Integrand size = 16, antiderivative size = 52 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {b^2 n^2}{4 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2} \]
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{4 x^2} \]
Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2742, 2741}
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.
\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx\) |
\(\Big \downarrow \) 2742 |
\(\displaystyle b n \int \frac {a+b \log \left (c x^n\right )}{x^3}dx-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}\) |
\(\Big \downarrow \) 2741 |
\(\displaystyle b n \left (-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}\right )-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}\) |
3.1.56.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* (p/(m + 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] && GtQ[p, 0]
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13
method | result | size |
parallelrisch | \(-\frac {2 b^{2} \ln \left (c \,x^{n}\right )^{2}+2 \ln \left (c \,x^{n}\right ) b^{2} n +b^{2} n^{2}+4 a b \ln \left (c \,x^{n}\right )+2 a b n +2 a^{2}}{4 x^{2}}\) | \(59\) |
risch | \(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{2 x^{2}}-\frac {\left (-i \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b^{2} \ln \left (c \right )+b^{2} n +2 a b \right ) \ln \left (x^{n}\right )}{2 x^{2}}-\frac {4 a^{2}-2 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-\pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-2 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-4 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-4 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+2 b^{2} n^{2}+8 \ln \left (c \right ) a b +4 \ln \left (c \right )^{2} b^{2}+4 b^{2} \ln \left (c \right ) n +4 a b n -\pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+4 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+4 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{8 x^{2}}\) | \(703\) |
Time = 0.32 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {2 \, b^{2} n^{2} \log \left (x\right )^{2} + b^{2} n^{2} + 2 \, b^{2} \log \left (c\right )^{2} + 2 \, a b n + 2 \, a^{2} + 2 \, {\left (b^{2} n + 2 \, a b\right )} \log \left (c\right ) + 2 \, {\left (b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, a b n\right )} \log \left (x\right )}{4 \, x^{2}} \]
-1/4*(2*b^2*n^2*log(x)^2 + b^2*n^2 + 2*b^2*log(c)^2 + 2*a*b*n + 2*a^2 + 2* (b^2*n + 2*a*b)*log(c) + 2*(b^2*n^2 + 2*b^2*n*log(c) + 2*a*b*n)*log(x))/x^ 2
Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=- \frac {a^{2}}{2 x^{2}} - \frac {a b n}{2 x^{2}} - \frac {a b \log {\left (c x^{n} \right )}}{x^{2}} - \frac {b^{2} n^{2}}{4 x^{2}} - \frac {b^{2} n \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {b^{2} \log {\left (c x^{n} \right )}^{2}}{2 x^{2}} \]
-a**2/(2*x**2) - a*b*n/(2*x**2) - a*b*log(c*x**n)/x**2 - b**2*n**2/(4*x**2 ) - b**2*n*log(c*x**n)/(2*x**2) - b**2*log(c*x**n)**2/(2*x**2)
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {1}{4} \, b^{2} {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {b^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b n}{2 \, x^{2}} - \frac {a b \log \left (c x^{n}\right )}{x^{2}} - \frac {a^{2}}{2 \, x^{2}} \]
-1/4*b^2*(n^2/x^2 + 2*n*log(c*x^n)/x^2) - 1/2*b^2*log(c*x^n)^2/x^2 - 1/2*a *b*n/x^2 - a*b*log(c*x^n)/x^2 - 1/2*a^2/x^2
Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {b^{2} n^{2} \log \left (x\right )^{2}}{2 \, x^{2}} - \frac {{\left (b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, a b n\right )} \log \left (x\right )}{2 \, x^{2}} - \frac {b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, b^{2} \log \left (c\right )^{2} + 2 \, a b n + 4 \, a b \log \left (c\right ) + 2 \, a^{2}}{4 \, x^{2}} \]
-1/2*b^2*n^2*log(x)^2/x^2 - 1/2*(b^2*n^2 + 2*b^2*n*log(c) + 2*a*b*n)*log(x )/x^2 - 1/4*(b^2*n^2 + 2*b^2*n*log(c) + 2*b^2*log(c)^2 + 2*a*b*n + 4*a*b*l og(c) + 2*a^2)/x^2
Time = 0.34 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {\frac {a^2}{2}+\frac {a\,b\,n}{2}+\frac {b^2\,n^2}{4}}{x^2}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {n\,b^2}{2}+a\,b\right )}{x^2}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2}{2\,x^2} \]