Integrand size = 17, antiderivative size = 128 \[ \int x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{3} x^3 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{6} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {(3-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{6} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \operatorname {ExpIntegralEi}\left (\frac {(3+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
1/3*x^3*Chi(d*(a+b*ln(c*x^n)))-1/6*x^3*Ei((-b*d*n+3)*(a+b*ln(c*x^n))/b/n)/ exp(3*a/b/n)/((c*x^n)^(3/n))-1/6*x^3*Ei((b*d*n+3)*(a+b*ln(c*x^n))/b/n)/exp (3*a/b/n)/((c*x^n)^(3/n))
Time = 1.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.76 \[ \int x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{6} x^3 \left (2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {(-3+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\operatorname {ExpIntegralEi}\left (\frac {(3+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )\right ) \]
(x^3*(2*CoshIntegral[d*(a + b*Log[c*x^n])] - (ExpIntegralEi[-(((-3 + b*d*n )*(a + b*Log[c*x^n]))/(b*n))] + ExpIntegralEi[((3 + b*d*n)*(a + b*Log[c*x^ n]))/(b*n)])/(E^((3*a)/(b*n))*(c*x^n)^(3/n))))/6
Time = 0.61 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.35, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, 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.
\(\displaystyle \int x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 7110 |
\(\displaystyle \frac {1}{3} x^3 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b d n \int \frac {x^2 \cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d \left (a+b \log \left (c x^n\right )\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} x^3 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b n \int \frac {x^2 \cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )}dx\) |
\(\Big \downarrow \) 6066 |
\(\displaystyle \frac {1}{3} x^3 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b n \left (\frac {1}{2} e^{-a d} x^{b d n} \left (c x^n\right )^{-b d} \int \frac {x^{2-b d n}}{a+b \log \left (c x^n\right )}dx+\frac {1}{2} e^{a d} x^{-b d n} \left (c x^n\right )^{b d} \int \frac {x^{b d n+2}}{a+b \log \left (c x^n\right )}dx\right )\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {1}{3} x^3 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b n \left (\frac {x^3 e^{-a d} \left (c x^n\right )^{-3/n} \int \frac {\left (c x^n\right )^{\frac {3-b d n}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n}+\frac {x^3 e^{a d} \left (c x^n\right )^{b d-\frac {b d n+3}{n}} \int \frac {\left (c x^n\right )^{\frac {b d n+3}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n}\right )\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {1}{3} x^3 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{3} b n \left (\frac {x^3 \left (c x^n\right )^{-3/n} e^{a \left (d-\frac {3}{b n}\right )-a d} \operatorname {ExpIntegralEi}\left (\frac {(3-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n}+\frac {x^3 e^{a d-a \left (\frac {3}{b n}+d\right )} \left (c x^n\right )^{b d-\frac {b d n+3}{n}} \operatorname {ExpIntegralEi}\left (\frac {(b d n+3) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n}\right )\) |
(x^3*CoshIntegral[d*(a + b*Log[c*x^n])])/3 - (b*n*((E^(-(a*d) + a*(d - 3/( b*n)))*x^3*ExpIntegralEi[((3 - b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*b*n*( c*x^n)^(3/n)) + (E^(a*d - a*(d + 3/(b*n)))*x^3*(c*x^n)^(b*d - (3 + b*d*n)/ n)*ExpIntegralEi[((3 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(2*b*n)))/3
3.1.100.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
\[\int x^{2} \operatorname {Chi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
\[ \int x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} {\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{2} \operatorname {Chi}\left (a d + b d \log {\left (c x^{n} \right )}\right )\, dx \]
\[ \int x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} {\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} {\rm Chi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
Timed out. \[ \int x^2 \text {Chi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^2\,\mathrm {coshint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]