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

3.1.38.1 Optimal result

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

output

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)
 

3.1.38.2 Mathematica [A] (verified)

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

input

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

output

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

3.1.38.3 Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.29, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {7109, 27, 6065, 2747, 2609}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7109

\(\displaystyle \frac {(e x)^{m+1} \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b d n \int \frac {(e x)^m \sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d \left (a+b \log \left (c x^n\right )\right )}dx}{m+1}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(e x)^{m+1} \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b n \int \frac {(e x)^m \sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )}dx}{m+1}\)

\(\Big \downarrow \) 6065

\(\displaystyle \frac {(e x)^{m+1} \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b n \left (\frac {1}{2} e^{a d} (e x)^m \left (c x^n\right )^{b d} x^{-b d n-m} \int \frac {x^{m+b d n}}{a+b \log \left (c x^n\right )}dx-\frac {1}{2} e^{-a d} (e x)^m \left (c x^n\right )^{-b d} x^{b d n-m} \int \frac {x^{m-b d n}}{a+b \log \left (c x^n\right )}dx\right )}{m+1}\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {(e x)^{m+1} \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b n \left (\frac {x e^{a d} (e x)^m \left (c x^n\right )^{b d-\frac {b d n+m+1}{n}} \int \frac {\left (c x^n\right )^{\frac {m+b d n+1}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n}-\frac {x e^{-a d} (e x)^m \left (c x^n\right )^{-\frac {-b d n+m+1}{n}-b d} \int \frac {\left (c x^n\right )^{\frac {m-b d n+1}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n}\right )}{m+1}\)

\(\Big \downarrow \) 2609

\(\displaystyle \frac {(e x)^{m+1} \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {b n \left (\frac {x (e x)^m e^{a d-\frac {a (b d n+m+1)}{b n}} \left (c x^n\right )^{b d-\frac {b d n+m+1}{n}} \operatorname {ExpIntegralEi}\left (\frac {(m+b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n}-\frac {x (e x)^m e^{-\frac {a (-b d n+m+1)}{b n}-a d} \left (c x^n\right )^{-\frac {-b d n+m+1}{n}-b d} \operatorname {ExpIntegralEi}\left (\frac {(m-b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n}\right )}{m+1}\)

input

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

output

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

3.1.38.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2609

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

rule 2747

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(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 6065

Int[(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.)*(x_))^(r_.)* 
Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[(-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] + Simp[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 7109

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

3.1.38.4 Maple [F]

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

input

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

output

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

3.1.38.5 Fricas [F]

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

input

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

output

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

3.1.38.6 Sympy [F]

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

input

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

output

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

3.1.38.7 Maxima [F]

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

input

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

output

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

3.1.38.8 Giac [F]

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

input

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

output

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

3.1.38.9 Mupad [F(-1)]

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

input

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

output

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