\(\int \text {Shi}(d (a+b \log (c x^n))) \, dx\) [34]

Optimal result

Integrand size = 13, antiderivative size = 119 \[ \int \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \] Output:

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))+x*Shi(d*(a+b* 
ln(c*x^n)))
 

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.80 \[ \int \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {(-1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \] Input:

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

Output:

(x*(ExpIntegralEi[-(((-1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))] - ExpIntegra 
lEi[((1 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)]))/(2*E^(a/(b*n))*(c*x^n)^n^(-1 
)) + x*SinhIntegral[d*(a + b*Log[c*x^n])]
 

Rubi [A] (verified)

Time = 0.60 (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 = {7106, 27, 6063, 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.

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

\(\Big \downarrow \) 7106

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6063

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

-(b*n*(-1/2*(E^(-(a*d) + a*(d - 1/(b*n)))*x*ExpIntegralEi[((1 - b*d*n)*(a 
+ b*Log[c*x^n]))/(b*n)])/(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)*ExpIntegralEi[((1 + b*d*n)*(a + b*Log[c*x 
^n]))/(b*n)])/(2*b*n))) + x*SinhIntegral[d*(a + b*Log[c*x^n])]
 

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 6063

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

rule 7106

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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