Integrand size = 17, antiderivative size = 56 \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {e^{a d} \left (c x^n\right )^{b d}}{b d n}+\frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n} \] Output:
-exp(a*d)*(c*x^n)^(b*d)/b/d/n+Ei(d*(a+b*ln(c*x^n)))*(a+b*ln(c*x^n))/b/n
Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93 \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {-e^{a d} \left (c x^n\right )^{b d}+d \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b d n} \] Input:
Integrate[ExpIntegralEi[d*(a + b*Log[c*x^n])]/x,x]
Output:
(-(E^(a*d)*(c*x^n)^(b*d)) + d*ExpIntegralEi[d*(a + b*Log[c*x^n])]*(a + b*L og[c*x^n]))/(b*d*n)
Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3039, 7281, 7036}
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.
\(\displaystyle \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {\int \operatorname {ExpIntegralEi}\left (a d+b \log \left (c x^n\right ) d\right )d\left (a d+b \log \left (c x^n\right ) d\right )}{b d n}\) |
\(\Big \downarrow \) 7036 |
\(\displaystyle \frac {\left (a d+b d \log \left (c x^n\right )\right ) \operatorname {ExpIntegralEi}\left (a d+b \log \left (c x^n\right ) d\right )-c x^n}{b d n}\) |
Input:
Int[ExpIntegralEi[d*(a + b*Log[c*x^n])]/x,x]
Output:
(-(c*x^n) + ExpIntegralEi[a*d + b*d*Log[c*x^n]]*(a*d + b*d*Log[c*x^n]))/(b *d*n)
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[ExpIntegralEi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(ExpInte gralEi[a + b*x]/b), x] - Simp[E^(a + b*x)/b, x] /; FreeQ[{a, b}, x]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\operatorname {expIntegral}\left (a d +\ln \left (c \,x^{n}\right ) b d \right ) \left (a d +\ln \left (c \,x^{n}\right ) b d \right )-{\mathrm e}^{a d +\ln \left (c \,x^{n}\right ) b d}}{n b d}\) | \(56\) |
default | \(\frac {\operatorname {expIntegral}\left (a d +\ln \left (c \,x^{n}\right ) b d \right ) \left (a d +\ln \left (c \,x^{n}\right ) b d \right )-{\mathrm e}^{a d +\ln \left (c \,x^{n}\right ) b d}}{n b d}\) | \(56\) |
parallelrisch | \(-\frac {-\ln \left (c \,x^{n}\right ) \operatorname {expIntegral}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right ) b^{2} d -\operatorname {expIntegral}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right ) a b d +{\mathrm e}^{d \left (a +b \ln \left (c \,x^{n}\right )\right )} b}{b^{2} d n}\) | \(71\) |
parts | \(\ln \left (x \right ) \operatorname {expIntegral}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-b n \left (\frac {\left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right ) \operatorname {expIntegral}_{1}\left (-\ln \left (x \right ) b d n -d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b \,n^{2}}+\frac {a \,\operatorname {expIntegral}_{1}\left (-\ln \left (x \right ) b d n -d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b^{2} n^{2}}+\frac {{\mathrm e}^{\ln \left (x \right ) b d n +d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )}}{b^{2} d \,n^{2}}\right )\) | \(144\) |
Input:
int(Ei(d*(a+b*ln(c*x^n)))/x,x,method=_RETURNVERBOSE)
Output:
1/n/b/d*(Ei(a*d+ln(c*x^n)*b*d)*(a*d+ln(c*x^n)*b*d)-exp(a*d+ln(c*x^n)*b*d))
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} {\rm Ei}\left (b d \log \left (c x^{n}\right ) + a d\right ) - e^{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}}{b d n} \] Input:
integrate(Ei(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")
Output:
((b*d*n*log(x) + b*d*log(c) + a*d)*Ei(b*d*log(c*x^n) + a*d) - e^(b*d*n*log (x) + b*d*log(c) + a*d))/(b*d*n)
\[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\operatorname {Ei}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:
integrate(Ei(d*(a+b*ln(c*x**n)))/x,x)
Output:
Integral(Ei(a*d + b*d*log(c*x**n))/x, x)
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d {\rm Ei}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) - e^{\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}}{b d n} \] Input:
integrate(Ei(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")
Output:
((b*log(c*x^n) + a)*d*Ei((b*log(c*x^n) + a)*d) - e^((b*log(c*x^n) + a)*d)) /(b*d*n)
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09 \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} {\rm Ei}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) - e^{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}}{b d n} \] Input:
integrate(Ei(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")
Output:
((b*d*n*log(x) + b*d*log(c) + a*d)*Ei(b*d*n*log(x) + b*d*log(c) + a*d) - e ^(b*d*n*log(x) + b*d*log(c) + a*d))/(b*d*n)
Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.27 \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\mathrm {ei}\left (a\,d+b\,d\,\ln \left (c\,x^n\right )\right )\,\ln \left (c\,x^n\right )}{n}+\frac {a\,\mathrm {ei}\left (a\,d+b\,d\,\ln \left (c\,x^n\right )\right )}{b\,n}-\frac {{\mathrm {e}}^{a\,d}\,{\left (c\,x^n\right )}^{b\,d}}{b\,d\,n} \] Input:
int(ei(d*(a + b*log(c*x^n)))/x,x)
Output:
(ei(a*d + b*d*log(c*x^n))*log(c*x^n))/n + (a*ei(a*d + b*d*log(c*x^n)))/(b* n) - (exp(a*d)*(c*x^n)^(b*d))/(b*d*n)
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.23 \[ \int \frac {\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\mathit {ei} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) \mathrm {log}\left (x^{n} c \right ) b d +\mathit {ei} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) a d -x^{b d n} e^{a d} c^{b d}}{b d n} \] Input:
int(Ei(d*(a+b*log(c*x^n)))/x,x)
Output:
(ei(log(x**n*c)*b*d + a*d)*log(x**n*c)*b*d + ei(log(x**n*c)*b*d + a*d)*a*d - x**(b*d*n)*e**(a*d)*c**(b*d))/(b*d*n)