Integrand size = 17, antiderivative size = 68 \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x} \] Output:
-Ei(d*(a+b*ln(c*x^n)))/x+exp(a/b/n)*(c*x^n)^(1/n)*Ei(-(-b*d*n+1)*(a+b*ln(c *x^n))/b/n)/x
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94 \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {-\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(-1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x} \] Input:
Integrate[ExpIntegralEi[d*(a + b*Log[c*x^n])]/x^2,x]
Output:
(-ExpIntegralEi[d*(a + b*Log[c*x^n])] + E^(a/(b*n))*(c*x^n)^n^(-1)*ExpInte gralEi[((-1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/x
Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, 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.
\(\displaystyle \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 7048 |
\(\displaystyle b n e^{a d} x^{-b d n} \left (c x^n\right )^{b d} \int \frac {x^{b d n-2}}{a+b \log \left (c x^n\right )}dx-\frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {b e^{a d} \left (c x^n\right )^{\frac {1}{n}} \int \frac {\left (c x^n\right )^{-\frac {1-b d n}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{x}-\frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x}-\frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}\) |
Input:
Int[ExpIntegralEi[d*(a + b*Log[c*x^n])]/x^2,x]
Output:
-(ExpIntegralEi[d*(a + b*Log[c*x^n])]/x) + (E^(a/(b*n))*(c*x^n)^n^(-1)*Exp IntegralEi[-(((1 - b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/x
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]
\[\int \frac {\operatorname {expIntegral}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}d x\]
Input:
int(Ei(d*(a+b*ln(c*x^n)))/x^2,x)
Output:
int(Ei(d*(a+b*ln(c*x^n)))/x^2,x)
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.22 \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {x {\rm Ei}\left (\frac {a b d n + {\left (b^{2} d n - b\right )} \log \left (c\right ) + {\left (b^{2} d n^{2} - b n\right )} \log \left (x\right ) - a}{b n}\right ) e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} - {\rm Ei}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x} \] Input:
integrate(Ei(d*(a+b*log(c*x^n)))/x^2,x, algorithm="fricas")
Output:
(x*Ei((a*b*d*n + (b^2*d*n - b)*log(c) + (b^2*d*n^2 - b*n)*log(x) - a)/(b*n ))*e^((b*log(c) + a)/(b*n)) - Ei(b*d*log(c*x^n) + a*d))/x
Timed out. \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\text {Timed out} \] Input:
integrate(Ei(d*(a+b*ln(c*x**n)))/x**2,x)
Output:
Timed out
\[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {{\rm Ei}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \] Input:
integrate(Ei(d*(a+b*log(c*x^n)))/x^2,x, algorithm="maxima")
Output:
integrate(Ei((b*log(c*x^n) + a)*d)/x^2, x)
\[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {{\rm Ei}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \] Input:
integrate(Ei(d*(a+b*log(c*x^n)))/x^2,x, algorithm="giac")
Output:
integrate(Ei((b*log(c*x^n) + a)*d)/x^2, x)
Timed out. \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\mathrm {ei}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \] Input:
int(ei(d*(a + b*log(c*x^n)))/x^2,x)
Output:
int(ei(d*(a + b*log(c*x^n)))/x^2, x)
\[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\mathit {ei} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}{x^{2}}d x \] Input:
int(Ei(d*(a+b*log(c*x^n)))/x^2,x)
Output:
int(ei(log(x**n*c)*b*d + a*d)/x**2,x)