Integrand size = 15, antiderivative size = 74 \[ \int x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} x^2 \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \] Output:
1/2*x^2*Ei(d*(a+b*ln(c*x^n)))-1/2*x^2*Ei((b*d*n+2)*(a+b*ln(c*x^n))/b/n)/ex p(2*a/b/n)/((c*x^n)^(2/n))
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93 \[ \int x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} x^2 \left (\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \] Input:
Integrate[x*ExpIntegralEi[d*(a + b*Log[c*x^n])],x]
Output:
(x^2*(ExpIntegralEi[d*(a + b*Log[c*x^n])] - ExpIntegralEi[((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)]/(E^((2*a)/(b*n))*(c*x^n)^(2/n))))/2
Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7048, 2747, 2609}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 7048 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} b n e^{a d} x^{-b d n} \left (c x^n\right )^{b d} \int \frac {x^{b d n+1}}{a+b \log \left (c x^n\right )}dx\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} b x^2 e^{a d} \left (c x^n\right )^{b d-\frac {b d n+2}{n}} \int \frac {\left (c x^n\right )^{\frac {b d n+2}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {1}{2} x^2 \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} x^2 e^{a d-a \left (\frac {2}{b n}+d\right )} \left (c x^n\right )^{b d-\frac {b d n+2}{n}} \operatorname {ExpIntegralEi}\left (\frac {(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\) |
Input:
Int[x*ExpIntegralEi[d*(a + b*Log[c*x^n])],x]
Output:
(x^2*ExpIntegralEi[d*(a + b*Log[c*x^n])])/2 - (E^(a*d - a*(d + 2/(b*n)))*x ^2*(c*x^n)^(b*d - (2 + b*d*n)/n)*ExpIntegralEi[((2 + b*d*n)*(a + b*Log[c*x ^n]))/(b*n)])/2
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n)) Subst[Int[E^(((m + 1)/n )*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
Int[ExpIntegralEi[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_)) ^(m_.), x_Symbol] :> Simp[(e*x)^(m + 1)*(ExpIntegralEi[d*(a + b*Log[c*x^n]) ]/(e*(m + 1))), x] - Simp[b*n*E^(a*d)*((c*x^n)^(b*d)/((m + 1)*(e*x)^(b*d*n) )) Int[(e*x)^(m + b*d*n)/(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
\[\int x \,\operatorname {expIntegral}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
Input:
int(x*Ei(d*(a+b*ln(c*x^n))),x)
Output:
int(x*Ei(d*(a+b*ln(c*x^n))),x)
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.12 \[ \int x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} \, x^{2} {\rm Ei}\left (b d \log \left (c x^{n}\right ) + a d\right ) - \frac {1}{2} \, {\rm Ei}\left (\frac {a b d n + {\left (b^{2} d n + 2 \, b\right )} \log \left (c\right ) + {\left (b^{2} d n^{2} + 2 \, b n\right )} \log \left (x\right ) + 2 \, a}{b n}\right ) e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \] Input:
integrate(x*Ei(d*(a+b*log(c*x^n))),x, algorithm="fricas")
Output:
1/2*x^2*Ei(b*d*log(c*x^n) + a*d) - 1/2*Ei((a*b*d*n + (b^2*d*n + 2*b)*log(c ) + (b^2*d*n^2 + 2*b*n)*log(x) + 2*a)/(b*n))*e^(-2*(b*log(c) + a)/(b*n))
Timed out. \[ \int x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(x*Ei(d*(a+b*ln(c*x**n))),x)
Output:
Timed out
\[ \int x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x {\rm Ei}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:
integrate(x*Ei(d*(a+b*log(c*x^n))),x, algorithm="maxima")
Output:
integrate(x*Ei((b*log(c*x^n) + a)*d), x)
Time = 0.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07 \[ \int x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} \, x^{2} {\rm Ei}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) - \frac {{\rm Ei}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d + \frac {2 \, \log \left (c\right )}{n} + \frac {2 \, a}{b n} + 2 \, \log \left (x\right )\right ) e^{\left (-\frac {2 \, a}{b n}\right )}}{2 \, c^{\frac {2}{n}}} \] Input:
integrate(x*Ei(d*(a+b*log(c*x^n))),x, algorithm="giac")
Output:
1/2*x^2*Ei(b*d*n*log(x) + b*d*log(c) + a*d) - 1/2*Ei(b*d*n*log(x) + b*d*lo g(c) + a*d + 2*log(c)/n + 2*a/(b*n) + 2*log(x))*e^(-2*a/(b*n))/c^(2/n)
Timed out. \[ \int x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x\,\mathrm {ei}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \] Input:
int(x*ei(d*(a + b*log(c*x^n))),x)
Output:
int(x*ei(d*(a + b*log(c*x^n))), x)
\[ \int x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathit {ei} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) x d x \] Input:
int(x*Ei(d*(a+b*log(c*x^n))),x)
Output:
int(ei(log(x**n*c)*b*d + a*d)*x,x)