3.101 \(\int \frac{\log ^3(c (a+b x^2)^p)}{x^4} \, dx\)

Optimal. Leaf size=253 \[ \frac{8 i b^{3/2} p^3 \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{a^{3/2}}-\frac{2 b^2 p \text{Unintegrable}\left (\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right )}{a}+\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^{3/2}}+\frac{8 i b^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{a^{3/2}}+\frac{16 b^{3/2} p^3 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{a x}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3} \]

[Out]

((8*I)*b^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/a^(3/2) + (16*b^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2
*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/a^(3/2) + (8*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])
/a^(3/2) - (2*b*p*Log[c*(a + b*x^2)^p]^2)/(a*x) - Log[c*(a + b*x^2)^p]^3/(3*x^3) + ((8*I)*b^(3/2)*p^3*PolyLog[
2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/a^(3/2) - (2*b^2*p*Unintegrable[Log[c*(a + b*x^2)^p]^2/(a + b*x^2
), x])/a

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Rubi [A]  time = 0.350779, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Log[c*(a + b*x^2)^p]^3/x^4,x]

[Out]

((8*I)*b^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/a^(3/2) + (16*b^(3/2)*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2
*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/a^(3/2) + (8*b^(3/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])
/a^(3/2) - (2*b*p*Log[c*(a + b*x^2)^p]^2)/(a*x) - Log[c*(a + b*x^2)^p]^3/(3*x^3) + ((8*I)*b^(3/2)*p^3*PolyLog[
2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/a^(3/2) - (2*b^2*p*Defer[Int][Log[c*(a + b*x^2)^p]^2/(a + b*x^2),
x])/a

Rubi steps

\begin{align*} \int \frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx &=-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+(2 b p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^2 \left (a+b x^2\right )} \, dx\\ &=-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+(2 b p) \int \left (\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a x^2}-\frac{b \log ^2\left (c \left (a+b x^2\right )^p\right )}{a \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac{(2 b p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx}{a}-\frac{\left (2 b^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{a}\\ &=-\frac{2 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{a x}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{\left (2 b^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{a}+\frac{\left (8 b^2 p^2\right ) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{a}\\ &=\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^{3/2}}-\frac{2 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{a x}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{\left (2 b^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{a}-\frac{\left (16 b^3 p^3\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}{a}\\ &=\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^{3/2}}-\frac{2 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{a x}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{\left (2 b^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{a}-\frac{\left (16 b^{5/2} p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a+b x^2} \, dx}{a^{3/2}}\\ &=\frac{8 i b^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{a^{3/2}}+\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^{3/2}}-\frac{2 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{a x}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{\left (2 b^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{a}+\frac{\left (16 b^2 p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{i-\frac{\sqrt{b} x}{\sqrt{a}}} \, dx}{a^2}\\ &=\frac{8 i b^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{a^{3/2}}+\frac{16 b^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{a^{3/2}}+\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^{3/2}}-\frac{2 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{a x}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{\left (2 b^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{a}-\frac{\left (16 b^2 p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{1+\frac{b x^2}{a}} \, dx}{a^2}\\ &=\frac{8 i b^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{a^{3/2}}+\frac{16 b^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{a^{3/2}}+\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^{3/2}}-\frac{2 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{a x}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}-\frac{\left (2 b^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{a}+\frac{\left (16 i b^{3/2} p^3\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 )}{a^{3/2}}\\ &=\frac{8 i b^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{a^{3/2}}+\frac{16 b^{3/2} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{a^{3/2}}+\frac{8 b^{3/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^{3/2}}-\frac{2 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{a x}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac{8 i b^{3/2} p^3 \text{Li}_2\left (1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{a^{3/2}}-\frac{\left (2 b^2 p\right ) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{a}\\ \end{align*}

