Integrand size = 17, antiderivative size = 130 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{4 x^2}-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]
1/4*exp(2*a/b/n)*(c*x^n)^(2/n)*Ei(-(-b*d*n+2)*(a+b*ln(c*x^n))/b/n)/x^2-1/4 *exp(2*a/b/n)*(c*x^n)^(2/n)*Ei(-(b*d*n+2)*(a+b*ln(c*x^n))/b/n)/x^2-1/2*Shi (d*(a+b*ln(c*x^n)))/x^2
Time = 1.25 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.14 \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {1}{4} e^{-\frac {(-2+b d n) \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{b n}} \left (\operatorname {ExpIntegralEi}\left (\frac {(-2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\operatorname {ExpIntegralEi}\left (-\frac {(2+b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \left (\cosh \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+\sinh \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2} \]
((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))])*(Cosh[d*(a + b*(-(n*Log[x]) + L og[c*x^n]))] + Sinh[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(4*E^(((-2 + b *d*n)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))) - SinhIntegral[d*(a + b* Log[c*x^n])]/(2*x^2)
Time = 0.61 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, 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.
\(\displaystyle \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 7109 |
\(\displaystyle \frac {1}{2} b d n \int \frac {\sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d x^3 \left (a+b \log \left (c x^n\right )\right )}dx-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} b n \int \frac {\sinh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3 \left (a+b \log \left (c x^n\right )\right )}dx-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\) |
\(\Big \downarrow \) 6065 |
\(\displaystyle \frac {1}{2} b n \left (\frac {1}{2} e^{a d} x^{-b d n} \left (c x^n\right )^{b d} \int \frac {x^{b d n-3}}{a+b \log \left (c x^n\right )}dx-\frac {1}{2} e^{-a d} x^{b d n} \left (c x^n\right )^{-b d} \int \frac {x^{-b d n-3}}{a+b \log \left (c x^n\right )}dx\right )-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {1}{2} b n \left (\frac {e^{a d} \left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-\frac {2-b d n}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n x^2}-\frac {e^{-a d} \left (c x^n\right )^{2/n} \int \frac {\left (c x^n\right )^{-\frac {b d n+2}{n}}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 n x^2}\right )-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {1}{2} b n \left (\frac {e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {(2-b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n x^2}-\frac {\left (c x^n\right )^{2/n} e^{a \left (\frac {2}{b n}+d\right )-a d} \operatorname {ExpIntegralEi}\left (-\frac {(b d n+2) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 b n x^2}\right )-\frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\) |
(b*n*((E^((2*a)/(b*n))*(c*x^n)^(2/n)*ExpIntegralEi[-(((2 - b*d*n)*(a + b*L og[c*x^n]))/(b*n))])/(2*b*n*x^2) - (E^(-(a*d) + a*(d + 2/(b*n)))*(c*x^n)^( 2/n)*ExpIntegralEi[-(((2 + b*d*n)*(a + b*Log[c*x^n]))/(b*n))])/(2*b*n*x^2) ))/2 - SinhIntegral[d*(a + b*Log[c*x^n])]/(2*x^2)
3.1.37.3.1 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]
\[\int \frac {\operatorname {Shi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]
\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\operatorname {Shi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]
\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
\[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {{\rm Shi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {Shi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {sinhint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]