Integrand size = 24, antiderivative size = 83 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\frac {b e n r}{27 x^3}-\frac {e r \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{27 x^3}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3} \] Output:
-1/27*b*e*n*r/x^3-1/27*e*r*(3*a+b*n+3*b*ln(c*x^n))/x^3-1/9*b*n*(d+e*ln(f*x ^r))/x^3-1/3*(a+b*ln(c*x^n))*(d+e*ln(f*x^r))/x^3
Time = 0.10 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\frac {9 a d+3 b d n+3 a e r+2 b e n r+3 e (3 a+b n) \log \left (f x^r\right )+3 b \log \left (c x^n\right ) \left (3 d+e r+3 e \log \left (f x^r\right )\right )}{27 x^3} \] Input:
Integrate[((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x^4,x]
Output:
-1/27*(9*a*d + 3*b*d*n + 3*a*e*r + 2*b*e*n*r + 3*e*(3*a + b*n)*Log[f*x^r] + 3*b*Log[c*x^n]*(3*d + e*r + 3*e*Log[f*x^r]))/x^3
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2813, 27, 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.
\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 2813 |
\(\displaystyle -e r \int -\frac {3 a+b n+3 b \log \left (c x^n\right )}{9 x^4}dx-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} e r \int \frac {3 a+b n+3 b \log \left (c x^n\right )}{x^4}dx-\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}\) |
\(\Big \downarrow \) 2741 |
\(\displaystyle -\frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{3 x^3}+\frac {1}{9} e r \left (-\frac {3 a+3 b \log \left (c x^n\right )+b n}{3 x^3}-\frac {b n}{3 x^3}\right )-\frac {b n \left (d+e \log \left (f x^r\right )\right )}{9 x^3}\) |
Input:
Int[((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/x^4,x]
Output:
(e*r*(-1/3*(b*n)/x^3 - (3*a + b*n + 3*b*Log[c*x^n])/(3*x^3)))/9 - (b*n*(d + e*Log[f*x^r]))/(9*x^3) - ((a + b*Log[c*x^n])*(d + e*Log[f*x^r]))/(3*x^3)
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_.))*((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 = 0.90 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.02
method | result | size |
parallelrisch | \(-\frac {9 \ln \left (c \,x^{n}\right ) \ln \left (f \,x^{r}\right ) b e +3 \ln \left (c \,x^{n}\right ) b e r +3 \ln \left (f \,x^{r}\right ) b e n +2 b e n r +9 \ln \left (c \,x^{n}\right ) b d +9 \ln \left (f \,x^{r}\right ) a e +3 a e r +3 b d n +9 d a}{27 x^{3}}\) | \(85\) |
risch | \(\text {Expression too large to display}\) | \(1451\) |
Input:
int((a+b*ln(c*x^n))*(d+e*ln(f*x^r))/x^4,x,method=_RETURNVERBOSE)
Output:
-1/27/x^3*(9*ln(c*x^n)*ln(f*x^r)*b*e+3*ln(c*x^n)*b*e*r+3*ln(f*x^r)*b*e*n+2 *b*e*n*r+9*ln(c*x^n)*b*d+9*ln(f*x^r)*a*e+3*a*e*r+3*b*d*n+9*d*a)
Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\frac {9 \, b e n r \log \left (x\right )^{2} + 3 \, b d n + 9 \, a d + {\left (2 \, b e n + 3 \, a e\right )} r + 3 \, {\left (b e r + 3 \, b d\right )} \log \left (c\right ) + 3 \, {\left (b e n + 3 \, b e \log \left (c\right ) + 3 \, a e\right )} \log \left (f\right ) + 3 \, {\left (3 \, b e r \log \left (c\right ) + 3 \, b e n \log \left (f\right ) + 3 \, b d n + {\left (2 \, b e n + 3 \, a e\right )} r\right )} \log \left (x\right )}{27 \, x^{3}} \] Input:
integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^4,x, algorithm="fricas")
Output:
-1/27*(9*b*e*n*r*log(x)^2 + 3*b*d*n + 9*a*d + (2*b*e*n + 3*a*e)*r + 3*(b*e *r + 3*b*d)*log(c) + 3*(b*e*n + 3*b*e*log(c) + 3*a*e)*log(f) + 3*(3*b*e*r* log(c) + 3*b*e*n*log(f) + 3*b*d*n + (2*b*e*n + 3*a*e)*r)*log(x))/x^3
