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

Optimal. Leaf size=175 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \log \left (d \left (a+c x^2\right )^n\right )}{\sqrt {a} \sqrt {c} e}+\frac {i n \text {Li}_2\left (1-\frac {2 \sqrt {a}}{i \sqrt {c} x+\sqrt {a}}\right )}{\sqrt {a} \sqrt {c} e}+\frac {i n \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )^2}{\sqrt {a} \sqrt {c} e}+\frac {2 n \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {c} x}\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c} e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, 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 12

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

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 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]

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 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 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 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]

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 = 0.05, size = 131, normalized size = 0.75 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (\log \left (d \left (a+c x^2\right )^n\right )+2 n \log \left (\frac {2 i}{-\frac {\sqrt {c} x}{\sqrt {a}}+i}\right )+i n \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )\right )+i n \text {Li}_2\left (\frac {\sqrt {c} x+i \sqrt {a}}{\sqrt {c} x-i \sqrt {a}}\right )}{\sqrt {a} \sqrt {c} e} \]

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)

________________________________________________________________________________________

fricas [F]  time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (c x^{2} + a\right )}^{n} d\right )}{c e x^{2} + a e}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (c x^{2} + a\right )}^{n} d\right )}{c e x^{2} + a e}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 1.05, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (d \left (c \,x^{2}+a \right )^{n}\right )}{c e \,x^{2}+a e}\, dx \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (c x^{2} + a\right )}^{n} d\right )}{c e x^{2} + a e}\,{d x} \]

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]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (d\,{\left (c\,x^2+a\right )}^n\right )}{c\,e\,x^2+a\,e} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\log {\left (d \left (a + c x^{2}\right )^{n} \right )}}{a + c x^{2}}\, dx}{e} \]

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

________________________________________________________________________________________