\(\int x^2 \text {Shi}(d (a+b \log (c x^n))) \, dx\) [32]

Optimal result

Integrand size = 17, antiderivative size = 128 \[ \int x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{6} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {(3-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{6} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {(3+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \] Output:

1/6*x^3*Ei((-b*d*n+3)*(a+b*ln(c*x^n))/b/n)/exp(3*a/b/n)/((c*x^n)^(3/n))-1/ 
6*x^3*Ei((b*d*n+3)*(a+b*ln(c*x^n))/b/n)/exp(3*a/b/n)/((c*x^n)^(3/n))+1/3*x 
^3*Shi(d*(a+b*ln(c*x^n)))
 

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.77 \[ \int x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{6} x^3 \left (e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {(-3+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (\frac {(3+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \] Input:

Integrate[x^2*SinhIntegral[d*(a + b*Log[c*x^n])],x]
 

Output:

(x^3*((ExpIntegralEi[-(((-3 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))] - ExpInte 
gralEi[((3 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^((3*a)/(b*n))*(c*x^n)^( 
3/n)) + 2*SinhIntegral[d*(a + b*Log[c*x^n])]))/6
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.35, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, 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.

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

\(\Big \downarrow \) 7109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6065

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

Int[x^2*SinhIntegral[d*(a + b*Log[c*x^n])],x]
 

Output:

-1/3*(b*n*(-1/2*(E^(-(a*d) + a*(d - 3/(b*n)))*x^3*ExpIntegralEi[((3 - b*d* 
n)*(a + b*Log[c*x^n]))/(b*n)])/(b*n*(c*x^n)^(3/n)) + (E^(a*d - a*(d + 3/(b 
*n)))*x^3*(c*x^n)^(b*d - (3 + b*d*n)/n)*ExpIntegralEi[((3 + b*d*n)*(a + b* 
Log[c*x^n]))/(b*n)])/(2*b*n))) + (x^3*SinhIntegral[d*(a + b*Log[c*x^n])])/ 
3
 

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]

\[\int x^{2} \operatorname {Shi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]

Input:

int(x^2*Shi(d*(a+b*ln(c*x^n))),x)
 

Output:

int(x^2*Shi(d*(a+b*ln(c*x^n))),x)
 

Fricas [F]

\[ \int x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} {\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:

integrate(x^2*Shi(d*(a+b*log(c*x^n))),x, algorithm="fricas")
 

Output:

integral(x^2*sinh_integral(b*d*log(c*x^n) + a*d), x)
 

Sympy [F]

\[ \int x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{2} \operatorname {Shi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \] Input:

integrate(x**2*Shi(d*(a+b*ln(c*x**n))),x)
 

Output:

Integral(x**2*Shi(a*d + b*d*log(c*x**n)), x)
 

Maxima [F]

\[ \int x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} {\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:

integrate(x^2*Shi(d*(a+b*log(c*x^n))),x, algorithm="maxima")
 

Output:

integrate(x^2*Shi((b*log(c*x^n) + a)*d), x)
 

Giac [F]

\[ \int x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} {\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \] Input:

integrate(x^2*Shi(d*(a+b*log(c*x^n))),x, algorithm="giac")
 

Output:

integrate(x^2*Shi((b*log(c*x^n) + a)*d), x)
 

Mupad [F(-1)]

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

int(x^2*sinhint(d*(a + b*log(c*x^n))),x)
 

Output:

int(x^2*sinhint(d*(a + b*log(c*x^n))), x)
 

Reduce [F]

\[ \int x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \mathit {shi} \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right ) x^{2}d x \] Input:

int(x^2*Shi(d*(a+b*log(c*x^n))),x)
 

Output:

int(shi(log(x**n*c)*b*d + a*d)*x**2,x)