Time = 1.88 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=- \frac {a d}{3 x^{3}} - \frac {a e r}{9 x^{3}} - \frac {a e \log {\left (f x^{r} \right )}}{3 x^{3}} - \frac {b d n}{9 x^{3}} - \frac {b d \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {2 b e n r}{27 x^{3}} - \frac {b e n \log {\left (f x^{r} \right )}}{9 x^{3}} - \frac {b e r \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {b e \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{3 x^{3}} \] Input:
integrate((a+b*ln(c*x**n))*(d+e*ln(f*x**r))/x**4,x)
Output:
-a*d/(3*x**3) - a*e*r/(9*x**3) - a*e*log(f*x**r)/(3*x**3) - b*d*n/(9*x**3) - b*d*log(c*x**n)/(3*x**3) - 2*b*e*n*r/(27*x**3) - b*e*n*log(f*x**r)/(9*x **3) - b*e*r*log(c*x**n)/(9*x**3) - b*e*log(c*x**n)*log(f*x**r)/(3*x**3)
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\frac {1}{9} \, b e {\left (\frac {r}{x^{3}} + \frac {3 \, \log \left (f x^{r}\right )}{x^{3}}\right )} \log \left (c x^{n}\right ) - \frac {b e n {\left (2 \, r + 3 \, \log \left (f\right ) + 3 \, \log \left (x^{r}\right )\right )}}{27 \, x^{3}} - \frac {b d n}{9 \, x^{3}} - \frac {a e r}{9 \, x^{3}} - \frac {b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a e \log \left (f x^{r}\right )}{3 \, x^{3}} - \frac {a d}{3 \, x^{3}} \] Input:
integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^4,x, algorithm="maxima")
Output:
-1/9*b*e*(r/x^3 + 3*log(f*x^r)/x^3)*log(c*x^n) - 1/27*b*e*n*(2*r + 3*log(f ) + 3*log(x^r))/x^3 - 1/9*b*d*n/x^3 - 1/9*a*e*r/x^3 - 1/3*b*d*log(c*x^n)/x ^3 - 1/3*a*e*log(f*x^r)/x^3 - 1/3*a*d/x^3
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\frac {b e n r \log \left (x\right )^{2}}{3 \, x^{3}} - \frac {{\left (2 \, b e n r + 3 \, b e r \log \left (c\right ) + 3 \, b e n \log \left (f\right ) + 3 \, b d n + 3 \, a e r\right )} \log \left (x\right )}{9 \, x^{3}} - \frac {2 \, b e n r + 3 \, b e r \log \left (c\right ) + 3 \, b e n \log \left (f\right ) + 9 \, b e \log \left (c\right ) \log \left (f\right ) + 3 \, b d n + 3 \, a e r + 9 \, b d \log \left (c\right ) + 9 \, a e \log \left (f\right ) + 9 \, a d}{27 \, x^{3}} \] Input:
integrate((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^4,x, algorithm="giac")
Output:
-1/3*b*e*n*r*log(x)^2/x^3 - 1/9*(2*b*e*n*r + 3*b*e*r*log(c) + 3*b*e*n*log( f) + 3*b*d*n + 3*a*e*r)*log(x)/x^3 - 1/27*(2*b*e*n*r + 3*b*e*r*log(c) + 3* b*e*n*log(f) + 9*b*e*log(c)*log(f) + 3*b*d*n + 3*a*e*r + 9*b*d*log(c) + 9* a*e*log(f) + 9*a*d)/x^3
Time = 25.51 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=-\ln \left (f\,x^r\right )\,\left (\frac {a\,e}{3\,x^3}+\frac {b\,e\,n}{9\,x^3}+\frac {b\,e\,\ln \left (c\,x^n\right )}{3\,x^3}\right )-\frac {\frac {a\,d}{3}+\frac {b\,d\,n}{9}+\frac {a\,e\,r}{9}+\frac {2\,b\,e\,n\,r}{27}}{x^3}-\frac {b\,\ln \left (c\,x^n\right )\,\left (3\,d+e\,r\right )}{9\,x^3} \] Input:
int(((d + e*log(f*x^r))*(a + b*log(c*x^n)))/x^4,x)
Output:
- log(f*x^r)*((a*e)/(3*x^3) + (b*e*n)/(9*x^3) + (b*e*log(c*x^n))/(3*x^3)) - ((a*d)/3 + (b*d*n)/9 + (a*e*r)/9 + (2*b*e*n*r)/27)/x^3 - (b*log(c*x^n)*( 3*d + e*r))/(9*x^3)
Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^4} \, dx=\frac {-9 \,\mathrm {log}\left (x^{n} c \right ) \mathrm {log}\left (x^{r} f \right ) b e -9 \,\mathrm {log}\left (x^{n} c \right ) b d -3 \,\mathrm {log}\left (x^{n} c \right ) b e r -9 \,\mathrm {log}\left (x^{r} f \right ) a e -3 \,\mathrm {log}\left (x^{r} f \right ) b e n -9 a d -3 a e r -3 b d n -2 b e n r}{27 x^{3}} \] Input:
int((a+b*log(c*x^n))*(d+e*log(f*x^r))/x^4,x)
Output:
( - 9*log(x**n*c)*log(x**r*f)*b*e - 9*log(x**n*c)*b*d - 3*log(x**n*c)*b*e* r - 9*log(x**r*f)*a*e - 3*log(x**r*f)*b*e*n - 9*a*d - 3*a*e*r - 3*b*d*n - 2*b*e*n*r)/(27*x**3)