\(\int \frac {\text {Shi}(d (a+b \log (c x^n)))}{x} \, dx\) [35]

Optimal result

Integrand size = 17, antiderivative size = 55 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n}+\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6663} \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b d n} \]

[In]

Int[SinhIntegral[d*(a + b*Log[c*x^n])]/x,x]

[Out]

-(Cosh[d*(a + b*Log[c*x^n])]/(b*d*n)) + ((a + b*Log[c*x^n])*SinhIntegral[d*(a + b*Log[c*x^n])])/(b*n)

Rule 6663

Int[SinhIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(SinhIntegral[a + b*x]/b), x] - Simp[Cosh[a
+ b*x]/b, x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {Shi}(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\text {Subst}\left (\int \text {Shi}(x) \, dx,x,a d+b d \log \left (c x^n\right )\right )}{b d n} \\ & = -\frac {\cosh \left (a d+b d \log \left (c x^n\right )\right )}{b d n}+\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Shi}\left (a d+b d \log \left (c x^n\right )\right )}{b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.75 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=-\frac {\cosh (a d) \cosh \left (b d \log \left (c x^n\right )\right )}{b d n}-\frac {\sinh (a d) \sinh \left (b d \log \left (c x^n\right )\right )}{b d n}+\frac {\log \left (c x^n\right ) \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n}+\frac {a \text {Shi}\left (a d+b d \log \left (c x^n\right )\right )}{b n} \]

[In]

Integrate[SinhIntegral[d*(a + b*Log[c*x^n])]/x,x]

[Out]

-((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]*SinhInteg
ral[d*(a + b*Log[c*x^n])])/n + (a*SinhIntegral[a*d + b*d*Log[c*x^n]])/(b*n)

Maple [A] (verified)

Time = 1.46 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\cosh \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n d b}\)	\(56\)
default	\(\frac {\operatorname {Shi}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )-\cosh \left (a d +b d \ln \left (c \,x^{n}\right )\right )}{n d b}\)	\(56\)
parts	\(\ln \left (x \right ) \operatorname {Shi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-n b \left (-\frac {\left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right ) \operatorname {Shi}\left (\ln \left (x \right ) b d n +d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b \,n^{2}}-\frac {a \,\operatorname {Shi}\left (\ln \left (x \right ) b d n +d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b^{2} n^{2}}+\frac {\cosh \left (\ln \left (x \right ) b d n +d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{b^{2} d \,n^{2}}\right )\)	\(140\)

[In]

int(Shi(d*(a+b*ln(c*x^n)))/x,x,method=_RETURNVERBOSE)

[Out]

1/n/d/b*(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)))

Fricas [F]

\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \]

[In]

integrate(Shi(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")

[Out]

integral(sinh_integral(b*d*log(c*x^n) + a*d)/x, x)

Sympy [F]

\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\operatorname {Shi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

[In]

integrate(Shi(d*(a+b*ln(c*x**n)))/x,x)

[Out]

Integral(Shi(a*d + b*d*log(c*x**n))/x, x)

Maxima [F]

\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \]

[In]

integrate(Shi(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")

[Out]

integrate(Shi((b*log(c*x^n) + a)*d)/x, x)

Giac [F]

\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x} \,d x } \]

[In]

integrate(Shi(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")

[Out]

integrate(Shi((b*log(c*x^n) + a)*d)/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,\ln \left (c\,x^n\right )}{n}+\frac {a\,\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b\,n}-\frac {{\mathrm {e}}^{a\,d}\,{\left (c\,x^n\right )}^{b\,d}}{2\,b\,d\,n}-\frac {{\mathrm {e}}^{-a\,d}}{2\,b\,d\,n\,{\left (c\,x^n\right )}^{b\,d}} \]

[In]

int(sinhint(d*(a + b*log(c*x^n)))/x,x)

[Out]

(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))