Integrand size = 23, antiderivative size = 115 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx=\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}-\frac {\log \left (1+\frac {d}{e x^2}\right ) \left (4 a-3 b n+4 b \log \left (c x^n\right )\right )}{8 d^3}+\frac {4 a-b n+4 b \log \left (c x^n\right )}{8 d^2 \left (d+e x^2\right )}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{4 d^3} \]
1/4*(a+b*ln(c*x^n))/d/(e*x^2+d)^2-1/8*ln(1+d/e/x^2)*(4*a-3*b*n+4*b*ln(c*x^ n))/d^3+1/8*(4*a-b*n+4*b*ln(c*x^n))/d^2/(e*x^2+d)+1/4*b*n*polylog(2,-d/e/x ^2)/d^3
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.44 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx=\frac {\frac {4 d^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2}+\frac {8 d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d+e x^2}+16 \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-8 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )-b n \left (\frac {d}{d-i \sqrt {d} \sqrt {e} x}+\frac {d}{d+i \sqrt {d} \sqrt {e} x}+2 \log (x)-\frac {d \log (x)}{\left (\sqrt {d}-i \sqrt {e} x\right )^2}-\frac {d \log (x)}{\left (\sqrt {d}+i \sqrt {e} x\right )^2}+\frac {5 \sqrt {e} x \log (x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {5 \sqrt {e} x \log (x)}{i \sqrt {d}+\sqrt {e} x}-8 \log ^2(x)-6 \log \left (i \sqrt {d}-\sqrt {e} x\right )-6 \log \left (i \sqrt {d}+\sqrt {e} x\right )+8 \log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+8 \log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+8 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+8 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )}{16 d^3} \]
((4*d^2*(a - b*n*Log[x] + b*Log[c*x^n]))/(d + e*x^2)^2 + (8*d*(a - b*n*Log [x] + b*Log[c*x^n]))/(d + e*x^2) + 16*Log[x]*(a - b*n*Log[x] + b*Log[c*x^n ]) - 8*(a - b*n*Log[x] + b*Log[c*x^n])*Log[d + e*x^2] - b*n*(d/(d - I*Sqrt [d]*Sqrt[e]*x) + d/(d + I*Sqrt[d]*Sqrt[e]*x) + 2*Log[x] - (d*Log[x])/(Sqrt [d] - I*Sqrt[e]*x)^2 - (d*Log[x])/(Sqrt[d] + I*Sqrt[e]*x)^2 + (5*Sqrt[e]*x *Log[x])/((-I)*Sqrt[d] + Sqrt[e]*x) + (5*Sqrt[e]*x*Log[x])/(I*Sqrt[d] + Sq rt[e]*x) - 8*Log[x]^2 - 6*Log[I*Sqrt[d] - Sqrt[e]*x] - 6*Log[I*Sqrt[d] + S qrt[e]*x] + 8*Log[x]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] + 8*Log[x]*Log[1 + (I* Sqrt[e]*x)/Sqrt[d]] + 8*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]] + 8*PolyLog[2 , (I*Sqrt[e]*x)/Sqrt[d]]))/(16*d^3)
Time = 0.49 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2785, 25, 2785, 27, 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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2785 |
| \(\displaystyle \frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}-\frac {\int -\frac {4 a-b n+4 b \log \left (c x^n\right )}{x \left (e x^2+d\right )^2}dx}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {4 a-b n+4 b \log \left (c x^n\right )}{x \left (e x^2+d\right )^2}dx}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 2785 |
