3.1.99 \(\int \log ^3(c (a+b x^2)^p) \, dx\) [99]

Optimal. Leaf size=290 \[ -48 p^3 x+\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {24 i \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{\sqrt {b}}-\frac {48 \sqrt {a} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+24 p^2 x \log \left (c \left (a+b x^2\right )^p\right )-\frac {24 \sqrt {a} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{\sqrt {b}}-6 p x \log ^2\left (c \left (a+b x^2\right )^p\right )+x \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {24 i \sqrt {a} p^3 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{\sqrt {b}}+6 a p \text {Int}\left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right ) \]

[Out]

-48*p^3*x+24*p^2*x*ln(c*(b*x^2+a)^p)-6*p*x*ln(c*(b*x^2+a)^p)^2+x*ln(c*(b*x^2+a)^p)^3+48*p^3*arctan(x*b^(1/2)/a
^(1/2))*a^(1/2)/b^(1/2)-24*I*p^3*arctan(x*b^(1/2)/a^(1/2))^2*a^(1/2)/b^(1/2)-24*p^2*arctan(x*b^(1/2)/a^(1/2))*
ln(c*(b*x^2+a)^p)*a^(1/2)/b^(1/2)-48*p^3*arctan(x*b^(1/2)/a^(1/2))*ln(2*a^(1/2)/(a^(1/2)+I*x*b^(1/2)))*a^(1/2)
/b^(1/2)-24*I*p^3*polylog(2,1-2*a^(1/2)/(a^(1/2)+I*x*b^(1/2)))*a^(1/2)/b^(1/2)+6*a*p*Unintegrable(ln(c*(b*x^2+
a)^p)^2/(b*x^2+a),x)

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Rubi [A]
time = 0.27, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx \end {gather*}

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 ) \text {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*}

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Mathematica [A] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(789\) vs. \(2(290)=580\).
time = 2.26, size = 789, normalized size = 2.72 \begin {gather*} \frac {6 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{\sqrt {b}}+3 p x \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2+x \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2 \left (-6 p-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )-\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} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2+4 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-2+2 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )+\log \left (a+b x^2\right )\right )+\sqrt {b} x \left (8-4 \log \left (a+b x^2\right )+\log ^2\left (a+b x^2\right )\right )+4 i \sqrt {a} \text {Li}_2\left (\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )}{\sqrt {b}}+\frac {p^3 \left (-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 (-48+24 \log \left (a+b x^2\right )-6 \log ^2\left (a+b x^2\right )+\log ^3\left (a+b x^2\right )\right )-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}{a+b x^2}\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}{a+b x^2}\right )+\sqrt {a+b x^2} \sin ^{-1}\left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right ) \log \left (a+b x^2\right )\right )\right )+24 a \sqrt {b x^2} \tanh ^{-1}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right ) \left (\log \left (a+b x^2\right )-\log \left (1+\frac {b x^2}{a}\right )\right )+6 (-a)^{3/2} \sqrt {-\frac {b x^2}{a}} \left (\log ^2\left (1+\frac {b x^2}{a}\right )-4 \log \left (1+\frac {b x^2}{a}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )-4 \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {b x^2}{a}}\right )\right )\right )}{\sqrt {-a} b x} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.04, size = 0, normalized size = 0.00 \[\int \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{3}\, dx\]

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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b*x^2)^p)^3,x)

[Out]

int(log(c*(a + b*x^2)^p)^3, x)

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