3.88 \(\int \frac{\log ^2(c (a+b x^2)^p)}{x^4} \, dx\)

Optimal. Leaf size=254 \[ -\frac{4 i b^{3/2} p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{3 a^{3/2}}-\frac{4 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac{4 i b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 a^{3/2}}+\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{8 b^{3/2} p^2 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x} \]

[Out]

(8*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3*a^(3/2)) - (((4*I)/3)*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2
)/a^(3/2) - (8*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(3*a^(3/2)) -
(4*b*p*Log[c*(a + b*x^2)^p])/(3*a*x) - (4*b^(3/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(3*a^(3
/2)) - Log[c*(a + b*x^2)^p]^2/(3*x^3) - (((4*I)/3)*b^(3/2)*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]
*x)])/a^(3/2)

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Rubi [A]  time = 0.28507, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {2457, 2476, 2455, 205, 2470, 12, 4920, 4854, 2402, 2315} \[ -\frac{4 i b^{3/2} p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{3 a^{3/2}}-\frac{4 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac{4 i b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 a^{3/2}}+\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{8 b^{3/2} p^2 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]^2/x^4,x]

[Out]

(8*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3*a^(3/2)) - (((4*I)/3)*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2
)/a^(3/2) - (8*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(3*a^(3/2)) -
(4*b*p*Log[c*(a + b*x^2)^p])/(3*a*x) - (4*b^(3/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(3*a^(3
/2)) - Log[c*(a + b*x^2)^p]^2/(3*x^3) - (((4*I)/3)*b^(3/2)*p^2*PolyLog[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]
*x)])/a^(3/2)

Rule 2457

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

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 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

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 ^2\left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac{1}{3} (4 b p) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{x^2 \left (a+b x^2\right )} \, dx\\ &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac{1}{3} (4 b p) \int \left (\frac{\log \left (c \left (a+b x^2\right )^p\right )}{a x^2}-\frac{b \log \left (c \left (a+b x^2\right )^p\right )}{a \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac{(4 b p) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx}{3 a}-\frac{\left (4 b^2 p\right ) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{3 a}\\ &=-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac{4 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac{\left (8 b^2 p^2\right ) \int \frac{1}{a+b x^2} \, dx}{3 a}+\frac{\left (8 b^3 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}{3 a}\\ &=\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac{4 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac{\left (8 b^{5/2} p^2\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a+b x^2} \, dx}{3 a^{3/2}}\\ &=\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{4 i b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 a^{3/2}}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac{4 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{\left (8 b^2 p^2\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{i-\frac{\sqrt{b} x}{\sqrt{a}}} \, dx}{3 a^2}\\ &=\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{4 i b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 a^{3/2}}-\frac{8 b^{3/2} 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 )}{3 a^{3/2}}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac{4 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac{\left (8 b^2 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}{3 a^2}\\ &=\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{4 i b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 a^{3/2}}-\frac{8 b^{3/2} 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 )}{3 a^{3/2}}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac{4 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{\left (8 i b^{3/2} 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 )}{3 a^{3/2}}\\ &=\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{4 i b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{3 a^{3/2}}-\frac{8 b^{3/2} 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 )}{3 a^{3/2}}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{3 a x}-\frac{4 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^{3/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{4 i b^{3/2} p^2 \text{Li}_2\left (1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{3 a^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0907811, size = 207, normalized size = 0.81 \[ \frac{-4 i b^{3/2} p^2 x^3 \text{PolyLog}\left (2,\frac{\sqrt{b} x+i \sqrt{a}}{\sqrt{b} x-i \sqrt{a}}\right )-4 b^{3/2} p x^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )+2 p \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )-2 p\right )-4 i b^{3/2} p^2 x^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2-\sqrt{a} \log \left (c \left (a+b x^2\right )^p\right ) \left (a \log \left (c \left (a+b x^2\right )^p\right )+4 b p x^2\right )}{3 a^{3/2} x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p]^2/x^4,x]

[Out]

((-4*I)*b^(3/2)*p^2*x^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 - 4*b^(3/2)*p*x^3*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[a]*Log[c*(a + b*x^2)^p]*(4*b*p*x^2 +
a*Log[c*(a + b*x^2)^p]) - (4*I)*b^(3/2)*p^2*x^3*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)
])/(3*a^(3/2)*x^3)

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Maple [F]  time = 0.964, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}}{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^2+a)^p)^2/x^4,x)

[Out]

int(ln(c*(b*x^2+a)^p)^2/x^4,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^4,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 (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{4}}, 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^4,x, algorithm="fricas")

[Out]

integral(log((b*x^2 + a)^p*c)^2/x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x**2+a)**p)**2/x**4,x)

[Out]

Integral(log(c*(a + b*x**2)**p)**2/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{4}}\,{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^4,x, algorithm="giac")

[Out]

integrate(log((b*x^2 + a)^p*c)^2/x^4, x)