| \(\displaystyle \frac {\frac {4 a+4 b \log \left (c x^n\right )-b n}{2 d \left (d+e x^2\right )}-\frac {\int -\frac {2 \left (4 a-3 b n+4 b \log \left (c x^n\right )\right )}{x \left (e x^2+d\right )}dx}{2 d}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {4 a-3 b n+4 b \log \left (c x^n\right )}{x \left (e x^2+d\right )}dx}{d}+\frac {4 a+4 b \log \left (c x^n\right )-b n}{2 d \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {\frac {\frac {2 b n \int \frac {\log \left (\frac {d}{e x^2}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x^2}+1\right ) \left (4 a+4 b \log \left (c x^n\right )-3 b n\right )}{2 d}}{d}+\frac {4 a+4 b \log \left (c x^n\right )-b n}{2 d \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{d}-\frac {\log \left (\frac {d}{e x^2}+1\right ) \left (4 a+4 b \log \left (c x^n\right )-3 b n\right )}{2 d}}{d}+\frac {4 a+4 b \log \left (c x^n\right )-b n}{2 d \left (d+e x^2\right )}}{4 d}+\frac {a+b \log \left (c x^n\right )}{4 d \left (d+e x^2\right )^2}\) |
(a + b*Log[c*x^n])/(4*d*(d + e*x^2)^2) + ((4*a - b*n + 4*b*Log[c*x^n])/(2* d*(d + e*x^2)) + (-1/2*(Log[1 + d/(e*x^2)]*(4*a - 3*b*n + 4*b*Log[c*x^n])) /d + (b*n*PolyLog[2, -(d/(e*x^2))])/d)/d)/(4*d)
3.3.34.3.1 Defintions of rubi rules used
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_.))^(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_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^2)^(q_.), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(q + 1)*((a + b*Log[c*x^n])/(2*d*f*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(f*x)^m*(d + e*x^2)^(q + 1)*(a*(m + 2*q + 3) + b*n + b*(m + 2*q + 3)*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && ILtQ[m, 0]
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 = 1.07 (sec) , antiderivative size = 390, normalized size of antiderivative = 3.39
| method | result | size |
| risch | \(-\frac {b \ln \left (x^{n}\right ) \ln \left (e \,x^{2}+d \right )}{2 d^{3}}+\frac {b \ln \left (x^{n}\right )}{2 d^{2} \left (e \,x^{2}+d \right )}+\frac {b \ln \left (x^{n}\right )}{4 d \left (e \,x^{2}+d \right )^{2}}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{3}}-\frac {b n \ln \left (x \right )^{2}}{2 d^{3}}+\frac {b n \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{3}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{3}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{3}}-\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{3}}-\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{3}}+\frac {3 b n \ln \left (e \,x^{2}+d \right )}{8 d^{3}}-\frac {b n}{8 d^{2} \left (e \,x^{2}+d \right )}-\frac {3 b n \ln \left (x \right )}{4 d^{3}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {e \left (\frac {\ln \left (e \,x^{2}+d \right )}{e}-\frac {d}{e \left (e \,x^{2}+d \right )}-\frac {d^{2}}{2 e \left (e \,x^{2}+d \right )^{2}}\right )}{2 d^{3}}+\frac {\ln \left (x \right )}{d^{3}}\right )\) | \(390\) |
-1/2*b*ln(x^n)/d^3*ln(e*x^2+d)+1/2*b*ln(x^n)/d^2/(e*x^2+d)+1/4*b*ln(x^n)/d /(e*x^2+d)^2+b*ln(x^n)/d^3*ln(x)-1/2*b*n/d^3*ln(x)^2+1/2*b*n/d^3*ln(x)*ln( e*x^2+d)-1/2*b*n/d^3*ln(x)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n/d^ 3*ln(x)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n/d^3*dilog((-e*x+(-d*e) ^(1/2))/(-d*e)^(1/2))-1/2*b*n/d^3*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+3 /8*b*n/d^3*ln(e*x^2+d)-1/8*b*n/d^2/(e*x^2+d)-3/4*b*n*ln(x)/d^3+(-1/2*I*b*P i*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)*(-1/2*e/d^3*(1/e*ln(e*x^2+d)-d/e/(e*x^2+d)-1/2*d^2/e/(e*x^2+d)^2)+1/d^ 3*ln(x))
\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
