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

Optimal. Leaf size=165 \[ \frac{i \text{PolyLog}\left (2,-1+\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right )}{\sqrt{a} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{c x^2}{a+b x^2}\right )}{\sqrt{a} \sqrt{b}}+\frac{i \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{b}}-\frac{2 \log \left (2-\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

[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])

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Rubi [A]  time = 0.191676, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {205, 2526, 12, 4924, 4868, 2447} \[ \frac{i \text{PolyLog}\left (2,-1+\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right )}{\sqrt{a} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{c x^2}{a+b x^2}\right )}{\sqrt{a} \sqrt{b}}+\frac{i \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{b}}-\frac{2 \log \left (2-\frac{2 \sqrt{a}}{\sqrt{a}-i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[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 205

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

Rule 2526

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 12

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

Rule 4924

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[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]

Rule 4868

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 2447

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]]

Rubi steps

\begin{align*} \int \frac{\log \left (\frac{c x^2}{a+b x^2}\right )}{a+b x^2} \, 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}}-\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]  time = 0.194038, size = 373, normalized size = 2.26 \[ \frac{4 \text{PolyLog}\left (2,\frac{\sqrt{b} x}{\sqrt{-a}}+1\right )-2 \text{PolyLog}\left (2,\frac{a-\sqrt{-a} \sqrt{b} x}{2 a}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{-a} \sqrt{b} x+a}{2 a}\right )-4 \text{PolyLog}\left (2,\frac{a \sqrt{b} x}{(-a)^{3/2}}+1\right )+2 \log \left (\sqrt{-a}-\sqrt{b} x\right ) \log \left (\frac{c x^2}{a+b x^2}\right )-2 \log \left (\sqrt{-a}+\sqrt{b} x\right ) \log \left (\frac{c x^2}{a+b x^2}\right )+\log ^2\left (\sqrt{-a}-\sqrt{b} x\right )-\log ^2\left (\sqrt{-a}+\sqrt{b} x\right )-4 \log \left (\frac{\sqrt{b} x}{\sqrt{-a}}\right ) \log \left (\sqrt{-a}-\sqrt{b} x\right )+2 \log \left (\frac{a-\sqrt{-a} \sqrt{b} x}{2 a}\right ) \log \left (\sqrt{-a}-\sqrt{b} x\right )+4 \log \left (\frac{a \sqrt{b} x}{(-a)^{3/2}}\right ) \log \left (\sqrt{-a}+\sqrt{b} x\right )-2 \log \left (\sqrt{-a}+\sqrt{b} x\right ) \log \left (\frac{\sqrt{-a} \sqrt{b} x+a}{2 a}\right )}{4 \sqrt{-a} \sqrt{b}} \]

Antiderivative was successfully verified.

[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])

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{2}+a}\ln \left ({\frac{c{x}^{2}}{b{x}^{2}+a}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\frac{c x^{2}}{b x^{2} + a}\right )}{b x^{2} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (\frac{c x^{2}}{a + b x^{2}} \right )}}{a + b x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{c x^{2}}{b x^{2} + a}\right )}{b x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)