3.86 \(\int \log ^2(c (a+b x^2)^p) \, dx\)

Optimal. Leaf size=237 \[ \frac{4 i \sqrt{a} p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{\sqrt{b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac{4 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}+\frac{4 i \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}-\frac{8 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{8 \sqrt{a} p^2 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}+8 p^2 x \]

[Out]

8*p^2*x - (8*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + ((4*I)*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]
^2)/Sqrt[b] + (8*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[b] - 4
*p*x*Log[c*(a + b*x^2)^p] + (4*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/Sqrt[b] + x*Log[c*(
a + b*x^2)^p]^2 + ((4*I)*Sqrt[a]*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[b]

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Rubi [A]  time = 0.267497, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {2450, 2476, 2448, 321, 205, 2470, 12, 4920, 4854, 2402, 2315} \[ \frac{4 i \sqrt{a} p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{\sqrt{b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac{4 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}+\frac{4 i \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}-\frac{8 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{8 \sqrt{a} p^2 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}+8 p^2 x \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]^2,x]

[Out]

8*p^2*x - (8*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + ((4*I)*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]
^2)/Sqrt[b] + (8*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[b] - 4
*p*x*Log[c*(a + b*x^2)^p] + (4*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/Sqrt[b] + x*Log[c*(
a + b*x^2)^p]^2 + ((4*I)*Sqrt[a]*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[b]

Rule 2450

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

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 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 \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx &=x \log ^2\left (c \left (a+b x^2\right )^p\right )-(4 b p) \int \frac{x^2 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=x \log ^2\left (c \left (a+b x^2\right )^p\right )-(4 b p) \int \left (\frac{\log \left (c \left (a+b x^2\right )^p\right )}{b}-\frac{a \log \left (c \left (a+b x^2\right )^p\right )}{b \left (a+b x^2\right )}\right ) \, dx\\ &=x \log ^2\left (c \left (a+b x^2\right )^p\right )-(4 p) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx+(4 a p) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac{4 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )+\left (8 b p^2\right ) \int \frac{x^2}{a+b x^2} \, dx-\left (8 a b p^2\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \left (a+b x^2\right )} \, dx\\ &=8 p^2 x-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac{4 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )-\left (8 a p^2\right ) \int \frac{1}{a+b x^2} \, dx-\left (8 \sqrt{a} \sqrt{b} p^2\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a+b x^2} \, dx\\ &=8 p^2 x-\frac{8 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{4 i \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac{4 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )+\left (8 p^2\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{i-\frac{\sqrt{b} x}{\sqrt{a}}} \, dx\\ &=8 p^2 x-\frac{8 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{4 i \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}+\frac{8 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{\sqrt{b}}-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac{4 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )-\left (8 p^2\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{1+\frac{b x^2}{a}} \, dx\\ &=8 p^2 x-\frac{8 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{4 i \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}+\frac{8 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{\sqrt{b}}-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac{4 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{\left (8 i \sqrt{a} p^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{\sqrt{b}}\\ &=8 p^2 x-\frac{8 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{4 i \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}+\frac{8 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{\sqrt{b}}-4 p x \log \left (c \left (a+b x^2\right )^p\right )+\frac{4 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}+x \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac{4 i \sqrt{a} p^2 \text{Li}_2\left (1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{\sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0812616, size = 193, normalized size = 0.81 \[ \frac{4 i \sqrt{a} p^2 \text{PolyLog}\left (2,\frac{\sqrt{b} x+i \sqrt{a}}{\sqrt{b} x-i \sqrt{a}}\right )+\sqrt{b} x \left (\log ^2\left (c \left (a+b x^2\right )^p\right )-4 p \log \left (c \left (a+b x^2\right )^p\right )+8 p^2\right )+4 \sqrt{a} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )+2 p \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )-2 p\right )+4 i \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p]^2,x]

[Out]

((4*I)*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + 4*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-2*p + 2*p*Log[(2*
Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)] + Log[c*(a + b*x^2)^p]) + Sqrt[b]*x*(8*p^2 - 4*p*Log[c*(a + b*x^2)^p] + Log[
c*(a + b*x^2)^p]^2) + (4*I)*Sqrt[a]*p^2*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)])/Sqrt[b
]

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Maple [F]  time = 0.844, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(ln(c*(b*x^2+a)^p)^2,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*(b*x^2+a)^p)^2,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 (\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((b*x^2 + a)^p*c)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((b*x^2 + a)^p*c)^2, x)