\(\int (e x)^m \text {Chi}(d (a+b \log (c x^n))) \, dx\) [106]

Optimal result

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

(e*x)^(1+m)*Chi(d*(a+b*ln(c*x^n)))/e/(1+m)-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))
 

Mathematica [A] (verified)

Time = 1.34 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.71 \[ \int (e x)^m \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^m \left (2 x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} x^{-m} \left (\operatorname {ExpIntegralEi}\left (\frac {(1+m-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {(1+m+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )\right )}{2 (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (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 = {7110, 27, 6066, 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.

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

\(\Big \downarrow \) 7110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6066

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

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

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 6066

Int[Cosh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*(((e_.) + Log[(g_.)*( 
x_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_.)*(x_))^(r_.), x_Symbol] :> Simp[((i*x) 
^r*(1/((c*x^n)^(b*d)*(2*x^(r - b*d*n)))))/E^(a*d)   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 7110

Int[CoshIntegral[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^ 
(m_.), x_Symbol] :> Simp[(e*x)^(m + 1)*(CoshIntegral[d*(a + b*Log[c*x^n])]/ 
(e*(m + 1))), x] - Simp[b*d*(n/(m + 1))   Int[(e*x)^m*(Cosh[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^m \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=e^{m} \left (\int x^{m} \chi \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )d x \right ) \] Input:

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

Output:

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