Mathematica [A]  time = 2.62218, size = 851, normalized size = 3.36 \[ \frac{\left (-a^2 \log ^3\left (b x^2+a\right )-6 a b x^2 \log ^2\left (b x^2+a\right )+6 \sqrt{a} \left (\frac{b x^2}{b x^2+a}\right )^{3/2} \left (b x^2+a\right )^{3/2} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{b x^2+a}}\right ) \log ^2\left (b x^2+a\right )+24 \sqrt{-a} \left (b x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x^2}}{\sqrt{-a}}\right ) \log \left (b x^2+a\right )+24 a b x^2 \sqrt{\frac{b x^2}{b x^2+a}} \, _3F_2\left (\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right ) \log \left (b x^2+a\right )-6 a^2 \left (-\frac{b x^2}{a}\right )^{3/2} \log ^2\left (\frac{b x^2}{a}+1\right )-12 a^2 \left (-\frac{b x^2}{a}\right )^{3/2} \log ^2\left (\frac{1}{2} \left (\sqrt{-\frac{b x^2}{a}}+1\right )\right )+48 a b x^2 \sqrt{\frac{b x^2}{b x^2+a}} \, _4F_3\left (\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2},\frac{3}{2};\frac{a}{b x^2+a}\right )-24 \sqrt{-a} \left (b x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x^2}}{\sqrt{-a}}\right ) \log \left (\frac{b x^2}{a}+1\right )+24 a^2 \left (-\frac{b x^2}{a}\right )^{3/2} \log \left (\frac{b x^2}{a}+1\right ) \log \left (\frac{1}{2} \left (\sqrt{-\frac{b x^2}{a}}+1\right )\right )+24 a^2 \left (-\frac{b x^2}{a}\right )^{3/2} \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{-\frac{b x^2}{a}}\right )\right ) p^3+3 \sqrt{a} \left (p \log \left (b x^2+a\right )-\log \left (c \left (b x^2+a\right )^p\right )\right ) \left (4 b \left (i \sqrt{b} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2+\sqrt{b} x \left (2 \log \left (\frac{2 \sqrt{a}}{i \sqrt{b} x+\sqrt{a}}\right )+\log \left (b x^2+a\right )-2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{a} \log \left (b x^2+a\right )+i \sqrt{b} x \text{PolyLog}\left (2,\frac{\sqrt{b} x+i \sqrt{a}}{\sqrt{b} x-i \sqrt{a}}\right )\right ) x^2+a^{3/2} \log ^2\left (b x^2+a\right )\right ) p^2-6 a b x^2 \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p-6 \sqrt{a} b^{3/2} x^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p-3 a^2 \log \left (b x^2+a\right ) \left (\log \left (c \left (b x^2+a\right )^p\right )-p \log \left (b x^2+a\right )\right )^2 p+a^2 \left (p \log \left (b x^2+a\right )-\log \left (c \left (b x^2+a\right )^p\right )\right )^3}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p]^3/x^4,x]

[Out]

(a^2*(p*Log[a + b*x^2] - Log[c*(a + b*x^2)^p])^3 - 6*a*b*p*x^2*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2
- 6*Sqrt[a]*b^(3/2)*p*x^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 - 3*a^2*p
*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 + 3*Sqrt[a]*p^2*(p*Log[a + b*x^2] - Log[c*(a +
b*x^2)^p])*(a^(3/2)*Log[a + b*x^2]^2 + 4*b*x^2*(I*Sqrt[b]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + Sqrt[a]*Log[a + b*
x^2] + Sqrt[b]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-2 + 2*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)] + Log[a + b*x^2]
) + I*Sqrt[b]*x*PolyLog[2, (I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)])) + p^3*(48*a*b*x^2*Sqrt[(b*x^2
)/(a + b*x^2)]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] + 24*Sqrt[-a]*(b*x^2)^(
3/2)*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*Log[a + b*x^2] + 24*a*b*x^2*Sqrt[(b*x^2)/(a + b*x^2)]*HypergeometricPFQ[{1/
2, 1/2, 1/2}, {3/2, 3/2}, a/(a + b*x^2)]*Log[a + b*x^2] - 6*a*b*x^2*Log[a + b*x^2]^2 + 6*Sqrt[a]*((b*x^2)/(a +
b*x^2))^(3/2)*(a + b*x^2)^(3/2)*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2]^2 - a^2*Log[a + b*x^2]^3 - 24*
Sqrt[-a]*(b*x^2)^(3/2)*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*Log[1 + (b*x^2)/a] - 6*a^2*(-((b*x^2)/a))^(3/2)*Log[1 + (
b*x^2)/a]^2 + 24*a^2*(-((b*x^2)/a))^(3/2)*Log[1 + (b*x^2)/a]*Log[(1 + Sqrt[-((b*x^2)/a)])/2] - 12*a^2*(-((b*x^
2)/a))^(3/2)*Log[(1 + Sqrt[-((b*x^2)/a)])/2]^2 + 24*a^2*(-((b*x^2)/a))^(3/2)*PolyLog[2, 1/2 - Sqrt[-((b*x^2)/a
)]/2]))/(3*a^2*x^3)

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Maple [A]  time = 6.689, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}}{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^2+a)^p)^3/x^4,x)

[Out]

int(ln(c*(b*x^2+a)^p)^3/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)^3/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{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)^3/x^4,x, algorithm="fricas")

[Out]

integral(log((b*x^2 + a)^p*c)^3/x^4, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x**2+a)**p)**3/x**4,x)

[Out]

Integral(log(c*(a + b*x**2)**p)**3/x**4, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{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)^3/x^4,x, algorithm="giac")

[Out]

integrate(log((b*x^2 + a)^p*c)^3/x^4, x)