Integrand size = 24, antiderivative size = 72 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=-\frac {b e n r}{x}-\frac {e r \left (a+b n+b \log \left (c x^n\right )\right )}{x}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \] Output:
-b*e*n*r/x-e*r*(a+b*n+b*ln(c*x^n))/x-b*n*(d+e*ln(f*x^r))/x-(a+b*ln(c*x^n)) *(d+e*ln(f*x^r))/x
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=-\frac {a d+b d n+a e r+2 b e n r+e (a+b n) \log \left (f x^r\right )+b \log \left (c x^n\right ) \left (d+e r+e \log \left (f x^r\right )\right )}{x} \] Input:
Integrate[((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x^2,x]
Output:
-((a*d + b*d*n + a*e*r + 2*b*e*n*r + e*(a + b*n)*Log[f*x^r] + b*Log[c*x^n] *(d + e*r + e*Log[f*x^r]))/x)
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2813, 25, 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.
\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 2813 |
\(\displaystyle -e r \int -\frac {a+b n+b \log \left (c x^n\right )}{x^2}dx-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle e r \int \frac {a+b n+b \log \left (c x^n\right )}{x^2}dx-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{x}\) |
\(\Big \downarrow \) 2741 |
\(\displaystyle -\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}+e r \left (-\frac {a+b \log \left (c x^n\right )+b n}{x}-\frac {b n}{x}\right )-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{x}\) |
Input:
Int[((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x^2,x]
Output:
e*r*(-((b*n)/x) - (a + b*n + b*Log[c*x^n])/x) - (b*n*(d + e*Log[f*x^r]))/x - ((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x
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_.) + Log[(f_.)*(x_)^(r_ .)]*(e_.))*((g_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Simp[(d + e*Log[f*x^r]) u, x] - Simp[e*r Int[Simp lifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] && !(EqQ[p, 1] && EqQ[a, 0] && NeQ[d, 0])
Time = 0.89 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.44
method | result | size |
parallelrisch | \(-\frac {\ln \left (c \,x^{n}\right ) \ln \left (f \,x^{r}\right ) b e \,n^{3}+\ln \left (c \,x^{n}\right ) b e \,n^{3} r +\ln \left (f \,x^{r}\right ) b e \,n^{4}+2 b e \,n^{4} r +\ln \left (c \,x^{n}\right ) b d \,n^{3}+\ln \left (f \,x^{r}\right ) a e \,n^{3}+a e \,n^{3} r +b d \,n^{4}+a d \,n^{3}}{x \,n^{3}}\) | \(104\) |
risch | \(\text {Expression too large to display}\) | \(1443\) |
Input:
int((a+b*ln(c*x^n))*(d+e*ln(f*x^r))/x^2,x,method=_RETURNVERBOSE)
Output:
-1/x*(ln(c*x^n)*ln(f*x^r)*b*e*n^3+ln(c*x^n)*b*e*n^3*r+ln(f*x^r)*b*e*n^4+2* b*e*n^4*r+ln(c*x^n)*b*d*n^3+ln(f*x^r)*a*e*n^3+a*e*n^3*r+b*d*n^4+a*d*n^3)/n ^3
Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=-\frac {b e n r \log \left (x\right )^{2} + b d n + a d + {\left (2 \, b e n + a e\right )} r + {\left (b e r + b d\right )} \log \left (c\right ) + {\left (b e n + b e \log \left (c\right ) + a e\right )} \log \left (f\right ) + {\left (b e r \log \left (c\right ) + b e n \log \left (f\right ) + b d n + {\left (2 \, b e n + a e\right )} r\right )} \log \left (x\right )}{x} \] Input:
integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^2,x, algorithm="fricas")
Output:
