\(\int \frac {a+b \log (c x^n)}{(d+\frac {e}{x}) x^2} \, dx\) [334]

Optimal result

Integrand size = 23, antiderivative size = 44 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^2} \, dx=-\frac {\log \left (1+\frac {e}{d x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e}{d x}\right )}{e} \]

[Out]

-ln(1+e/d/x)*(a+b*ln(c*x^n))/e+b*n*polylog(2,-e/d/x)/e

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2375, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^2} \, dx=\frac {b n \operatorname {PolyLog}\left (2,-\frac {e}{d x}\right )}{e}-\frac {\log \left (\frac {e}{d x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e} \]

[In]

Int[(a + b*Log[c*x^n])/((d + e/x)*x^2),x]

[Out]

-((Log[1 + e/(d*x)]*(a + b*Log[c*x^n]))/e) + (b*n*PolyLog[2, -(e/(d*x))])/e

Rule 2375

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c*x^n])^p/(e*r)), x] - Dist[b*f^m*n*(p/(e*r)), Int[Log[1 + e*(x^r/d)]*((
a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2438

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (1+\frac {e}{d x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {(b n) \int \frac {\log \left (1+\frac {e}{d x}\right )}{x} \, dx}{e} \\ & = -\frac {\log \left (1+\frac {e}{d x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \text {Li}_2\left (-\frac {e}{d x}\right )}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.43 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^2} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (1+\frac {d x}{e}\right )\right )}{2 b e n}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{e} \]

[In]

Integrate[(a + b*Log[c*x^n])/((d + e/x)*x^2),x]

[Out]

((a + b*Log[c*x^n])*(a + b*Log[c*x^n] - 2*b*n*Log[1 + (d*x)/e]))/(2*b*e*n) - (b*n*PolyLog[2, -((d*x)/e)])/e

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 4.11

method	result	size
risch	\(-\frac {b \ln \left (x^{n}\right ) \ln \left (d x +e \right )}{e}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{e}-\frac {b n \ln \left (x \right )^{2}}{2 e}+\frac {b n \ln \left (d x +e \right ) \ln \left (-\frac {d x}{e}\right )}{e}+\frac {b n \operatorname {dilog}\left (-\frac {d x}{e}\right )}{e}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (d x +e \right )}{e}+\frac {\ln \left (x \right )}{e}\right )\)	\(181\)

[In]

int((a+b*ln(c*x^n))/(d+e/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

-b*ln(x^n)/e*ln(d*x+e)+b*ln(x^n)/e*ln(x)-1/2*b*n/e*ln(x)^2+b*n/e*ln(d*x+e)*ln(-d*x/e)+b*n/e*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)*(-1/e*ln(d*x+e)+1/e*ln(x))

Fricas [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{2}} \,d x } \]

[In]

integrate((a+b*log(c*x^n))/(d+e/x)/x^2,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(d*x^2 + e*x), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 3.93 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^2} \, dx=\frac {2 a d \left (\begin {cases} - \frac {x}{e} - \frac {1}{2 d} & \text {for}\: d = 0 \\\frac {\log {\left (2 d x \right )}}{2 d} & \text {otherwise} \end {cases}\right )}{e} - \frac {2 a d \left (\begin {cases} \frac {x}{e} + \frac {1}{2 d} & \text {for}\: d = 0 \\\frac {\log {\left (2 d x + 2 e \right )}}{2 d} & \text {otherwise} \end {cases}\right )}{e} + b n \left (\begin {cases} - \frac {1}{d x} & \text {for}\: e = 0 \\\frac {\begin {cases} \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right ) - b \left (\begin {cases} \frac {1}{d x} & \text {for}\: e = 0 \\\frac {\log {\left (d + \frac {e}{x} \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]

[In]

integrate((a+b*ln(c*x**n))/(d+e/x)/x**2,x)

[Out]

2*a*d*Piecewise((-x/e - 1/(2*d), Eq(d, 0)), (log(2*d*x)/(2*d), True))/e - 2*a*d*Piecewise((x/e + 1/(2*d), Eq(d
, 0)), (log(2*d*x + 2*e)/(2*d), True))/e + b*n*Piecewise((-1/(d*x), Eq(e, 0)), (Piecewise((polylog(2, e*exp_po
lar(I*pi)/(d*x)), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log(x) + polylog(2, e*exp_polar(I*pi)/(d*x)), Abs(x)
< 1), (-log(d)*log(1/x) + polylog(2, e*exp_polar(I*pi)/(d*x)), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0)
, ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) + polylog(2, e*exp_polar(I*pi)/(d*x)), True))
/e, True)) - b*Piecewise((1/(d*x), Eq(e, 0)), (log(d + e/x)/e, True))*log(c*x**n)

Maxima [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{2}} \,d x } \]

[In]

integrate((a+b*log(c*x^n))/(d+e/x)/x^2,x, algorithm="maxima")

[Out]

-a*(log(d*x + e)/e - log(x)/e) + b*integrate((log(c) + log(x^n))/(d*x^2 + e*x), x)

Giac [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^2} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{2}} \,d x } \]

[In]

integrate((a+b*log(c*x^n))/(d+e/x)/x^2,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((d + e/x)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x^2} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,\left (d+\frac {e}{x}\right )} \,d x \]

[In]

int((a + b*log(c*x^n))/(x^2*(d + e/x)),x)

[Out]

int((a + b*log(c*x^n))/(x^2*(d + e/x)), x)