Integrand size = 20, antiderivative size = 101 \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {-a-b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2 \sqrt {d x}}-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \]
-1/2*exp(1/2*a/b/n)*(c*x^n)^(1/2/n)*Ei(1/2*(-a-b*ln(c*x^n))/b/n)/b^2/d/n^2 /(d*x)^(1/2)-1/b/d/n/(a+b*ln(c*x^n))/(d*x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {x \left (2 b n+e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{2 b^2 n^2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \]
-1/2*(x*(2*b*n + E^(a/(2*b*n))*(c*x^n)^(1/(2*n))*ExpIntegralEi[-1/2*(a + b *Log[c*x^n])/(b*n)]*(a + b*Log[c*x^n])))/(b^2*n^2*(d*x)^(3/2)*(a + b*Log[c *x^n]))
Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2743, 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.
| \(\displaystyle \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle -\frac {\int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}dx}{2 b n}-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}\) |
| \(\Big \downarrow \) 2747 |
| \(\displaystyle -\frac {\left (c x^n\right )^{\left .\frac {1}{2}\right /n} \int \frac {\left (c x^n\right )^{\left .-\frac {1}{2}\right /n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 b d n^2 \sqrt {d x}}-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle -\frac {e^{\frac {a}{2 b n}} \left (c x^n\right )^{\left .\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (-\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2 \sqrt {d x}}-\frac {1}{b d n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}\) |
-1/2*(E^(a/(2*b*n))*(c*x^n)^(1/(2*n))*ExpIntegralEi[-1/2*(a + b*Log[c*x^n] )/(b*n)])/(b^2*d*n^2*Sqrt[d*x]) - 1/(b*d*n*Sqrt[d*x]*(a + b*Log[c*x^n]))
3.2.11.3.1 Defintions of rubi rules used
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)*((a + b*Log[c*x^n])^(p + 1)/(b*d*n*(p + 1))), x] - Simp[(m + 1)/(b*n*(p + 1)) Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.93 (sec) , antiderivative size = 429, normalized size of antiderivative = 4.25
| method | result | size |
| risch | \(-\frac {2}{b n \sqrt {d x}\, \left (2 a +2 b \ln \left (c \right )+2 b \ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}\right ) d}+\frac {{\mathrm e}^{-\frac {i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}} \operatorname {Ei}_{1}\left (\frac {\ln \left (d x \right )}{2}-\frac {i \left (b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\operatorname {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \operatorname {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}\right )}{2 b^{2} n^{2} d}\) | \(429\) |
-2/b/n/(d*x)^(1/2)/(2*a+2*b*ln(c)+2*b*ln(exp(n*ln(x)))-I*b*Pi*csgn(I*c)*cs gn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))+I*b*Pi*csgn(I*c)*csgn(I*c*exp(n* ln(x)))^2+I*b*Pi*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2-I*b*Pi*csgn (I*c*exp(n*ln(x)))^3)/d+1/2/b^2/n^2*exp(-1/4*I*(b*Pi*csgn(I*c)*csgn(I*exp( n*ln(x)))*csgn(I*c*exp(n*ln(x)))-b*Pi*csgn(I*c)*csgn(I*c*exp(n*ln(x)))^2-b *Pi*csgn(I*exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2+b*Pi*csgn(I*c*exp(n*ln(x )))^3+2*I*b*n*(ln(x)-ln(d*x))+2*I*b*ln(c)+2*I*b*(ln(exp(n*ln(x)))-n*ln(x)) +2*I*a)/b/n)*Ei(1,1/2*ln(d*x)-1/4*I*(b*Pi*csgn(I*c)*csgn(I*exp(n*ln(x)))*c sgn(I*c*exp(n*ln(x)))-b*Pi*csgn(I*c)*csgn(I*c*exp(n*ln(x)))^2-b*Pi*csgn(I* exp(n*ln(x)))*csgn(I*c*exp(n*ln(x)))^2+b*Pi*csgn(I*c*exp(n*ln(x)))^3+2*I*b *n*(ln(x)-ln(d*x))+2*I*b*ln(c)+2*I*b*(ln(exp(n*ln(x)))-n*ln(x))+2*I*a)/b/n )/d
\[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
\[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
-4*b*n*integrate(1/((b^3*d^(3/2)*log(c)^3 + b^3*d^(3/2)*log(x^n)^3 + 3*a*b ^2*d^(3/2)*log(c)^2 + 3*a^2*b*d^(3/2)*log(c) + a^3*d^(3/2) + 3*(b^3*d^(3/2 )*log(c) + a*b^2*d^(3/2))*log(x^n)^2 + 3*(b^3*d^(3/2)*log(c)^2 + 2*a*b^2*d ^(3/2)*log(c) + a^2*b*d^(3/2))*log(x^n))*x^(3/2)), x) - 2/((b^2*d^(3/2)*lo g(c)^2 + b^2*d^(3/2)*log(x^n)^2 + 2*a*b*d^(3/2)*log(c) + a^2*d^(3/2) + 2*( b^2*d^(3/2)*log(c) + a*b*d^(3/2))*log(x^n))*sqrt(x))
Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (85) = 170\).
