3.99 \(\int \log ^3(c (a+b x^2)^p) \, dx\)

Optimal. Leaf size=289 \[ 6 a p \text{Unintegrable}\left (\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right )-\frac{24 i \sqrt{a} p^3 \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{\sqrt{b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac{24 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{24 i \sqrt{a} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}+\frac{48 \sqrt{a} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{48 \sqrt{a} p^3 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-48 p^3 x \]

[Out]

-48*p^3*x + (48*Sqrt[a]*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] - ((24*I)*Sqrt[a]*p^3*ArcTan[(Sqrt[b]*x)/Sqrt
[a]]^2)/Sqrt[b] - (48*Sqrt[a]*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[b
] + 24*p^2*x*Log[c*(a + b*x^2)^p] - (24*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/Sqrt[b]
- 6*p*x*Log[c*(a + b*x^2)^p]^2 + x*Log[c*(a + b*x^2)^p]^3 - ((24*I)*Sqrt[a]*p^3*PolyLog[2, 1 - (2*Sqrt[a])/(Sq
rt[a] + I*Sqrt[b]*x)])/Sqrt[b] + 6*a*p*Unintegrable[Log[c*(a + b*x^2)^p]^2/(a + b*x^2), x]

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Rubi [A]  time = 0.43435, 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 = {} \[ \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Log[c*(a + b*x^2)^p]^3,x]

[Out]

-48*p^3*x + (48*Sqrt[a]*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] - ((24*I)*Sqrt[a]*p^3*ArcTan[(Sqrt[b]*x)/Sqrt
[a]]^2)/Sqrt[b] - (48*Sqrt[a]*p^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/Sqrt[b
] + 24*p^2*x*Log[c*(a + b*x^2)^p] - (24*Sqrt[a]*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/Sqrt[b]
- 6*p*x*Log[c*(a + b*x^2)^p]^2 + x*Log[c*(a + b*x^2)^p]^3 - ((24*I)*Sqrt[a]*p^3*PolyLog[2, 1 - (2*Sqrt[a])/(Sq
rt[a] + I*Sqrt[b]*x)])/Sqrt[b] + 6*a*p*Defer[Int][Log[c*(a + b*x^2)^p]^2/(a + b*x^2), x]

