Integrand size = 15, antiderivative size = 128 \[ \int x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{4} e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{4} e^{-\frac {2 a}{b n}} x^2 \left (c x^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\frac {1}{2} x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]
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))+1/2*x ^2*Shi(d*(a+b*ln(c*x^n)))
Time = 1.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.77 \[ \int x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{4} x^2 \left (e^{-\frac {2 a}{b n}} \left (c x^n\right )^{-2/n} \left (\operatorname {ExpIntegralEi}\left (-\frac {(-2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )+2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \]
(x^2*((ExpIntegralEi[-(((-2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))] - ExpInte gralEi[((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n)])/(E^((2*a)/(b*n))*(c*x^n)^( 2/n)) + 2*SinhIntegral[d*(a + b*Log[c*x^n])]))/4
Time = 0.59 (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 = {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.
\(\displaystyle \int x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 7109 |
\(\displaystyle \frac {1}{2} x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} b d n \int \frac {x \sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d \left (a+b \log \left (c x^n\right )\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} b n \int \frac {x \sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )}dx\) |
\(\Big \downarrow \) 6065 |
\(\displaystyle \frac {1}{2} x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} b n \left (\frac {1}{2} e^{a d} x^{-b d n} \left (c x^n\right )^{b d} \int \frac {x^{b d n+1}}{a+b \log \left (c x^n\right )}dx-\frac {1}{2} e^{-a d} x^{b d n} \left (c x^n\right )^{-b d} \int \frac {x^{1-b d n}}{a+b \log \left (c x^n\right )}dx\right )\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {1}{2} x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} b n \left (\frac {x^2 e^{a d} \left (c x^n\right )^{b d-\frac {b d n+2}{n}} \int \frac {\left (c x^n\right )^{\frac {b d n+2}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n}-\frac {x^2 e^{-a d} \left (c x^n\right )^{-2/n} \int \frac {\left (c x^n\right )^{\frac {2-b d n}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n}\right )\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {1}{2} x^2 \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} b n \left (\frac {x^2 e^{a d-a \left (\frac {2}{b n}+d\right )} \left (c x^n\right )^{b d-\frac {b d n+2}{n}} \operatorname {ExpIntegralEi}\left (\frac {(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n}-\frac {x^2 \left (c x^n\right )^{-2/n} e^{a \left (d-\frac {2}{b n}\right )-a d} \operatorname {ExpIntegralEi}\left (\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n}\right )\) |
-1/2*(b*n*(-1/2*(E^(-(a*d) + a*(d - 2/(b*n)))*x^2*ExpIntegralEi[((2 - b*d* n)*(a + b*Log[c*x^n]))/(b*n)])/(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))) + (x^2*SinhIntegral[d*(a + b*Log[c*x^n])])/ 2
3.1.33.3.1 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[(((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]
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]
\[\int x \,\operatorname {Shi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
\[ \int x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x {\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x \operatorname {Shi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
\[ \int x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x {\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
\[ \int x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x {\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
Timed out. \[ \int x \text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x\,\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]