erfi(d*(a+b*ln(c*x**n))),x)

[Out]

Integral((e*x)**m*erfi(a*d + b*d*log(c*x**n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*erfi((b*log(c*x^n) + a)*d), x)