Integrand size = 21, antiderivative size = 74 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=-\frac {b n}{d x}-\frac {a+b \log \left (c x^n\right )}{d x}+\frac {e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2} \]
Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=-\frac {\frac {2 b d n}{x}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{b n}-2 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-2 b e n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{2 d^2} \]
-1/2*((2*b*d*n)/x + (2*d*(a + b*Log[c*x^n]))/x + (e*(a + b*Log[c*x^n])^2)/ (b*n) - 2*e*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] - 2*b*e*n*PolyLog[2, -((e* x)/d)])/d^2
Time = 0.40 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {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.
\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx\) |
\(\Big \downarrow \) 2780 |
\(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x^2}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}\) |
\(\Big \downarrow \) 2741 |
\(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}\) |
\(\Big \downarrow \) 2779 |
\(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}}{d}-\frac {e \left (\frac {b n \int \frac {\log \left (\frac {d}{e x}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}\right )}{d}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {-\frac {a+b \log \left (c x^n\right )}{x}-\frac {b n}{x}}{d}-\frac {e \left (\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}\right )}{d}\) |
(-((b*n)/x) - (a + b*Log[c*x^n])/x)/d - (e*(-((Log[1 + d/(e*x)]*(a + b*Log [c*x^n]))/d) + (b*n*PolyLog[2, -(d/(e*x))])/d))/d
3.1.36.3.1 Defintions of rubi rules used
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 = 0.76 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.99
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) e \ln \left (e x +d \right )}{d^{2}}-\frac {b \ln \left (x^{n}\right )}{d x}-\frac {b \ln \left (x^{n}\right ) e \ln \left (x \right )}{d^{2}}-\frac {b n e \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{2}}-\frac {b n e \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{2}}-\frac {b n}{d x}+\frac {b n e \ln \left (x \right )^{2}}{2 d^{2}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )\) | \(221\) |
b*ln(x^n)*e/d^2*ln(e*x+d)-b*ln(x^n)/d/x-b*ln(x^n)*e/d^2*ln(x)-b*n*e/d^2*ln (e*x+d)*ln(-e*x/d)-b*n*e/d^2*dilog(-e*x/d)-b*n/d/x+1/2*b*n*e/d^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)*(e/d^2*ln(e*x+d)-1/d/x-e/d^2*ln(x))
\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
Time = 32.45 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.92 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=- \frac {a}{d x} + \frac {a e^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{2}} - \frac {a e \log {\left (x \right )}}{d^{2}} - \frac {b n}{d x} - \frac {b \log {\left (c x^{n} \right )}}{d x} - \frac {b e^{2} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {b e^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{2}} + \frac {b e n \log {\left (x \right )}^{2}}{2 d^{2}} - \frac {b e \log {\left (x \right )} \log {\left (c x^{n} \right )}}{d^{2}} \]
-a/(d*x) + a*e**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/d**2 - a*e*log(x)/d**2 - b*n/(d*x) - b*log(c*x**n)/(d*x) - b*e**2*n*Piecewise(( x/d, Eq(e, 0)), (Piecewise((-polylog(2, e*x*exp_polar(I*pi)/d), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log(x) - polylog(2, e*x*exp_polar(I*pi)/d), Abs(x) < 1), (-log(d)*log(1/x) - polylog(2, e*x*exp_polar(I*pi)/d), 1/Abs( x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((1, 1) , ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x*exp_polar(I*pi)/d), True)) /e, True))/d**2 + b*e**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True) )*log(c*x**n)/d**2 + b*e*n*log(x)**2/(2*d**2) - b*e*log(x)*log(c*x**n)/d** 2
\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
a*(e*log(e*x + d)/d^2 - e*log(x)/d^2 - 1/(d*x)) + b*integrate((log(c) + lo g(x^n))/(e*x^3 + d*x^2), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,\left (d+e\,x\right )} \,d x \]