Integrand size = 17, antiderivative size = 55 \[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]
Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.75 \[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {a \text {Chi}\left (a d+b d \log \left (c x^n\right )\right )}{b n}+\frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \log \left (c x^n\right )}{n}-\frac {\cosh \left (b d \log \left (c x^n\right )\right ) \sinh (a d)}{b d n}-\frac {\cosh (a d) \sinh \left (b d \log \left (c x^n\right )\right )}{b d n} \]
(a*CoshIntegral[a*d + b*d*Log[c*x^n]])/(b*n) + (CoshIntegral[d*(a + b*Log[ c*x^n])]*Log[c*x^n])/n - (Cosh[b*d*Log[c*x^n]]*Sinh[a*d])/(b*d*n) - (Cosh[ a*d]*Sinh[b*d*Log[c*x^n]])/(b*d*n)
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3039, 7281, 7083}
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.
\(\displaystyle \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {\int \text {Chi}\left (a d+b \log \left (c x^n\right ) d\right )d\left (a d+b \log \left (c x^n\right ) d\right )}{b d n}\) |
\(\Big \downarrow \) 7083 |
\(\displaystyle \frac {\left (a d+b d \log \left (c x^n\right )\right ) \text {Chi}\left (a d+b \log \left (c x^n\right ) d\right )-\frac {x^{-n} \left (c^2 x^{2 n}-1\right )}{2 c}}{b d n}\) |
(-1/2*(-1 + c^2*x^(2*n))/(c*x^n) + CoshIntegral[a*d + b*d*Log[c*x^n]]*(a*d + b*d*Log[c*x^n]))/(b*d*n)
3.2.3.3.1 Defintions of rubi rules used
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[CoshIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(CoshInte gral[a + b*x]/b), x] - Simp[Sinh[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 = 1.51 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {\operatorname {Chi}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\sinh \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n d b}\) | \(56\) |
default | \(\frac {\operatorname {Chi}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\sinh \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n d b}\) | \(56\) |
parts | \(\ln \left (x \right ) \operatorname {Chi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-n b \left (-\frac {\left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right ) \operatorname {Chi}\left (\ln \left (x \right ) b d n +d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b \,n^{2}}-\frac {a \,\operatorname {Chi}\left (\ln \left (x \right ) b d n +d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b^{2} n^{2}}+\frac {\sinh \left (\ln \left (x \right ) b d n +d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b^{2} d \,n^{2}}\right )\) | \(140\) |
\[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {{\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \]
\[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\operatorname {Chi}\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x}\, dx \]
\[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {{\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \]
\[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {{\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\ln \left (c\,x^n\right )\,\mathrm {coshint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{n}+\frac {a\,\mathrm {coshint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}-\frac {{\mathrm {e}}^{a\,d}\,{\left (c\,x^n\right )}^{b\,d}}{2\,b\,d\,n}+\frac {{\mathrm {e}}^{-a\,d}}{2\,b\,d\,n\,{\left (c\,x^n\right )}^{b\,d}} \]