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

Optimal result

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

[Out]

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))

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6691, 12, 5651, 2347, 2209} \[ \int x \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{4} x^2 e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]

[In]

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

[Out]

(x^2*CoshIntegral[d*(a + b*Log[c*x^n])])/2 - (x^2*ExpIntegralEi[((2 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(4*E^
((2*a)/(b*n))*(c*x^n)^(2/n)) - (x^2*ExpIntegralEi[((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(4*E^((2*a)/(b*n))*
(c*x^n)^(2/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 5651

Int[Cosh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*(((e_.) + Log[(g_.)*(x_)^(m_.)]*(f_.))*(h_.))^(q_.)*((i_
.)*(x_))^(r_.), x_Symbol] :> Dist[((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] + 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 6691

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] - Dist[b*d*(n/(m + 1)), Int[(e*x)^m*(Cosh[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 {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 \\ & = \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 \\ & = \frac {1}{2} x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{4} \left (b e^{-a d} n x^{b d n} \left (c x^n\right )^{-b d}\right ) \int \frac {x^{1-b d n}}{a+b \log \left (c x^n\right )} \, dx-\frac {1}{4} \left (b e^{a d} n x^{-b d n} \left (c x^n\right )^{b d}\right ) \int \frac {x^{1+b d n}}{a+b \log \left (c x^n\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} \left (b e^{-a d} x^2 \left (c x^n\right )^{-b d-\frac {2-b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(2-b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )-\frac {1}{4} \left (b e^{a d} x^2 \left (c x^n\right )^{b d-\frac {2+b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(2+b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right ) \\ & = \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 ) \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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