3.2.1 \(\int \frac {\log ^3(c (a+b x^2)^p)}{x^4} \, dx\) [101]

Optimal. Leaf size=254 \[ \frac {8 i b^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{a^{3/2}}+\frac {16 b^{3/2} p^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{a^{3/2}}+\frac {8 b^{3/2} p^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^{3/2}}-\frac {2 b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{a x}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{3 x^3}+\frac {8 i b^{3/2} p^3 \text {Li}_2\left (1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{a^{3/2}}-\frac {2 b^2 p \text {Int}\left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right )}{a} \]

[Out]

8*I*b^(3/2)*p^3*arctan(x*b^(1/2)/a^(1/2))^2/a^(3/2)+8*b^(3/2)*p^2*arctan(x*b^(1/2)/a^(1/2))*ln(c*(b*x^2+a)^p)/
a^(3/2)-2*b*p*ln(c*(b*x^2+a)^p)^2/a/x-1/3*ln(c*(b*x^2+a)^p)^3/x^3+16*b^(3/2)*p^3*arctan(x*b^(1/2)/a^(1/2))*ln(
2*a^(1/2)/(a^(1/2)+I*x*b^(1/2)))/a^(3/2)+8*I*b^(3/2)*p^3*polylog(2,1-2*a^(1/2)/(a^(1/2)+I*x*b^(1/2)))/a^(3/2)-
2*b^2*p*Unintegrable(ln(c*(b*x^2+a)^p)^2/(b*x^2+a),x)/a

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Rubi [A]
time = 0.22, 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 \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx \end {gather*}

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 ) \text {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*}

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Mathematica [A] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(851\) vs. \(2(254)=508\).
time = 1.83, size = 851, normalized size = 3.35 \begin {gather*} \frac {a^2 \left (p \log \left (a+b x^2\right )-\log \left (c \left (a+b x^2\right )^p\right )\right )^3-6 a b p x^2 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2-6 \sqrt {a} b^{3/2} p x^3 \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 )^2-3 a^2 p \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+3 \sqrt {a} p^2 \left (p \log \left (a+b x^2\right )-\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (a^{3/2} \log ^2\left (a+b x^2\right )+4 b x^2 \left (i \sqrt {b} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2+\sqrt {a} \log \left (a+b x^2\right )+\sqrt {b} x \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 )+i \sqrt {b} x \text {Li}_2\left (\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )\right )+p^3 \left (48 a b x^2 \sqrt {\frac {b x^2}{a+b x^2}} \, _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 )+24 \sqrt {-a} \left (b x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right ) \log \left (a+b x^2\right )+24 a b x^2 \sqrt {\frac {b x^2}{a+b x^2}} \, _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 ) \log \left (a+b x^2\right )-6 a b x^2 \log ^2\left (a+b x^2\right )+6 \sqrt {a} \left (\frac {b x^2}{a+b x^2}\right )^{3/2} \left (a+b x^2\right )^{3/2} \sin ^{-1}\left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right ) \log ^2\left (a+b x^2\right )-a^2 \log ^3\left (a+b x^2\right )-24 \sqrt {-a} \left (b x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right ) \log \left (1+\frac {b x^2}{a}\right )-6 a^2 \left (-\frac {b x^2}{a}\right )^{3/2} \log ^2\left (1+\frac {b x^2}{a}\right )+24 a^2 \left (-\frac {b x^2}{a}\right )^{3/2} \log \left (1+\frac {b x^2}{a}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )-12 a^2 \left (-\frac {b x^2}{a}\right )^{3/2} \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )+24 a^2 \left (-\frac {b x^2}{a}\right )^{3/2} \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {b x^2}{a}}\right )\right )}{3 a^2 x^3} \end {gather*}

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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{x^{4}}\, dx\]

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 [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^4,x, algorithm="maxima")

[Out]

-1/3*p^3*log(b*x^2 + a)^3/x^3 + integrate((b*x^2*log(c)^3 + a*log(c)^3 + ((2*p^3 + 3*p^2*log(c))*b*x^2 + 3*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^6 + a*x^4), 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^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{4}}\, dx \end {gather*}

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 = 0.00, 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^4,x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x^4} \,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^4,x)

[Out]

int(log(c*(a + b*x^2)^p)^3/x^4, x)

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