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

Optimal result

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

[Out]

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)))

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6690, 12, 5650, 2347, 2209} \[ \int x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{6} x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {(3-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{6} x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {(b d n+3) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]

[In]

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

[Out]

(x^3*ExpIntegralEi[((3 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(6*E^((3*a)/(b*n))*(c*x^n)^(3/n)) - (x^3*ExpIntegr
alEi[((3 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(6*E^((3*a)/(b*n))*(c*x^n)^(3/n)) + (x^3*SinhIntegral[d*(a + b*L
og[c*x^n])])/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(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 5650

Int[(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.)*(x_))^(r_.)*Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]
*(b_.))*(d_.)], x_Symbol] :> Dist[(-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] + Dist[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 6690

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] - Dist[b*d*(n/(m + 1)), Int[(e*x)^m*(Sinh[d*(a + b*Lo
g[c*x^n])]/(d*(a + b*Log[c*x^n]))), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \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 \\ & = \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 \\ & = \frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {1}{6} \left (b e^{-a d} n x^{b d n} \left (c x^n\right )^{-b d}\right ) \int \frac {x^{2-b d n}}{a+b \log \left (c x^n\right )} \, dx-\frac {1}{6} \left (b e^{a d} n x^{-b d n} \left (c x^n\right )^{b d}\right ) \int \frac {x^{2+b d n}}{a+b \log \left (c x^n\right )} \, dx \\ & = \frac {1}{3} x^3 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {1}{6} \left (b e^{-a d} x^3 \left (c x^n\right )^{-b d-\frac {3-b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(3-b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )-\frac {1}{6} \left (b e^{a d} x^3 \left (c x^n\right )^{b d-\frac {3+b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(3+b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\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}{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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.77 \[ \int x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{6} x^3 \left (e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {(-3+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (\frac {(3+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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