\(\int \frac {\log (\frac {c x^2}{a+b x^2})}{a+b x^2} \, dx\) [279]

Optimal result

Integrand size = 25, antiderivative size = 165 \[ \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx=\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}-\frac {2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (2-\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right )}{\sqrt {a} \sqrt {b}}+\frac {i \operatorname {PolyLog}\left (2,-1+\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right )}{\sqrt {a} \sqrt {b}} \]

[Out]

I*arctan(x*b^(1/2)/a^(1/2))^2/a^(1/2)/b^(1/2)+arctan(x*b^(1/2)/a^(1/2))*ln(c*x^2/(b*x^2+a))/a^(1/2)/b^(1/2)-2*
arctan(x*b^(1/2)/a^(1/2))*ln(2-2*a^(1/2)/(a^(1/2)-I*x*b^(1/2)))/a^(1/2)/b^(1/2)+I*polylog(2,-1+2*a^(1/2)/(a^(1
/2)-I*x*b^(1/2)))/a^(1/2)/b^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {211, 2606, 12, 5044, 4988, 2497} \[ \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {c x^2}{a+b x^2}\right )}{\sqrt {a} \sqrt {b}}+\frac {i \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {b}}-\frac {2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (2-\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}\right )}{\sqrt {a} \sqrt {b}}+\frac {i \operatorname {PolyLog}\left (2,\frac {2 \sqrt {a}}{\sqrt {a}-i \sqrt {b} x}-1\right )}{\sqrt {a} \sqrt {b}} \]

[In]

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

[Out]

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

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 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 2606

Int[Log[(c_.)*(RFx_)^(n_.)]/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2), x]}, Simp[u*L
og[c*RFx^n], x] - Dist[n, Int[SimplifyIntegrand[u*(D[RFx, x]/RFx), x], x], x]] /; FreeQ[{c, d, e, n}, x] && Ra
tionalFunctionQ[RFx, x] &&  !PolynomialQ[RFx, x]

Rule 4988

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTan[c*x])
^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))
]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 5044

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*d*(p + 1))), x] + Dist[I/d, Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(373\) vs. \(2(165)=330\).

Time = 0.16 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.26 \[ \int \frac {\log \left (\frac {c x^2}{a+b x^2}\right )}{a+b x^2} \, dx=\frac {-4 \log \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ) \log \left (\sqrt {-a}-\sqrt {b} x\right )+\log ^2\left (\sqrt {-a}-\sqrt {b} x\right )+4 \log \left (\frac {a \sqrt {b} x}{(-a)^{3/2}}\right ) \log \left (\sqrt {-a}+\sqrt {b} x\right )-\log ^2\left (\sqrt {-a}+\sqrt {b} x\right )+2 \log \left (\sqrt {-a}-\sqrt {b} x\right ) \log \left (\frac {a-\sqrt {-a} \sqrt {b} x}{2 a}\right )-2 \log \left (\sqrt {-a}+\sqrt {b} x\right ) \log \left (\frac {a+\sqrt {-a} \sqrt {b} x}{2 a}\right )+2 \log \left (\sqrt {-a}-\sqrt {b} x\right ) \log \left (\frac {c x^2}{a+b x^2}\right )-2 \log \left (\sqrt {-a}+\sqrt {b} x\right ) \log \left (\frac {c x^2}{a+b x^2}\right )+4 \operatorname {PolyLog}\left (2,1+\frac {\sqrt {b} x}{\sqrt {-a}}\right )-2 \operatorname {PolyLog}\left (2,\frac {a-\sqrt {-a} \sqrt {b} x}{2 a}\right )+2 \operatorname {PolyLog}\left (2,\frac {a+\sqrt {-a} \sqrt {b} x}{2 a}\right )-4 \operatorname {PolyLog}\left (2,1+\frac {a \sqrt {b} x}{(-a)^{3/2}}\right )}{4 \sqrt {-a} \sqrt {b}} \]

[In]

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

[Out]

(-4*Log[(Sqrt[b]*x)/Sqrt[-a]]*Log[Sqrt[-a] - Sqrt[b]*x] + Log[Sqrt[-a] - Sqrt[b]*x]^2 + 4*Log[(a*Sqrt[b]*x)/(-
a)^(3/2)]*Log[Sqrt[-a] + Sqrt[b]*x] - Log[Sqrt[-a] + Sqrt[b]*x]^2 + 2*Log[Sqrt[-a] - Sqrt[b]*x]*Log[(a - Sqrt[
-a]*Sqrt[b]*x)/(2*a)] - 2*Log[Sqrt[-a] + Sqrt[b]*x]*Log[(a + Sqrt[-a]*Sqrt[b]*x)/(2*a)] + 2*Log[Sqrt[-a] - Sqr
t[b]*x]*Log[(c*x^2)/(a + b*x^2)] - 2*Log[Sqrt[-a] + Sqrt[b]*x]*Log[(c*x^2)/(a + b*x^2)] + 4*PolyLog[2, 1 + (Sq
rt[b]*x)/Sqrt[-a]] - 2*PolyLog[2, (a - Sqrt[-a]*Sqrt[b]*x)/(2*a)] + 2*PolyLog[2, (a + Sqrt[-a]*Sqrt[b]*x)/(2*a
)] - 4*PolyLog[2, 1 + (a*Sqrt[b]*x)/(-a)^(3/2)])/(4*Sqrt[-a]*Sqrt[b])

Maple [C] (verified)

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

Time = 1.68 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {c \,x^{2}}{b \,x^{2}+a}\right )+b \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha b}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}\right )-4 \operatorname {dilog}\left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )-4 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 b}\)	\(121\)
risch	\(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {c \,x^{2}}{b \,x^{2}+a}\right )+b \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{\underline {\hspace {1.25 ex}}\alpha b}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )}{a}\right )-4 \operatorname {dilog}\left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )-4 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 b}\)	\(121\)

[In]

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

[Out]

1/4/b*sum(1/_alpha*(2*ln(x-_alpha)*ln(c*x^2/(b*x^2+a))+b*(1/_alpha/b*ln(x-_alpha)^2+2*_alpha/a*ln(x-_alpha)*ln
(1/2*(x+_alpha)/_alpha)+2*_alpha/a*dilog(1/2*(x+_alpha)/_alpha))-4*dilog(x/_alpha)-4*ln(x-_alpha)*ln(x/_alpha)
),_alpha=RootOf(_Z^2*b+a))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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