Integrand size = 17, antiderivative size = 55 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n}+\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \] Output:
-cosh(d*(a+b*ln(c*x^n)))/b/d/n+(a+b*ln(c*x^n))*Shi(d*(a+b*ln(c*x^n)))/b/n
Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.75 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {\cosh (a d) \cosh \left (b d \log \left (c x^n\right )\right )}{b d n}-\frac {\sinh (a d) \sinh \left (b d \log \left (c x^n\right )\right )}{b d n}+\frac {\log \left (c x^n\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n}+\frac {a \text {Shi}\left (a d+b d \log \left (c x^n\right )\right )}{b n} \] Input:
Integrate[SinhIntegral[d*(a + b*Log[c*x^n])]/x,x]
Output:
-((Cosh[a*d]*Cosh[b*d*Log[c*x^n]])/(b*d*n)) - (Sinh[a*d]*Sinh[b*d*Log[c*x^ n]])/(b*d*n) + (Log[c*x^n]*SinhIntegral[d*(a + b*Log[c*x^n])])/n + (a*Sinh Integral[a*d + b*d*Log[c*x^n]])/(b*n)
Time = 0.29 (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, 7082}
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.
\(\displaystyle \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {\int \text {Shi}\left (a d+b \log \left (c x^n\right ) d\right )d\left (a d+b \log \left (c x^n\right ) d\right )}{b d n}\) |
\(\Big \downarrow \) 7082 |
\(\displaystyle \frac {\left (a d+b d \log \left (c x^n\right )\right ) \text {Shi}\left (a d+b \log \left (c x^n\right ) d\right )-\frac {x^{-n} \left (c^2 x^{2 n}+1\right )}{2 c}}{b d n}\) |
Input:
Int[SinhIntegral[d*(a + b*Log[c*x^n])]/x,x]
Output:
(-1/2*(1 + c^2*x^(2*n))/(c*x^n) + (a*d + b*d*Log[c*x^n])*SinhIntegral[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[SinhIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(SinhInte gral[a + b*x]/b), x] - Simp[Cosh[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 = 1.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {\operatorname {Shi}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\cosh \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d}\) | \(56\) |
default | \(\frac {\operatorname {Shi}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\cosh \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n b d}\) | \(56\) |
parts | \(\ln \left (x \right ) \operatorname {Shi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-b n \left (-\frac {\left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right ) \operatorname {Shi}\left (\ln \left (x \right ) b d n +d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b \,n^{2}}-\frac {a \,\operatorname {Shi}\left (\ln \left (x \right ) b d n +d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b^{2} n^{2}}+\frac {\cosh \left (\ln \left (x \right ) b d n +d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b^{2} d \,n^{2}}\right )\) | \(140\) |
Input:
int(Shi(d*(a+b*ln(c*x^n)))/x,x,method=_RETURNVERBOSE)
Output:
1/n/b/d*(Shi(a*d+b*d*ln(c*x^n))*(a*d+b*d*ln(c*x^n))-cosh(a*d+b*d*ln(c*x^n) ))
\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \] Input:
integrate(Shi(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")
Output:
integral(sinh_integral(b*d*log(c*x^n) + a*d)/x, x)
\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\operatorname {Shi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:
integrate(Shi(d*(a+b*ln(c*x**n)))/x,x)
Output:
Integral(Shi(a*d + b*d*log(c*x**n))/x, x)
\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \] Input:
integrate(Shi(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")
Output:
integrate(Shi((b*log(c*x^n) + a)*d)/x, x)
\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \] Input:
integrate(Shi(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")
Output:
integrate(Shi((b*log(c*x^n) + a)*d)/x, x)
Timed out. \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,\ln \left (c\,x^n\right )}{n}+\frac {a\,\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}-\frac {{\mathrm {e}}^{a\,d}\,{\left (c\,x^n\right )}^{b\,d}}{2\,b\,d\,n}-\frac {{\mathrm {e}}^{-a\,d}}{2\,b\,d\,n\,{\left (c\,x^n\right )}^{b\,d}} \] Input:
int(sinhint(d*(a + b*log(c*x^n)))/x,x)
Output:
(sinhint(d*(a + b*log(c*x^n)))*log(c*x^n))/n + (a*sinhint(d*(a + b*log(c*x ^n))))/(b*n) - (exp(a*d)*(c*x^n)^(b*d))/(2*b*d*n) - exp(-a*d)/(2*b*d*n*(c* x^n)^(b*d))
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.22 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {-\cosh \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )+\mathrm {log}\left (x^{n} c \right ) \mathit {shi} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) b d +\mathit {shi} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) a d}{b d n} \] Input:
int(Shi(d*(a+b*log(c*x^n)))/x,x)
Output:
( - cosh(log(x**n*c)*b*d + a*d) + log(x**n*c)*shi(log(x**n*c)*b*d + a*d)*b *d + shi(log(x**n*c)*b*d + a*d)*a*d)/(b*d*n)