-(b*e*n*r*log(x)^2 + b*d*n + a*d + (2*b*e*n + a*e)*r + (b*e*r + b*d)*log(c ) + (b*e*n + b*e*log(c) + a*e)*log(f) + (b*e*r*log(c) + b*e*n*log(f) + b*d *n + (2*b*e*n + a*e)*r)*log(x))/x
Time = 0.39 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=- \frac {a d}{x} - \frac {a e r}{x} - \frac {a e \log {\left (f x^{r} \right )}}{x} - \frac {b d n}{x} - \frac {b d \log {\left (c x^{n} \right )}}{x} - \frac {2 b e n r}{x} - \frac {b e n \log {\left (f x^{r} \right )}}{x} - \frac {b e r \log {\left (c x^{n} \right )}}{x} - \frac {b e \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{x} \] Input:
integrate((a+b*ln(c*x**n))*(d+e*ln(f*x**r))/x**2,x)
Output:
-a*d/x - a*e*r/x - a*e*log(f*x**r)/x - b*d*n/x - b*d*log(c*x**n)/x - 2*b*e *n*r/x - b*e*n*log(f*x**r)/x - b*e*r*log(c*x**n)/x - b*e*log(c*x**n)*log(f *x**r)/x
Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=-b e {\left (\frac {r}{x} + \frac {\log \left (f x^{r}\right )}{x}\right )} \log \left (c x^{n}\right ) - \frac {b e n {\left (2 \, r + \log \left (f\right ) + \log \left (x^{r}\right )\right )}}{x} - \frac {b d n}{x} - \frac {a e r}{x} - \frac {b d \log \left (c x^{n}\right )}{x} - \frac {a e \log \left (f x^{r}\right )}{x} - \frac {a d}{x} \] Input:
integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^2,x, algorithm="maxima")
Output:
-b*e*(r/x + log(f*x^r)/x)*log(c*x^n) - b*e*n*(2*r + log(f) + log(x^r))/x - b*d*n/x - a*e*r/x - b*d*log(c*x^n)/x - a*e*log(f*x^r)/x - a*d/x
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=-\frac {b e n r \log \left (x\right )^{2}}{x} - \frac {{\left (2 \, b e n r + b e r \log \left (c\right ) + b e n \log \left (f\right ) + b d n + a e r\right )} \log \left (x\right )}{x} - \frac {2 \, b e n r + b e r \log \left (c\right ) + b e n \log \left (f\right ) + b e \log \left (c\right ) \log \left (f\right ) + b d n + a e r + b d \log \left (c\right ) + a e \log \left (f\right ) + a d}{x} \] Input:
integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^2,x, algorithm="giac")
Output:
-b*e*n*r*log(x)^2/x - (2*b*e*n*r + b*e*r*log(c) + b*e*n*log(f) + b*d*n + a *e*r)*log(x)/x - (2*b*e*n*r + b*e*r*log(c) + b*e*n*log(f) + b*e*log(c)*log (f) + b*d*n + a*e*r + b*d*log(c) + a*e*log(f) + a*d)/x
Time = 25.73 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=-\ln \left (f\,x^r\right )\,\left (\frac {a\,e}{x}+\frac {b\,e\,n}{x}+\frac {b\,e\,\ln \left (c\,x^n\right )}{x}\right )-\frac {a\,d+b\,d\,n+a\,e\,r+2\,b\,e\,n\,r}{x}-\frac {b\,\ln \left (c\,x^n\right )\,\left (d+e\,r\right )}{x} \] Input:
int(((d + e*log(f*x^r))*(a + b*log(c*x^n)))/x^2,x)
Output:
- log(f*x^r)*((a*e)/x + (b*e*n)/x + (b*e*log(c*x^n))/x) - (a*d + b*d*n + a *e*r + 2*b*e*n*r)/x - (b*log(c*x^n)*(d + e*r))/x
Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx=\frac {-\mathrm {log}\left (x^{n} c \right ) \mathrm {log}\left (x^{r} f \right ) b e -\mathrm {log}\left (x^{n} c \right ) b d -\mathrm {log}\left (x^{n} c \right ) b e r -\mathrm {log}\left (x^{r} f \right ) a e -\mathrm {log}\left (x^{r} f \right ) b e n -a d -a e r -b d n -2 b e n r}{x} \] Input:
int((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^2,x)
Output:
( - log(x**n*c)*log(x**r*f)*b*e - log(x**n*c)*b*d - log(x**n*c)*b*e*r - lo g(x**r*f)*a*e - log(x**r*f)*b*e*n - a*d - a*e*r - b*d*n - 2*b*e*n*r)/x