3.2.1 \(\int x \text {Chi}(d (a+b \log (c x^n))) \, dx\) [101]

3.2.1.1 Optimal result

Integrand size = 15, antiderivative size = 128 \[ \int x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{4} e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]

output

1/2*x^2*Chi(d*(a+b*ln(c*x^n)))-1/4*x^2*Ei((-b*d*n+2)*(a+b*ln(c*x^n))/b/n)/ 
exp(2*a/b/n)/((c*x^n)^(2/n))-1/4*x^2*Ei((b*d*n+2)*(a+b*ln(c*x^n))/b/n)/exp 
(2*a/b/n)/((c*x^n)^(2/n))
 

3.2.1.2 Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.76 \[ \int x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{4} x^2 \left (2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {(-2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )\right ) \]

input

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

output

(x^2*(2*CoshIntegral[d*(a + b*Log[c*x^n])] - (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)])/(E^((2*a)/(b*n))*(c*x^n)^(2/n))))/4
 

3.2.1.3 Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.35, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, 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.

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

\(\Big \downarrow \) 7110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6066

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

input

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

output

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

3.2.1.3.1 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 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]
 

3.2.1.4 Maple [F]

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

input

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

output

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

3.2.1.5 Fricas [F]

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

input

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

output

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

3.2.1.6 Sympy [F]

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

input

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

output

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

3.2.1.7 Maxima [F]

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

input

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

output

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

3.2.1.8 Giac [F]

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

input

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

output

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

3.2.1.9 Mupad [F(-1)]

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

input

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

output

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