Rubi steps

\begin{align*} \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx &=x \log ^3\left (c \left (a+b x^2\right )^p\right )-(6 b p) \int \frac{x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=x \log ^3\left (c \left (a+b x^2\right )^p\right )-(6 b p) \int \left (\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac{a \log ^2\left (c \left (a+b x^2\right )^p\right )}{b \left (a+b x^2\right )}\right ) \, dx\\ &=x \log ^3\left (c \left (a+b x^2\right )^p\right )-(6 p) \int \log ^2\left (c \left (a+b x^2\right )^p\right ) \, dx+(6 a p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (24 b p^2\right ) \int \frac{x^2 \log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (24 b p^2\right ) \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\\ &=-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (24 p^2\right ) \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx-\left (24 a p^2\right ) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ &=24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac{24 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx-\left (48 b p^3\right ) \int \frac{x^2}{a+b x^2} \, dx+\left (48 a b 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\\ &=-48 p^3 x+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac{24 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (48 a p^3\right ) \int \frac{1}{a+b x^2} \, dx+\left (48 \sqrt{a} \sqrt{b} p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a+b x^2} \, dx\\ &=-48 p^3 x+\frac{48 \sqrt{a} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{24 i \sqrt{a} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac{24 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx-\left (48 p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{i-\frac{\sqrt{b} x}{\sqrt{a}}} \, dx\\ &=-48 p^3 x+\frac{48 \sqrt{a} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{24 i \sqrt{a} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}-\frac{48 \sqrt{a} 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 )}{\sqrt{b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac{24 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx+\left (48 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\\ &=-48 p^3 x+\frac{48 \sqrt{a} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{24 i \sqrt{a} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}-\frac{48 \sqrt{a} 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 )}{\sqrt{b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac{24 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )+(6 a p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx-\frac{\left (48 i \sqrt{a} 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 )}{\sqrt{b}}\\ &=-48 p^3 x+\frac{48 \sqrt{a} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}-\frac{24 i \sqrt{a} p^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{\sqrt{b}}-\frac{48 \sqrt{a} 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 )}{\sqrt{b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac{24 \sqrt{a} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt{b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac{24 i \sqrt{a} p^3 \text{Li}_2\left (1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{\sqrt{b}}+(6 a p) \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 3.35596, size = 789, normalized size = 2.73 \[ \frac{p^3 \left (-6 \sqrt{-a^2} \sqrt{\frac{b x^2}{a+b x^2}} \left (8 \sqrt{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 )+\log \left (a+b x^2\right ) \left (4 \sqrt{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 )+\sqrt{a+b x^2} \log \left (a+b x^2\right ) \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b x^2}}\right )\right )\right )+6 (-a)^{3/2} \sqrt{-\frac{b x^2}{a}} \left (-4 \text{PolyLog}\left (2,\frac{1}{2}-\frac{1}{2} \sqrt{-\frac{b x^2}{a}}\right )+\log ^2\left (\frac{b x^2}{a}+1\right )+2 \log ^2\left (\frac{1}{2} \left (\sqrt{-\frac{b x^2}{a}}+1\right )\right )-4 \log \left (\frac{1}{2} \left (\sqrt{-\frac{b x^2}{a}}+1\right )\right ) \log \left (\frac{b x^2}{a}+1\right )\right )-48 \sqrt{-a^2} \sqrt{\frac{b x^2}{a+b x^2}} \sqrt{a+b x^2} \sin ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b x^2}}\right )+\sqrt{-a} b x^2 \left (\log ^3\left (a+b x^2\right )-6 \log ^2\left (a+b x^2\right )+24 \log \left (a+b x^2\right )-48\right )+24 a \sqrt{b x^2} \left (\log \left (a+b x^2\right )-\log \left (\frac{b x^2}{a}+1\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b x^2}}{\sqrt{-a}}\right )\right )}{\sqrt{-a} b x}-\frac{3 p^2 \left (p \log \left (a+b x^2\right )-\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (4 i \sqrt{a} \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 (a+b x^2\right )-4 \log \left (a+b x^2\right )+8\right )+4 \sqrt{a} \left (\log \left (a+b x^2\right )+2 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )-2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+4 i \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2\right )}{\sqrt{b}}+3 p x \log \left (a+b x^2\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2+x \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2 \left (\log \left (c \left (a+b x^2\right )^p\right )+p \left (-\log \left (a+b x^2\right )\right )-6 p\right )+\frac{6 \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 )-p \log \left (a+b x^2\right )\right )^2}{\sqrt{b}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Log[c*(a + b*x^2)^p]^3,x]

[Out]

(6*Sqrt[a]*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/Sqrt[b] + 3*p*x*Log[a
+ b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 + x*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2*(-
6*p - p*Log[a + b*x^2] + Log[c*(a + b*x^2)^p]) - (3*p^2*(p*Log[a + b*x^2] - Log[c*(a + b*x^2)^p])*((4*I)*Sqrt[
a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2 + 4*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-2 + 2*Log[(2*Sqrt[a])/(Sqrt[a] + I*
Sqrt[b]*x)] + Log[a + b*x^2]) + Sqrt[b]*x*(8 - 4*Log[a + b*x^2] + Log[a + b*x^2]^2) + (4*I)*Sqrt[a]*PolyLog[2,
(I*Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)]))/Sqrt[b] + (p^3*(-48*Sqrt[-a^2]*Sqrt[(b*x^2)/(a + b*x^2)
]*Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]] + Sqrt[-a]*b*x^2*(-48 + 24*Log[a + b*x^2] - 6*Log[a + b*x^2]
^2 + Log[a + b*x^2]^3) - 6*Sqrt[-a^2]*Sqrt[(b*x^2)/(a + b*x^2)]*(8*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2, 1
/2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] + Log[a + b*x^2]*(4*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}
, a/(a + b*x^2)] + Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^2])) + 24*a*Sqrt[b*x^2]*ArcTanh
[Sqrt[b*x^2]/Sqrt[-a]]*(Log[a + b*x^2] - Log[1 + (b*x^2)/a]) + 6*(-a)^(3/2)*Sqrt[-((b*x^2)/a)]*(Log[1 + (b*x^2
)/a]^2 - 4*Log[1 + (b*x^2)/a]*Log[(1 + Sqrt[-((b*x^2)/a)])/2] + 2*Log[(1 + Sqrt[-((b*x^2)/a)])/2]^2 - 4*PolyLo
g[2, 1/2 - Sqrt[-((b*x^2)/a)]/2])))/(Sqrt[-a]*b*x)

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Maple [A]  time = 0.866, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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

[Out]

integral(log((b*x^2 + a)^p*c)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(log((b*x^2 + a)^p*c)^3, x)