\(\int \frac {\text {Shi}(d (a+b \log (c x^n)))}{x^3} \, dx\) [37]

Optimal result

Integrand size = 17, antiderivative size = 130 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \] Output:

1/4*exp(2*a/b/n)*(c*x^n)^(2/n)*Ei(-(-b*d*n+2)*(a+b*ln(c*x^n))/b/n)/x^2-1/4 
*exp(2*a/b/n)*(c*x^n)^(2/n)*Ei(-(b*d*n+2)*(a+b*ln(c*x^n))/b/n)/x^2-1/2*Shi 
(d*(a+b*ln(c*x^n)))/x^2
 

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.14 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {1}{4} e^{-\frac {(-2+b d n) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} \left (\operatorname {ExpIntegralEi}\left (\frac {(-2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (-\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \left (\cosh \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+\sinh \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \] Input:

Integrate[SinhIntegral[d*(a + b*Log[c*x^n])]/x^3,x]
 

Output:

((ExpIntegralEi[((-2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)] - ExpIntegralEi[- 
(((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))])*(Cosh[d*(a + b*(-(n*Log[x]) + L 
og[c*x^n]))] + Sinh[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(4*E^(((-2 + b 
*d*n)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))) - SinhIntegral[d*(a + b* 
Log[c*x^n])]/(2*x^2)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {7109, 27, 6065, 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.

\(\displaystyle \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\)

\(\Big \downarrow \) 7109

\(\displaystyle \frac {1}{2} b d n \int \frac {\sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d x^3 \left (a+b \log \left (c x^n\right )\right )}dx-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} b n \int \frac {\sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3 \left (a+b \log \left (c x^n\right )\right )}dx-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 6065

\(\displaystyle \frac {1}{2} b n \left (\frac {1}{2} e^{a d} x^{-b d n} \left (c x^n\right )^{b d} \int \frac {x^{b d n-3}}{a+b \log \left (c x^n\right )}dx-\frac {1}{2} e^{-a d} x^{b d n} \left (c x^n\right )^{-b d} \int \frac {x^{-b d n-3}}{a+b \log \left (c x^n\right )}dx\right )-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {1}{2} b n \left (\frac {e^{a d} \left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-\frac {2-b d n}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n x^2}-\frac {e^{-a d} \left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-\frac {b d n+2}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n x^2}\right )-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {1}{2} b n \left (\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n x^2}-\frac {\left (c x^n\right )^{2/n} e^{a \left (\frac {2}{b n}+d\right )-a d} \operatorname {ExpIntegralEi}\left (-\frac {(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n x^2}\right )-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

Input:

Int[SinhIntegral[d*(a + b*Log[c*x^n])]/x^3,x]
 

Output:

(b*n*((E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 - b*d*n)*(a + b*L 
og[c*x^n]))/(b*n))])/(2*b*n*x^2) - (E^(-(a*d) + a*(d + 2/(b*n)))*(c*x^n)^( 
2/n)*ExpIntegralEi[-(((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*b*n*x^2) 
))/2 - SinhIntegral[d*(a + b*Log[c*x^n])]/(2*x^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2609

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]
 

rule 2747

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]
 

rule 6065

Int[(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.)*(x_))^(r_.)* 
Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[(-E^(( 
-a)*d))*(i*x)^r*(1/((c*x^n)^(b*d)*(2*x^(r - b*d*n))))   Int[x^(r - b*d*n)*( 
h*(e + f*Log[g*x^m]))^q, x], x] + Simp[E^(a*d)*(i*x)^r*((c*x^n)^(b*d)/(2*x^ 
(r + b*d*n)))   Int[x^(r + b*d*n)*(h*(e + f*Log[g*x^m]))^q, x], x] /; FreeQ 
[{a, b, c, d, e, f, g, h, i, m, n, q, r}, x]
 

rule 7109

Int[((e_.)*(x_))^(m_.)*SinhIntegral[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*( 
d_.)], x_Symbol] :> Simp[(e*x)^(m + 1)*(SinhIntegral[d*(a + b*Log[c*x^n])]/ 
(e*(m + 1))), x] - Simp[b*d*(n/(m + 1))   Int[(e*x)^m*(Sinh[d*(a + b*Log[c* 
x^n])]/(d*(a + b*Log[c*x^n]))), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] & 
& NeQ[m, -1]
 

Maple [F]

\[\int \frac {\operatorname {Shi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]

Input:

int(Shi(d*(a+b*ln(c*x^n)))/x^3,x)
 

Output:

int(Shi(d*(a+b*ln(c*x^n)))/x^3,x)
 

Fricas [F]

\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate(Shi(d*(a+b*log(c*x^n)))/x^3,x, algorithm="fricas")
 

Output:

integral(sinh_integral(b*d*log(c*x^n) + a*d)/x^3, x)
 

Sympy [F]

\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\operatorname {Shi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \] Input:

integrate(Shi(d*(a+b*ln(c*x**n)))/x**3,x)
 

Output:

Integral(Shi(a*d + b*d*log(c*x**n))/x**3, x)
 

Maxima [F]

\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate(Shi(d*(a+b*log(c*x^n)))/x^3,x, algorithm="maxima")
 

Output:

integrate(Shi((b*log(c*x^n) + a)*d)/x^3, x)
 

Giac [F]

\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate(Shi(d*(a+b*log(c*x^n)))/x^3,x, algorithm="giac")
 

Output:

integrate(Shi((b*log(c*x^n) + a)*d)/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \] Input:

int(sinhint(d*(a + b*log(c*x^n)))/x^3,x)
 

Output:

int(sinhint(d*(a + b*log(c*x^n)))/x^3, x)
 

Reduce [F]

\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathit {shi} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}{x^{3}}d x \] Input:

int(Shi(d*(a+b*log(c*x^n)))/x^3,x)
 

Output:

int(shi(log(x**n*c)*b*d + a*d)/x**3,x)