3.246 \(\int \frac{a+b \log (c x^n)}{1-e x^2} \, dx\)

Optimal. Leaf size=66 \[ \frac{b n \text{PolyLog}\left (2,-\sqrt{e} x\right )}{2 \sqrt{e}}-\frac{b n \text{PolyLog}\left (2,\sqrt{e} x\right )}{2 \sqrt{e}}+\frac{\tanh ^{-1}\left (\sqrt{e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.0387247, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {206, 2324, 12, 5912} \[ \frac{b n \text{PolyLog}\left (2,-\sqrt{e} x\right )}{2 \sqrt{e}}-\frac{b n \text{PolyLog}\left (2,\sqrt{e} x\right )}{2 \sqrt{e}}+\frac{\tanh ^{-1}\left (\sqrt{e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(1 - e*x^2),x]

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2324

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 12

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

Rule 5912

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

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{1-e x^2} \, dx &=\frac{\tanh ^{-1}\left (\sqrt{e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}-(b n) \int \frac{\tanh ^{-1}\left (\sqrt{e} x\right )}{\sqrt{e} x} \, dx\\ &=\frac{\tanh ^{-1}\left (\sqrt{e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}-\frac{(b n) \int \frac{\tanh ^{-1}\left (\sqrt{e} x\right )}{x} \, dx}{\sqrt{e}}\\ &=\frac{\tanh ^{-1}\left (\sqrt{e} x\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt{e}}+\frac{b n \text{Li}_2\left (-\sqrt{e} x\right )}{2 \sqrt{e}}-\frac{b n \text{Li}_2\left (\sqrt{e} x\right )}{2 \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 0.0247137, size = 72, normalized size = 1.09 \[ \frac{b n \text{PolyLog}\left (2,-\sqrt{e} x\right )-b n \text{PolyLog}\left (2,\sqrt{e} x\right )+\left (\log \left (1-\sqrt{e} x\right )-\log \left (\sqrt{e} x+1\right )\right ) \left (-\left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.145, size = 200, normalized size = 3. \begin{align*} -{(-{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -b\ln \left ( c \right ) -a){\it Artanh} \left ( x\sqrt{e} \right ){\frac{1}{\sqrt{e}}}}-{\ln \left ( x \right ) bn{\it Artanh} \left ( x\sqrt{e} \right ){\frac{1}{\sqrt{e}}}}+{b\ln \left ({x}^{n} \right ){\it Artanh} \left ( x\sqrt{e} \right ){\frac{1}{\sqrt{e}}}}-{\frac{\ln \left ( x \right ) bn}{2}\ln \left ( 1-x\sqrt{e} \right ){\frac{1}{\sqrt{e}}}}+{\frac{\ln \left ( x \right ) bn}{2}\ln \left ( x\sqrt{e}+1 \right ){\frac{1}{\sqrt{e}}}}-{\frac{bn}{2}{\it dilog} \left ( 1-x\sqrt{e} \right ){\frac{1}{\sqrt{e}}}}+{\frac{bn}{2}{\it dilog} \left ( x\sqrt{e}+1 \right ){\frac{1}{\sqrt{e}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/2*I*b*Pi*csgn(I*c*x
^n)^3-1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)-b*ln(c)-a)/e^(1/2)*arctanh(x*e^(1/2))-b/e^(1/2)*arctanh(x*e^(1/2))*
n*ln(x)+b/e^(1/2)*arctanh(x*e^(1/2))*ln(x^n)-1/2*b*n/e^(1/2)*ln(x)*ln(1-x*e^(1/2))+1/2*b*n/e^(1/2)*ln(x)*ln(x*
e^(1/2)+1)-1/2*b*n/e^(1/2)*dilog(1-x*e^(1/2))+1/2*b*n/e^(1/2)*dilog(x*e^(1/2)+1)

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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((a+b*log(c*x^n))/(-e*x^2+1),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{b \log \left (c x^{n}\right ) + a}{e x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(-e*x^2+1),x, algorithm="fricas")

[Out]

integral(-(b*log(c*x^n) + a)/(e*x^2 - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{a}{e x^{2} - 1}\, dx - \int \frac{b \log{\left (c x^{n} \right )}}{e x^{2} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(a/(e*x**2 - 1), x) - Integral(b*log(c*x**n)/(e*x**2 - 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \log \left (c x^{n}\right ) + a}{e x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(-e*x^2+1),x, algorithm="giac")

[Out]

integrate(-(b*log(c*x^n) + a)/(e*x^2 - 1), x)