Time = 119.75 (sec) , antiderivative size = 403, normalized size of antiderivative = 3.50 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx=- \frac {a e \left (\begin {cases} \frac {x^{2}}{2 d^{3}} & \text {for}\: e = 0 \\- \frac {1}{4 e \left (d + e x^{2}\right )^{2}} & \text {otherwise} \end {cases}\right )}{d} - \frac {a e \left (\begin {cases} \frac {x^{2}}{2 d^{2}} & \text {for}\: e = 0 \\- \frac {1}{2 d e + 2 e^{2} x^{2}} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {a \log {\left (x \right )}}{d^{3}} - \frac {a \log {\left (d + e x^{2} \right )}}{2 d^{3}} + \frac {b e^{2} n \left (\begin {cases} - \frac {1}{2 e^{3} x^{2}} & \text {for}\: d = 0 \\- \frac {1}{4 d e^{2} + 4 e^{3} x^{2}} - \frac {\log {\left (d + e x^{2} \right )}}{4 d e^{2}} & \text {otherwise} \end {cases}\right )}{2 d^{2}} - \frac {b e^{2} \left (\begin {cases} \frac {1}{e^{3} x^{2}} & \text {for}\: d = 0 \\- \frac {1}{2 d \left (\frac {d}{x^{2}} + e\right )^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{2 d^{2}} - \frac {b e n \left (\begin {cases} - \frac {1}{2 e^{2} x^{2}} & \text {for}\: d = 0 \\- \frac {\log {\left (d + e x^{2} \right )}}{2 d e} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {b e \left (\begin {cases} \frac {1}{e^{2} x^{2}} & \text {for}\: d = 0 \\- \frac {1}{\frac {d^{2}}{x^{2}} + d e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{2}} + \frac {b n \left (\begin {cases} - \frac {1}{2 e x^{2}} & \text {for}\: d = 0 \\\frac {\begin {cases} \frac {\operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x^{2}}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} + \frac {\operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x^{2}}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x^{2}}\right )}{2} & \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 )} + \frac {\operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x^{2}}\right )}{2} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{2 d^{2}} - \frac {b \left (\begin {cases} \frac {1}{e x^{2}} & \text {for}\: d = 0 \\\frac {\log {\left (\frac {d}{x^{2}} + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{2 d^{2}} \]
-a*e*Piecewise((x**2/(2*d**3), Eq(e, 0)), (-1/(4*e*(d + e*x**2)**2), True) )/d - a*e*Piecewise((x**2/(2*d**2), Eq(e, 0)), (-1/(2*d*e + 2*e**2*x**2), True))/d**2 + a*log(x)/d**3 - a*log(d + e*x**2)/(2*d**3) + b*e**2*n*Piecew ise((-1/(2*e**3*x**2), Eq(d, 0)), (-1/(4*d*e**2 + 4*e**3*x**2) - log(d + e *x**2)/(4*d*e**2), True))/(2*d**2) - b*e**2*Piecewise((1/(e**3*x**2), Eq(d , 0)), (-1/(2*d*(d/x**2 + e)**2), True))*log(c*x**n)/(2*d**2) - b*e*n*Piec ewise((-1/(2*e**2*x**2), Eq(d, 0)), (-log(d + e*x**2)/(2*d*e), True))/d**2 + b*e*Piecewise((1/(e**2*x**2), Eq(d, 0)), (-1/(d**2/x**2 + d*e), True))* log(c*x**n)/d**2 + b*n*Piecewise((-1/(2*e*x**2), Eq(d, 0)), (Piecewise((po lylog(2, d*exp_polar(I*pi)/(e*x**2))/2, (Abs(x) < 1) & (1/Abs(x) < 1)), (l og(e)*log(x) + polylog(2, d*exp_polar(I*pi)/(e*x**2))/2, Abs(x) < 1), (-lo g(e)*log(1/x) + polylog(2, d*exp_polar(I*pi)/(e*x**2))/2, 1/Abs(x) < 1), ( -meijerg(((), (1, 1)), ((0, 0), ()), x)*log(e) + meijerg(((1, 1), ()), (() , (0, 0)), x)*log(e) + polylog(2, d*exp_polar(I*pi)/(e*x**2))/2, True))/d, True))/(2*d**2) - b*Piecewise((1/(e*x**2), Eq(d, 0)), (log(d/x**2 + e)/d, True))*log(c*x**n)/(2*d**2)
\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
1/4*a*((2*e*x^2 + 3*d)/(d^2*e^2*x^4 + 2*d^3*e*x^2 + d^4) - 2*log(e*x^2 + d )/d^3 + 4*log(x)/d^3) + b*integrate((log(c) + log(x^n))/(e^3*x^7 + 3*d*e^2 *x^5 + 3*d^2*e*x^3 + d^3*x), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (e\,x^2+d\right )}^3} \,d x \]