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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 130, 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} \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]

[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 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 {\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 ) \text {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 ) \text {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} \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} \\ \end{align*}

Mathematica [A] (verified)

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

[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]

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

[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)

Fricas [F]

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

[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]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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