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

Optimal. Leaf size=125 \[ \frac{x (e x)^m \left (c x^n\right )^{-\frac{m+1}{n}} \exp \left (\frac{(m+1) \left (-4 a b d^2 n+m+1\right )}{4 b^2 d^2 n^2}\right ) \text{Erf}\left (\frac{-2 a b d^2 n-2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right )}{m+1}+\frac{(e x)^{m+1} \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.312641, antiderivative size = 125, 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 = {6401, 2278, 2274, 15, 20, 2276, 2234, 2205} \[ \frac{x (e x)^m \left (c x^n\right )^{-\frac{m+1}{n}} \exp \left (\frac{(m+1) \left (-4 a b d^2 n+m+1\right )}{4 b^2 d^2 n^2}\right ) \text{Erf}\left (\frac{-2 a b d^2 n-2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right )}{m+1}+\frac{(e x)^{m+1} \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e*x)^(1 + m)*Erf[d*(a + b*Log[c*x^n])])/(e*(1 + m)) + (E^(((1 + m)*(1 + m - 4*a*b*d^2*n))/(4*b^2*d^2*n^2))*x
*(e*x)^m*Erf[(1 + m - 2*a*b*d^2*n - 2*b^2*d^2*n*Log[c*x^n])/(2*b*d*n)])/((1 + m)*(c*x^n)^((1 + m)/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 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 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 (e x)^m \text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac{(e x)^{1+m} \text{erf}\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{erf}\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{erf}\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{erf}\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{erf}\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{erf}\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{erf}\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{erf}\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{erf}\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 = 0.509761, size = 127, normalized size = 1.02 \[ \frac{(e x)^m \left (x \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x^{-m} \text{Erf}\left (a d-\frac{-2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right ) \exp \left (\frac{(m+1) \left (-4 a b d^2 n-4 b^2 d^2 n \log \left (c x^n\right )+4 b^2 d^2 n^2 \log (x)+m+1\right )}{4 b^2 d^2 n^2}\right )\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}{\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((e*x)^m*erf(d*(a+b*ln(c*x^n))),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{e^{m} x x^{m} \operatorname{erf}\left (b d \log \left (x^{n}\right ) +{\left (b \log \left (c\right ) + a\right )} d\right )}{m + 1} - \frac{-\frac{\sqrt{\pi } c^{2 \, a b d^{2}} e^{m} \operatorname{erf}\left (-b d n \log \left (x\right ) - b d \log \left (c\right ) - a d + \frac{m}{2 \, b d n} + \frac{1}{2 \, b d n}\right ) e^{\left (-\frac{a m}{b n} - \frac{a}{b n} + \frac{m^{2}}{4 \, b^{2} d^{2} n^{2}} + \frac{m}{2 \, b^{2} d^{2} n^{2}} + \frac{1}{4 \, b^{2} d^{2} n^{2}}\right )}}{c^{\frac{m}{n}} c^{\left (\frac{1}{n}\right )}}}{\sqrt{\pi } c^{2 \, a b d^{2}}{\left (m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.87312, size = 427, normalized size = 3.42 \begin{align*} \frac{x \operatorname{erf}\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{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 - 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*erf(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

(x*erf(b*d*log(c*x^n) + a*d)*e^(m*log(e) + m*log(x)) - 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 - 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(-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((e*x)**m*erf(d*(a+b*ln(c*x**n))),x)

[Out]

Timed out

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Giac [A]  time = 1.62525, size = 211, normalized size = 1.69 \begin{align*} \frac{x^{m + 1} \operatorname{erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) e^{m}}{m + 1} + \frac{\pi \operatorname{erf}\left (-b d n \log \left (x\right ) - b d \log \left (c\right ) - a d + \frac{m}{2 \, b d n} + \frac{1}{2 \, b d n}\right ) e^{\left (m - \frac{a m}{b n} - \frac{a}{b n} + \frac{m^{2}}{4 \, b^{2} d^{2} n^{2}} + \frac{m}{2 \, b^{2} d^{2} n^{2}} + \frac{1}{4 \, b^{2} d^{2} n^{2}}\right )}}{{\left (\pi + \pi m\right )} c^{\frac{m}{n}} c^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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