Integrand size = 24, antiderivative size = 57 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \, dx=-\frac {e r \left (a+b \log \left (c x^n\right )\right )^3}{6 b^2 n^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n} \] Output:
-1/6*e*r*(a+b*ln(c*x^n))^3/b^2/n^2+1/2*(a+b*ln(c*x^n))^2*(d+e*ln(f*x^r))/b /n
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \, dx=\frac {1}{6} \log (x) \left (2 b e n r \log ^2(x)+6 \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )-3 \log (x) \left (b d n+a e r+b e r \log \left (c x^n\right )+b e n \log \left (f x^r\right )\right )\right ) \] Input:
Integrate[((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x,x]
Output:
(Log[x]*(2*b*e*n*r*Log[x]^2 + 6*(a + b*Log[c*x^n])*(d + e*Log[f*x^r]) - 3* Log[x]*(b*d*n + a*e*r + b*e*r*Log[c*x^n] + b*e*n*Log[f*x^r])))/6
Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2813, 27, 2739, 15}
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.
\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 2813 |
\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n}-e r \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n}-\frac {e r \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x}dx}{2 b n}\) |
\(\Big \downarrow \) 2739 |
\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n}-\frac {e r \int \left (a+b \log \left (c x^n\right )\right )^2d\left (a+b \log \left (c x^n\right )\right )}{2 b^2 n^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{2 b n}-\frac {e r \left (a+b \log \left (c x^n\right )\right )^3}{6 b^2 n^2}\) |
Input:
Int[((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x,x]
Output:
-1/6*(e*r*(a + b*Log[c*x^n])^3)/(b^2*n^2) + ((a + b*Log[c*x^n])^2*(d + e*L og[f*x^r]))/(2*b*n)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( b*n) Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} , x]
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 = 1.00 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.81
method | result | size |
parallelrisch | \(\frac {-\ln \left (c \,x^{n}\right )^{3} b e \,n^{2} r +3 \ln \left (c \,x^{n}\right )^{2} \ln \left (f \,x^{r}\right ) b e \,n^{3}+6 \ln \left (x \right ) a d \,n^{4}-3 \ln \left (c \,x^{n}\right )^{2} a e \,n^{2} r +3 \ln \left (c \,x^{n}\right )^{2} b d \,n^{3}+6 \ln \left (c \,x^{n}\right ) \ln \left (f \,x^{r}\right ) a e \,n^{3}}{6 n^{4}}\) | \(103\) |
risch | \(\text {Expression too large to display}\) | \(1597\) |
Input:
int((a+b*ln(c*x^n))*(d+e*ln(f*x^r))/x,x,method=_RETURNVERBOSE)
Output:
1/6*(-ln(c*x^n)^3*b*e*n^2*r+3*ln(c*x^n)^2*ln(f*x^r)*b*e*n^3+6*ln(x)*a*d*n^ 4-3*ln(c*x^n)^2*a*e*n^2*r+3*ln(c*x^n)^2*b*d*n^3+6*ln(c*x^n)*ln(f*x^r)*a*e* n^3)/n^4
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \, dx=\frac {1}{3} \, b e n r \log \left (x\right )^{3} + \frac {1}{2} \, {\left (b e r \log \left (c\right ) + b e n \log \left (f\right ) + b d n + a e r\right )} \log \left (x\right )^{2} + {\left (b d \log \left (c\right ) + a d + {\left (b e \log \left (c\right ) + a e\right )} \log \left (f\right )\right )} \log \left (x\right ) \] Input:
integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x,x, algorithm="fricas")
Output:
1/3*b*e*n*r*log(x)^3 + 1/2*(b*e*r*log(c) + b*e*n*log(f) + b*d*n + a*e*r)*l og(x)^2 + (b*d*log(c) + a*d + (b*e*log(c) + a*e)*log(f))*log(x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e \log {\left (f x^{r} \right )}\right )}{x}\, dx \] Input:
integrate((a+b*ln(c*x**n))*(d+e*ln(f*x**r))/x,x)
Output:
Integral((a + b*log(c*x**n))*(d + e*log(f*x**r))/x, x)
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \, dx=\frac {b e \log \left (c x^{n}\right ) \log \left (f x^{r}\right )^{2}}{2 \, r} - \frac {b e n \log \left (f x^{r}\right )^{3}}{6 \, r^{2}} + \frac {b d \log \left (c x^{n}\right )^{2}}{2 \, n} + \frac {a e \log \left (f x^{r}\right )^{2}}{2 \, r} + a d \log \left (x\right ) \] Input:
integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x,x, algorithm="maxima")
Output:
1/2*b*e*log(c*x^n)*log(f*x^r)^2/r - 1/6*b*e*n*log(f*x^r)^3/r^2 + 1/2*b*d*l og(c*x^n)^2/n + 1/2*a*e*log(f*x^r)^2/r + a*d*log(x)
Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \, dx=\frac {1}{3} \, b e n r \log \left (x\right )^{3} + \frac {1}{2} \, b e r \log \left (c\right ) \log \left (x\right )^{2} + \frac {1}{2} \, b e n \log \left (f\right ) \log \left (x\right )^{2} + b e \log \left (c\right ) \log \left (f\right ) \log \left (x\right ) + \frac {1}{2} \, b d n \log \left (x\right )^{2} + \frac {1}{2} \, a e r \log \left (x\right )^{2} + b d \log \left (c\right ) \log \left (x\right ) + a e \log \left (f\right ) \log \left (x\right ) + a d \log \left (x\right ) \] Input:
integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x,x, algorithm="giac")
Output:
1/3*b*e*n*r*log(x)^3 + 1/2*b*e*r*log(c)*log(x)^2 + 1/2*b*e*n*log(f)*log(x) ^2 + b*e*log(c)*log(f)*log(x) + 1/2*b*d*n*log(x)^2 + 1/2*a*e*r*log(x)^2 + b*d*log(c)*log(x) + a*e*log(f)*log(x) + a*d*log(x)
Time = 25.72 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \, dx=a\,d\,\ln \left (x\right )+\frac {b\,d\,{\ln \left (c\,x^n\right )}^2}{2\,n}+\frac {a\,e\,{\ln \left (f\,x^r\right )}^2}{2\,r}-\frac {b\,e\,r\,{\ln \left (c\,x^n\right )}^3}{6\,n^2}+\frac {b\,e\,{\ln \left (c\,x^n\right )}^2\,\ln \left (f\,x^r\right )}{2\,n} \] Input:
int(((d + e*log(f*x^r))*(a + b*log(c*x^n)))/x,x)
Output:
a*d*log(x) + (b*d*log(c*x^n)^2)/(2*n) + (a*e*log(f*x^r)^2)/(2*r) - (b*e*r* log(c*x^n)^3)/(6*n^2) + (b*e*log(c*x^n)^2*log(f*x^r))/(2*n)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x} \, dx=\frac {2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right ) \mathrm {log}\left (x^{r} f \right )}{x}d x \right ) b e n r +\mathrm {log}\left (x^{n} c \right )^{2} b d r +\mathrm {log}\left (x^{r} f \right )^{2} a e n +2 \,\mathrm {log}\left (x \right ) a d n r}{2 n r} \] Input:
int((a+b*log(c*x^n))*(d+e*log(f*x^r))/x,x)
Output:
(2*int((log(x**n*c)*log(x**r*f))/x,x)*b*e*n*r + log(x**n*c)**2*b*d*r + log (x**r*f)**2*a*e*n + 2*log(x)*a*d*n*r)/(2*n*r)