Integrand size = 13, antiderivative size = 65 \[ \int \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \] Output:
x*Ei(d*(a+b*ln(c*x^n)))-x*Ei((b*d*n+1)*(a+b*ln(c*x^n))/b/n)/exp(a/b/n)/((c *x^n)^(1/n))
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \left (\operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \] Input:
Integrate[ExpIntegralEi[d*(a + b*Log[c*x^n])],x]
Output:
x*(ExpIntegralEi[d*(a + b*Log[c*x^n])] - ExpIntegralEi[((1 + b*d*n)*(a + b *Log[c*x^n]))/(b*n)]/(E^(a/(b*n))*(c*x^n)^n^(-1)))
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.26, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {7046, 34, 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.
\(\displaystyle \int \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 7046 |
\(\displaystyle x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b n e^{a d} \int \frac {\left (c x^n\right )^{b d}}{a+b \log \left (c x^n\right )}dx\) |
\(\Big \downarrow \) 34 |
\(\displaystyle x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b n e^{a d} x^{-b d n} \left (c x^n\right )^{b d} \int \frac {x^{b d n}}{a+b \log \left (c x^n\right )}dx\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b x e^{a d} \left (c x^n\right )^{b d-\frac {b d n+1}{n}} \int \frac {\left (c x^n\right )^{\frac {b d n+1}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle x \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x e^{a d-a \left (\frac {1}{b n}+d\right )} \left (c x^n\right )^{b d-\frac {b d n+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\) |
Input:
Int[ExpIntegralEi[d*(a + b*Log[c*x^n])],x]
Output:
x*ExpIntegralEi[d*(a + b*Log[c*x^n])] - E^(a*d - a*(d + 1/(b*n)))*x*(c*x^n )^(b*d - (1 + b*d*n)/n)*ExpIntegralEi[((1 + b*d*n)*(a + b*Log[c*x^n]))/(b* n)]
Int[(u_.)*((a_.)*(x_)^(m_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*x^m)^F racPart[p]/x^(m*FracPart[p])) Int[u*x^(m*p), x], x] /; FreeQ[{a, m, p}, x ] && !IntegerQ[p]
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_.)], x_Symbol] : > Simp[x*ExpIntegralEi[d*(a + b*Log[c*x^n])], x] - Simp[b*n*E^(a*d) Int[( c*x^n)^(b*d)/(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, n}, x]
\[\int \operatorname {expIntegral}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
Input:
int(Ei(d*(a+b*ln(c*x^n))),x)
Output:
int(Ei(d*(a+b*ln(c*x^n))),x)
Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.15 \[ \int \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x {\rm Ei}\left (b d \log \left (c x^{n}\right ) + a d\right ) - {\rm Ei}\left (\frac {a b d n + {\left (b^{2} d n + b\right )} \log \left (c\right ) + {\left (b^{2} d n^{2} + b n\right )} \log \left (x\right ) + a}{b n}\right ) e^{\left (-\frac {b \log \left (c\right ) + a}{b n}\right )} \] Input:
integrate(Ei(d*(a+b*log(c*x^n))),x, algorithm="fricas")
Output:
x*Ei(b*d*log(c*x^n) + a*d) - 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))
\[ \int \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \operatorname {Ei}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \] Input:
integrate(Ei(d*(a+b*ln(c*x**n))),x)
Output:
Integral(Ei(d*(a + b*log(c*x**n))), x)
\[ \int \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { {\rm Ei}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:
integrate(Ei(d*(a+b*log(c*x^n))),x, algorithm="maxima")
Output:
integrate(Ei((b*log(c*x^n) + a)*d), x)
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.08 \[ \int \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x {\rm Ei}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) - \frac {{\rm Ei}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d + \frac {\log \left (c\right )}{n} + \frac {a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac {a}{b n}\right )}}{c^{\left (\frac {1}{n}\right )}} \] Input:
integrate(Ei(d*(a+b*log(c*x^n))),x, algorithm="giac")
Output:
x*Ei(b*d*n*log(x) + b*d*log(c) + a*d) - Ei(b*d*n*log(x) + b*d*log(c) + a*d + log(c)/n + a/(b*n) + log(x))*e^(-a/(b*n))/c^(1/n)
Timed out. \[ \int \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {ei}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \] Input:
int(ei(d*(a + b*log(c*x^n))),x)
Output:
int(ei(d*(a + b*log(c*x^n))), x)
\[ \int \operatorname {ExpIntegralEi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathit {ei} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )d x \] Input:
int(Ei(d*(a+b*log(c*x^n))),x)
Output:
int(ei(log(x**n*c)*b*d + a*d),x)