\(\int \frac {a+b \log (c x^n)}{(d+e x^2)^2} \, dx\) [228]

Optimal result

Integrand size = 20, antiderivative size = 164 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt {e}}-\frac {i b n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} \sqrt {e}}+\frac {i b n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} \sqrt {e}} \]

[Out]

1/2*x*(a+b*ln(c*x^n))/d/(e*x^2+d)-1/2*b*n*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)/e^(1/2)+1/2*arctan(x*e^(1/2)/d^(1/
2))*(a+b*ln(c*x^n))/d^(3/2)/e^(1/2)-1/4*I*b*n*polylog(2,-I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(1/2)+1/4*I*b*n*polylo
g(2,I*x*e^(1/2)/d^(1/2))/d^(3/2)/e^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2360, 211, 2361, 12, 4940, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}-\frac {b n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}-\frac {i b n \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} \sqrt {e}}+\frac {i b n \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} \sqrt {e}} \]

[In]

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

[Out]

-1/2*(b*n*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (x*(a + b*Log[c*x^n]))/(2*d*(d + e*x^2)) + (ArcTan[
(Sqrt[e]*x)/Sqrt[d]]*(a + b*Log[c*x^n]))/(2*d^(3/2)*Sqrt[e]) - ((I/4)*b*n*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]]
)/(d^(3/2)*Sqrt[e]) + ((I/4)*b*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2360

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(q +
1)*((a + b*Log[c*x^n])/(2*d*(q + 1))), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*Log[
c*x^n]), x], x] + Dist[b*(n/(2*d*(q + 1))), Int[(d + e*x^2)^(q + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] &
& LtQ[q, -1]

Rule 2361

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2),
x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]

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]

Rule 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac {\int \frac {a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{2 d}-\frac {(b n) \int \frac {1}{d+e x^2} \, dx}{2 d} \\ & = -\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt {e}}-\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{2 d} \\ & = -\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt {e}}-\frac {(b n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{2 d^{3/2} \sqrt {e}} \\ & = -\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt {e}}-\frac {(i b n) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 d^{3/2} \sqrt {e}}+\frac {(i b n) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{4 d^{3/2} \sqrt {e}} \\ & = -\frac {b n \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{2 d \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^{3/2} \sqrt {e}}-\frac {i b n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} \sqrt {e}}+\frac {i b n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{4 d^{3/2} \sqrt {e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.76 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=\frac {1}{4} \left (\frac {a+b \log \left (c x^n\right )}{d \left (\sqrt {-d} \sqrt {e}+e x\right )}+\frac {a+b \log \left (c x^n\right )}{(-d)^{3/2} \sqrt {e}+d e x}+\frac {b d n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{5/2} \sqrt {e}}+\frac {b n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{3/2} \sqrt {e}}+\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2} \sqrt {e}}+\frac {b d n \operatorname {PolyLog}\left (2,\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{5/2} \sqrt {e}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2} \sqrt {e}}\right ) \]

[In]

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

[Out]

((a + b*Log[c*x^n])/(d*(Sqrt[-d]*Sqrt[e] + e*x)) + (a + b*Log[c*x^n])/((-d)^(3/2)*Sqrt[e] + d*e*x) + (b*d*n*(L
og[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/((-d)^(5/2)*Sqrt[e]) + (b*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]))/((-d)^(3
/2)*Sqrt[e]) + ((a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/((-d)^(3/2)*Sqrt[e]) + (d*(a + b*Log[c*x^n])
*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/((-d)^(5/2)*Sqrt[e]) + (b*d*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/((-d)^(5/2
)*Sqrt[e]) + (b*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/((-d)^(3/2)*Sqrt[e]))/4

Maple [C] (warning: unable to verify)

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

Time = 0.52 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.74

method	result	size
risch	\(\frac {b x \ln \left (x^{n}\right )}{2 d \left (e \,x^{2}+d \right )}-\frac {b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) n \ln \left (x \right )}{2 d \sqrt {d e}}+\frac {b \arctan \left (\frac {x e}{\sqrt {d e}}\right ) \ln \left (x^{n}\right )}{2 d \sqrt {d e}}-\frac {b n \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2} e}{4 d \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right ) x^{2} e}{4 d \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \left (e \,x^{2}+d \right ) \sqrt {-d e}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \left (e \,x^{2}+d \right ) \sqrt {-d e}}+\frac {b n \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \sqrt {-d e}\, d}-\frac {b n \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{4 \sqrt {-d e}\, d}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{2 d \left (e \,x^{2}+d \right )}+\frac {\arctan \left (\frac {x e}{\sqrt {d e}}\right )}{2 d \sqrt {d e}}\right )\)	\(449\)

[In]

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

[Out]

1/2*b*x/d/(e*x^2+d)*ln(x^n)-1/2*b/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*n*ln(x)+1/2*b/d/(d*e)^(1/2)*arctan(x*e
/(d*e)^(1/2))*ln(x^n)-1/2*b*n/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/4*b*n*ln(x)/d/(e*x^2+d)/(-d*e)^(1/2)*ln(
(-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))*x^2*e-1/4*b*n*ln(x)/d/(e*x^2+d)/(-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1
/2))*x^2*e+1/4*b*n*ln(x)/(e*x^2+d)/(-d*e)^(1/2)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/4*b*n*ln(x)/(e*x^2+d)/(
-d*e)^(1/2)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/4*b*n/(-d*e)^(1/2)/d*dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))
-1/4*b*n/(-d*e)^(1/2)/d*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+(-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)*(1/2*x/d/(e*x^2+d)+1/2/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

integral((b*log(c*x^n) + a)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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