### 3.92 $$\int \frac{\log (d (a+c x^2)^n)}{a e+c e x^2} \, dx$$

Optimal. Leaf size=175 $\frac{i n \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{c} x}\right )}{\sqrt{a} \sqrt{c} e}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt{a} \sqrt{c} e}+\frac{i n \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{c} e}+\frac{2 n \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{c} x}\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c} e}$

[Out]

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

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Rubi [A]  time = 0.157902, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.28, Rules used = {205, 2470, 12, 4920, 4854, 2402, 2315} $\frac{i n \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{c} x}\right )}{\sqrt{a} \sqrt{c} e}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt{a} \sqrt{c} e}+\frac{i n \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )^2}{\sqrt{a} \sqrt{c} e}+\frac{2 n \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{c} x}\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c} e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

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

Rule 4920

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

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.0430713, size = 131, normalized size = 0.75 $\frac{i n \text{PolyLog}\left (2,\frac{\sqrt{c} x+i \sqrt{a}}{\sqrt{c} x-i \sqrt{a}}\right )+\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (\log \left (d \left (a+c x^2\right )^n\right )+2 n \log \left (\frac{2 i}{-\frac{\sqrt{c} x}{\sqrt{a}}+i}\right )+i n \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )\right )}{\sqrt{a} \sqrt{c} e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [F]  time = 0.101, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( d \left ( c{x}^{2}+a \right ) ^{n} \right ) }{ce{x}^{2}+ea}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(ln(d*(c*x^2+a)^n)/(c*e*x^2+a*e),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(d*(c*x^2+a)^n)/(c*e*x^2+a*e),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 ({\left (c x^{2} + a\right )}^{n} d\right )}{c e x^{2} + a e}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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