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

Optimal. Leaf size=130 \[ \frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{ExpIntegralEi}\left (-\frac{(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{ExpIntegralEi}\left (-\frac{(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.24719, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6555, 12, 5539, 2310, 2178} \[ \frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6555

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]

Rule 12

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

Rule 5539

Int[(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.)*(x_))^(r_.)*Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]
*(b_.))*(d_.)], x_Symbol] :> -Dist[(i*x)^r/(E^(a*d)*(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 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2178

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

Rubi steps

\begin{align*} \int \frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\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}+\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}-\frac{1}{4} \left (b e^{-a d} n x^{b d n} \left (c x^n\right )^{-b d}\right ) \int \frac{x^{-3-b d n}}{a+b \log \left (c x^n\right )} \, dx+\frac{1}{4} \left (b e^{a d} n x^{-b d n} \left (c x^n\right )^{b d}\right ) \int \frac{x^{-3+b d n}}{a+b \log \left (c x^n\right )} \, dx\\ &=-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}-\frac{\left (b e^{-a d} \left (c x^n\right )^{-b d-\frac{-2-b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(-2-b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{4 x^2}+\frac{\left (b e^{a d} \left (c x^n\right )^{b d-\frac{-2+b d n}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(-2+b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{4 x^2}\\ &=\frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\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} \text{Ei}\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}\\ \end{align*}

Mathematica [A]  time = 1.90219, size = 148, normalized size = 1.14 \[ \frac{1}{4} \exp \left (-\frac{(b d n-2) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left (\text{ExpIntegralEi}\left (\frac{(b d n-2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\text{ExpIntegralEi}\left (-\frac{(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \left (\sinh \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )+\cosh \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )-\frac{\text{Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((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]) + Log[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)

________________________________________________________________________________________

Maple [F]  time = 0.106, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Shi} \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{Shi}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Shi}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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