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

Optimal result

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

[Out]

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)/ex
p(a/b/n)/((c*x^n)^(1/n))+x*Shi(d*(a+b*ln(c*x^n)))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6687, 12, 5648, 2347, 2209} \[ \int \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(1-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{2} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {(b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]

[In]

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

[Out]

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

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 5648

Int[(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_S
ymbol] :> Dist[(-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]
+ Dist[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 6687

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

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

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.80 \[ \int \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {(-1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (\frac {(1+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(sinhint(d*(a + b*log(c*x^n))),x)

[Out]

int(sinhint(d*(a + b*log(c*x^n))), x)