Integrand size = 13, antiderivative size = 119 \[ \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
x*Chi(d*(a+b*ln(c*x^n)))-1/2*x*Ei((-b*d*n+1)*(a+b*ln(c*x^n))/b/n)/exp(a/b/ n)/((c*x^n)^(1/n))-1/2*x*Ei((b*d*n+1)*(a+b*ln(c*x^n))/b/n)/exp(a/b/n)/((c* x^n)^(1/n))
Time = 1.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.78 \[ \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {(-1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \]
x*CoshIntegral[d*(a + b*Log[c*x^n])] - (x*(ExpIntegralEi[-(((-1 + b*d*n)*( a + b*Log[c*x^n]))/(b*n))] + ExpIntegralEi[((1 + b*d*n)*(a + b*Log[c*x^n]) )/(b*n)]))/(2*E^(a/(b*n))*(c*x^n)^n^(-1))
Time = 0.55 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.35, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {7107, 27, 6064, 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.
\(\displaystyle \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 7107 |
\(\displaystyle x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \int \frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d \left (a+b \log \left (c x^n\right )\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b n \int \frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )}dx\) |
\(\Big \downarrow \) 6064 |
\(\displaystyle x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b n \left (\frac {1}{2} e^{-a d} x^{b d n} \left (c x^n\right )^{-b d} \int \frac {x^{-b d n}}{a+b \log \left (c x^n\right )}dx+\frac {1}{2} e^{a d} x^{-b d n} \left (c x^n\right )^{b d} \int \frac {x^{b d n}}{a+b \log \left (c x^n\right )}dx\right )\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b n \left (\frac {x e^{-a d} \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{\frac {1-b d n}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n}+\frac {x e^{a d} \left (c x^n\right )^{b d-\frac {b d n+1}{n}} \int \frac {\left (c x^n\right )^{\frac {b d n+1}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n}\right )\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-b n \left (\frac {x \left (c x^n\right )^{-1/n} e^{a \left (d-\frac {1}{b n}\right )-a d} \operatorname {ExpIntegralEi}\left (\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n}+\frac {x e^{a d-a \left (\frac {1}{b n}+d\right )} \left (c x^n\right )^{b d-\frac {b d n+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n}\right )\) |
x*CoshIntegral[d*(a + b*Log[c*x^n])] - b*n*((E^(-(a*d) + a*(d - 1/(b*n)))* x*ExpIntegralEi[((1 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*b*n*(c*x^n)^n^ (-1)) + (E^(a*d - a*(d + 1/(b*n)))*x*(c*x^n)^(b*d - (1 + b*d*n)/n)*ExpInte gralEi[((1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*b*n))
3.2.2.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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[Cosh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*(((e_.) + Log[(g_.)*( x_)^(m_.)]*(f_.))*(h_.))^(q_.), x_Symbol] :> Simp[1/((c*x^n)^(b*d)*(2/x^(b* d*n)))/E^(a*d) Int[(h*(e + f*Log[g*x^m]))^q/x^(b*d*n), x], x] + Simp[E^(a *d)*((c*x^n)^(b*d)/(2*x^(b*d*n))) Int[x^(b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, q}, x]
Int[CoshIntegral[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[x*CoshIntegral[d*(a + b*Log[c*x^n])], x] - Simp[b*d*n Int[Cosh[d*(a + b*Log[c*x^n])]/(d*(a + b*Log[c*x^n])), x], x] /; FreeQ[{a, b, c, d, n}, x]
\[\int \operatorname {Chi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
\[ \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { {\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \operatorname {Chi}\left (d \left (a + b \log {\left (c x^{n} \right )}\right )\right )\, dx \]
\[ \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { {\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { {\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
Timed out. \[ \int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathrm {coshint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]