Integrand size = 23, antiderivative size = 135 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^4} \, dx=-\frac {b n}{4 e x^2}+\frac {b d n}{e^2 x}-\frac {a+b \log \left (c x^n\right )}{2 e x^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )}{e^2 x}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^3 n}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{e^3}-\frac {b d^2 n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{e^3} \]
-1/4*b*n/e/x^2+b*d*n/e^2/x+1/2*(-a-b*ln(c*x^n))/e/x^2+d*(a+b*ln(c*x^n))/e^ 2/x+1/2*d^2*(a+b*ln(c*x^n))^2/b/e^3/n-d^2*(a+b*ln(c*x^n))*ln(1+d*x/e)/e^3- b*d^2*n*polylog(2,-d*x/e)/e^3
Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^4} \, dx=-\frac {\frac {b e^2 n}{x^2}-\frac {4 b d e n}{x}+\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac {4 d e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+4 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )+4 b d^2 n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{4 e^3} \]
-1/4*((b*e^2*n)/x^2 - (4*b*d*e*n)/x + (2*e^2*(a + b*Log[c*x^n]))/x^2 - (4* d*e*(a + b*Log[c*x^n]))/x - (2*d^2*(a + b*Log[c*x^n])^2)/(b*n) + 4*d^2*(a + b*Log[c*x^n])*Log[1 + (d*x)/e] + 4*b*d^2*n*PolyLog[2, -((d*x)/e)])/e^3
Time = 0.60 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2005, 2780, 2741, 2780, 2741, 2779, 2838}
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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+\frac {e}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^3 (d x+e)}dx\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x^3}dx}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{x^2 (e+d x)}dx}{e}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{x^2 (e+d x)}dx}{e}\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}}{e}-\frac {d \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x^2}dx}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{x (e+d x)}dx}{e}\right )}{e}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}}{e}-\frac {d \left (\frac {-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{x (e+d x)}dx}{e}\right )}{e}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}}{e}-\frac {d \left (\frac {-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}}{e}-\frac {d \left (\frac {b n \int \frac {\log \left (\frac {e}{d x}+1\right )}{x}dx}{e}-\frac {\log \left (\frac {e}{d x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}\right )}{e}\right )}{e}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{2 x^2}-\frac {b n}{4 x^2}}{e}-\frac {d \left (\frac {-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}}{e}-\frac {d \left (\frac {b n \operatorname {PolyLog}\left (2,-\frac {e}{d x}\right )}{e}-\frac {\log \left (\frac {e}{d x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}\right )}{e}\right )}{e}\) |
(-1/4*(b*n)/x^2 - (a + b*Log[c*x^n])/(2*x^2))/e - (d*((-((b*n)/x) - (a + b *Log[c*x^n])/x)/e - (d*(-((Log[1 + e/(d*x)]*(a + b*Log[c*x^n]))/e) + (b*n* PolyLog[2, -(e/(d*x))])/e))/e))/e
3.4.36.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
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_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.))/((d_) + (e_.)* (x_)^(r_.)), x_Symbol] :> Simp[1/d Int[x^m*(a + b*Log[c*x^n])^p, x], x] - Simp[e/d Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /; Fre eQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -1]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.96
| method | result | size |
| risch | \(-\frac {b \ln \left (x^{n}\right ) d^{2} \ln \left (d x +e \right )}{e^{3}}-\frac {b \ln \left (x^{n}\right )}{2 e \,x^{2}}+\frac {b \ln \left (x^{n}\right ) d^{2} \ln \left (x \right )}{e^{3}}+\frac {b \ln \left (x^{n}\right ) d}{e^{2} x}+\frac {b d n}{e^{2} x}-\frac {b n}{4 e \,x^{2}}-\frac {b n \,d^{2} \ln \left (x \right )^{2}}{2 e^{3}}+\frac {b n \,d^{2} \ln \left (d x +e \right ) \ln \left (-\frac {d x}{e}\right )}{e^{3}}+\frac {b n \,d^{2} \operatorname {dilog}\left (-\frac {d x}{e}\right )}{e^{3}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {d^{2} \ln \left (d x +e \right )}{e^{3}}-\frac {1}{2 e \,x^{2}}+\frac {d^{2} \ln \left (x \right )}{e^{3}}+\frac {d}{e^{2} x}\right )\) | \(264\) |
-b*ln(x^n)*d^2/e^3*ln(d*x+e)-1/2*b*ln(x^n)/e/x^2+b*ln(x^n)*d^2/e^3*ln(x)+b *ln(x^n)*d/e^2/x+b*d*n/e^2/x-1/4*b*n/e/x^2-1/2*b*n*d^2/e^3*ln(x)^2+b*n*d^2 /e^3*ln(d*x+e)*ln(-d*x/e)+b*n*d^2/e^3*dilog(-d*x/e)+(-1/2*I*b*Pi*csgn(I*c) *csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b*Pi *csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*b*Pi*csgn(I*c*x^n)^3+b*ln(c)+a)*(-d^2/e ^3*ln(d*x+e)-1/2/e/x^2+d^2/e^3*ln(x)+d/e^2/x)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^4} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{4}} \,d x } \]
Time = 42.97 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.96 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^4} \, dx=- \frac {a d^{3} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{e^{3}} + \frac {a d^{2} \log {\left (x \right )}}{e^{3}} + \frac {a d}{e^{2} x} - \frac {a}{2 e x^{2}} + \frac {b d^{3} n \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{e^{3}} - \frac {b d^{3} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{3}} - \frac {b d^{2} n \log {\left (x \right )}^{2}}{2 e^{3}} + \frac {b d^{2} \log {\left (x \right )} \log {\left (c x^{n} \right )}}{e^{3}} + \frac {b d n}{e^{2} x} + \frac {b d \log {\left (c x^{n} \right )}}{e^{2} x} - \frac {b n}{4 e x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 e x^{2}} \]
-a*d**3*Piecewise((x/e, Eq(d, 0)), (log(d*x + e)/d, True))/e**3 + a*d**2*l og(x)/e**3 + a*d/(e**2*x) - a/(2*e*x**2) + b*d**3*n*Piecewise((x/e, Eq(d, 0)), (Piecewise((-polylog(2, d*x*exp_polar(I*pi)/e), (Abs(x) < 1) & (1/Abs (x) < 1)), (log(e)*log(x) - polylog(2, d*x*exp_polar(I*pi)/e), Abs(x) < 1) , (-log(e)*log(1/x) - polylog(2, d*x*exp_polar(I*pi)/e), 1/Abs(x) < 1), (- meijerg(((), (1, 1)), ((0, 0), ()), x)*log(e) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(e) - polylog(2, d*x*exp_polar(I*pi)/e), True))/d, True))/ e**3 - b*d**3*Piecewise((x/e, Eq(d, 0)), (log(d*x + e)/d, True))*log(c*x** n)/e**3 - b*d**2*n*log(x)**2/(2*e**3) + b*d**2*log(x)*log(c*x**n)/e**3 + b *d*n/(e**2*x) + b*d*log(c*x**n)/(e**2*x) - b*n/(4*e*x**2) - b*log(c*x**n)/ (2*e*x**2)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^4} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{4}} \,d x } \]
-1/2*a*(2*d^2*log(d*x + e)/e^3 - 2*d^2*log(x)/e^3 - (2*d*x - e)/(e^2*x^2)) + b*integrate((log(c) + log(x^n))/(d*x^4 + e*x^3), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^4} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^4} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,\left (d+\frac {e}{x}\right )} \,d x \]