Integrand size = 20, antiderivative size = 98 \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{2 b n}} \sqrt {d x} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
1/2*Ei(1/2*(a+b*ln(c*x^n))/b/n)*(d*x)^(1/2)/b^2/d/exp(1/2*a/b/n)/n^2/((c*x ^n)^(1/2/n))-(d*x)^(1/2)/b/d/n/(a+b*ln(c*x^n))
Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\frac {x \left (e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )-\frac {2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2 \sqrt {d x}} \]
(x*(ExpIntegralEi[(a + b*Log[c*x^n])/(2*b*n)]/(E^(a/(2*b*n))*(c*x^n)^(1/(2 *n))) - (2*b*n)/(a + b*Log[c*x^n])))/(2*b^2*n^2*Sqrt[d*x])
Time = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx\) |
| \(\Big \downarrow \) 2743 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}dx}{2 b n}-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )}\) |
| \(\Big \downarrow \) 2747 |
| \(\displaystyle \frac {\sqrt {d x} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \int \frac {\left (c x^n\right )^{\left .\frac {1}{2}\right /n}}{a+b \log \left (c x^n\right )}d\log \left (c x^n\right )}{2 b d n^2}-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle \frac {\sqrt {d x} e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac {\sqrt {d x}}{b d n \left (a+b \log \left (c x^n\right )\right )}\) |
(Sqrt[d*x]*ExpIntegralEi[(a + b*Log[c*x^n])/(2*b*n)])/(2*b^2*d*E^(a/(2*b*n ))*n^2*(c*x^n)^(1/(2*n))) - Sqrt[d*x]/(b*d*n*(a + b*Log[c*x^n]))
3.2.10.3.1 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 = 0.82 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.36
| method | result | size |
| risch | \(-\frac {2 x}{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 )}-\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 d \,b^{2} n^{2}}\) | \(427\) |
-2/b/n*x/(d*x)^(1/2)/(2*a+2*b*ln(c)+2*b*ln(exp(n*ln(x)))-I*b*Pi*csgn(I*c)* csgn(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*cs gn(I*c*exp(n*ln(x)))^3)-1/2/d/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))) *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)
\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\sqrt {d x} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
4*b*n*integrate(1/((b^3*sqrt(d)*log(c)^3 + b^3*sqrt(d)*log(x^n)^3 + 3*a*b^ 2*sqrt(d)*log(c)^2 + 3*a^2*b*sqrt(d)*log(c) + a^3*sqrt(d) + 3*(b^3*sqrt(d) *log(c) + a*b^2*sqrt(d))*log(x^n)^2 + 3*(b^3*sqrt(d)*log(c)^2 + 2*a*b^2*sq rt(d)*log(c) + a^2*b*sqrt(d))*log(x^n))*sqrt(x)), x) + 2*sqrt(x)/(b^2*sqrt (d)*log(c)^2 + b^2*sqrt(d)*log(x^n)^2 + 2*a*b*sqrt(d)*log(c) + a^2*sqrt(d) + 2*(b^2*sqrt(d)*log(c) + a*b*sqrt(d))*log(x^n))
\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\int \frac {1}{\sqrt {d\,x}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]