Time = 0.34 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.90 \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {\frac {b c^{\frac {1}{2 \, n}} n {\rm Ei}\left (-\frac {\log \left (c\right )}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \left (x\right )\right ) e^{\left (\frac {a}{2 \, b n}\right )} \log \left (x\right )}{b^{3} \sqrt {d} n^{3} \log \left (x\right ) + b^{3} \sqrt {d} n^{2} \log \left (c\right ) + a b^{2} \sqrt {d} n^{2}} + \frac {b c^{\frac {1}{2 \, n}} {\rm Ei}\left (-\frac {\log \left (c\right )}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \left (x\right )\right ) e^{\left (\frac {a}{2 \, b n}\right )} \log \left (c\right )}{b^{3} \sqrt {d} n^{3} \log \left (x\right ) + b^{3} \sqrt {d} n^{2} \log \left (c\right ) + a b^{2} \sqrt {d} n^{2}} + \frac {a c^{\frac {1}{2 \, n}} {\rm Ei}\left (-\frac {\log \left (c\right )}{2 \, n} - \frac {a}{2 \, b n} - \frac {1}{2} \, \log \left (x\right )\right ) e^{\left (\frac {a}{2 \, b n}\right )}}{b^{3} \sqrt {d} n^{3} \log \left (x\right ) + b^{3} \sqrt {d} n^{2} \log \left (c\right ) + a b^{2} \sqrt {d} n^{2}} + \frac {2 \, b n}{{\left (b^{3} \sqrt {d} n^{3} \log \left (x\right ) + b^{3} \sqrt {d} n^{2} \log \left (c\right ) + a b^{2} \sqrt {d} n^{2}\right )} \sqrt {x}}}{2 \, d} \]
-1/2*(b*c^(1/2/n)*n*Ei(-1/2*log(c)/n - 1/2*a/(b*n) - 1/2*log(x))*e^(1/2*a/ (b*n))*log(x)/(b^3*sqrt(d)*n^3*log(x) + b^3*sqrt(d)*n^2*log(c) + a*b^2*sqr t(d)*n^2) + b*c^(1/2/n)*Ei(-1/2*log(c)/n - 1/2*a/(b*n) - 1/2*log(x))*e^(1/ 2*a/(b*n))*log(c)/(b^3*sqrt(d)*n^3*log(x) + b^3*sqrt(d)*n^2*log(c) + a*b^2 *sqrt(d)*n^2) + a*c^(1/2/n)*Ei(-1/2*log(c)/n - 1/2*a/(b*n) - 1/2*log(x))*e ^(1/2*a/(b*n))/(b^3*sqrt(d)*n^3*log(x) + b^3*sqrt(d)*n^2*log(c) + a*b^2*sq rt(d)*n^2) + 2*b*n/((b^3*sqrt(d)*n^3*log(x) + b^3*sqrt(d)*n^2*log(c) + a*b ^2*sqrt(d)*n^2)*sqrt(x)))/d
Timed out. \[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{{\left (d\,x\right )}^{3/2}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]