\(\int (e x)^m \text {Shi}(d (a+b \log (c x^n))) \, dx\) [38]

Optimal result

Integrand size = 19, antiderivative size = 167 \[ \int (e x)^m \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}-\frac {e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \operatorname {ExpIntegralEi}\left (\frac {(1+m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}+\frac {(e x)^{1+m} \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)} \]

[Out]

1/2*x*(e*x)^m*Ei((-b*d*n+m+1)*(a+b*ln(c*x^n))/b/n)/exp(a*(1+m)/b/n)/(1+m)/((c*x^n)^((1+m)/n))-1/2*x*(e*x)^m*Ei
((b*d*n+m+1)*(a+b*ln(c*x^n))/b/n)/exp(a*(1+m)/b/n)/(1+m)/((c*x^n)^((1+m)/n))+(e*x)^(1+m)*Shi(d*(a+b*ln(c*x^n))
)/e/(1+m)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6690, 12, 5650, 2347, 2209} \[ \int (e x)^m \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x (e x)^m e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m-b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}-\frac {x (e x)^m e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m+b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}+\frac {(e x)^{m+1} \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.98 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.72 \[ \int (e x)^m \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^m \left (e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} x^{-m} \left (\operatorname {ExpIntegralEi}\left (\frac {(1+m-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (\frac {(1+m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+2 x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 (1+m)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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