Integrand size = 23, antiderivative size = 95 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^3} \, dx=-\frac {b n}{e x}-\frac {a+b \log \left (c x^n\right )}{e x}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^2 n}+\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{e^2}+\frac {b d n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{e^2} \]
-b*n/e/x+(-a-b*ln(c*x^n))/e/x-1/2*d*(a+b*ln(c*x^n))^2/b/e^2/n+d*(a+b*ln(c* x^n))*ln(1+d*x/e)/e^2+b*d*n*polylog(2,-d*x/e)/e^2
Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^3} \, dx=-\frac {\frac {2 b e n}{x}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{b n}-2 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )-2 b d n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{2 e^2} \]
-1/2*((2*b*e*n)/x + (2*e*(a + b*Log[c*x^n]))/x + (d*(a + b*Log[c*x^n])^2)/ (b*n) - 2*d*(a + b*Log[c*x^n])*Log[1 + (d*x)/e] - 2*b*d*n*PolyLog[2, -((d* x)/e)])/e^2
Time = 0.42 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2005, 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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+\frac {e}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^2 (d x+e)}dx\) |
| \(\Big \downarrow \) 2780 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x^2}dx}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{x (e+d x)}dx}{e}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{x (e+d x)}dx}{e}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}}{e}-\frac {d \left (\frac {b n \int \frac {\log \left (\frac {e}{d x}+1\right )}{x}dx}{e}-\frac {\log \left (\frac {e}{d x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}\right )}{e}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}}{e}-\frac {d \left (\frac {b n \operatorname {PolyLog}\left (2,-\frac {e}{d x}\right )}{e}-\frac {\log \left (\frac {e}{d x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}\right )}{e}\) |
(-((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
3.4.35.3.1 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 = 0.13 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.33
| method | result | size |
| risch | \(\frac {b \ln \left (x^{n}\right ) d \ln \left (d x +e \right )}{e^{2}}-\frac {b \ln \left (x^{n}\right )}{e x}-\frac {b \ln \left (x^{n}\right ) d \ln \left (x \right )}{e^{2}}-\frac {b n d \ln \left (d x +e \right ) \ln \left (-\frac {d x}{e}\right )}{e^{2}}-\frac {b n d \operatorname {dilog}\left (-\frac {d x}{e}\right )}{e^{2}}-\frac {b n}{e x}+\frac {b n d \ln \left (x \right )^{2}}{2 e^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {d \ln \left (d x +e \right )}{e^{2}}-\frac {1}{e x}-\frac {d \ln \left (x \right )}{e^{2}}\right )\) | \(221\) |
b*ln(x^n)*d/e^2*ln(d*x+e)-b*ln(x^n)/e/x-b*ln(x^n)*d/e^2*ln(x)-b*n*d/e^2*ln (d*x+e)*ln(-d*x/e)-b*n*d/e^2*dilog(-d*x/e)-b*n/e/x+1/2*b*n*d/e^2*ln(x)^2+( -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/e^2*ln(d*x+e)-1/e/x-d/e^2*ln(x))
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{3}} \,d x } \]
Time = 37.15 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.27 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^3} \, dx=\frac {a d^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {a d \log {\left (x \right )}}{e^{2}} - \frac {a}{e x} - \frac {b d^{2} n \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {b d^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e^{2}} + \frac {b d n \log {\left (x \right )}^{2}}{2 e^{2}} - \frac {b d \log {\left (x \right )} \log {\left (c x^{n} \right )}}{e^{2}} - \frac {b n}{e x} - \frac {b \log {\left (c x^{n} \right )}}{e x} \]
a*d**2*Piecewise((x/e, Eq(d, 0)), (log(d*x + e)/d, True))/e**2 - a*d*log(x )/e**2 - a/(e*x) - b*d**2*n*Piecewise((x/e, Eq(d, 0)), (Piecewise((-polylo g(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) - po lylog(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) - po lylog(2, d*x*exp_polar(I*pi)/e), True))/d, True))/e**2 + b*d**2*Piecewise( (x/e, Eq(d, 0)), (log(d*x + e)/d, True))*log(c*x**n)/e**2 + b*d*n*log(x)** 2/(2*e**2) - b*d*log(x)*log(c*x**n)/e**2 - b*n/(e*x) - b*log(c*x**n)/(e*x)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{3}} \,d x } \]
a*(d*log(d*x + e)/e^2 - d*log(x)/e^2 - 1/(e*x)) + b*integrate((log(c) + lo g(x^n))/(d*x^3 + e*x^2), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^3} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,\left (d+\frac {e}{x}\right )} \,d x \]