3.647 \(\int x^3 \tan ^{-1}(x)^2 \, dx\)

Optimal. Leaf size=53 \[ \frac{1}{4} x^4 \tan ^{-1}(x)^2-\frac{1}{6} x^3 \tan ^{-1}(x)+\frac{x^2}{12}-\frac{1}{3} \log \left (x^2+1\right )+\frac{1}{2} x \tan ^{-1}(x)-\frac{1}{4} \tan ^{-1}(x)^2 \]

[Out]

x^2/12 + (x*ArcTan[x])/2 - (x^3*ArcTan[x])/6 - ArcTan[x]^2/4 + (x^4*ArcTan[x]^2)
/4 - Log[1 + x^2]/3

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Rubi [A]  time = 0.179567, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875 \[ \frac{1}{4} x^4 \tan ^{-1}(x)^2-\frac{1}{6} x^3 \tan ^{-1}(x)+\frac{x^2}{12}-\frac{1}{3} \log \left (x^2+1\right )+\frac{1}{2} x \tan ^{-1}(x)-\frac{1}{4} \tan ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]  Int[x^3*ArcTan[x]^2,x]

[Out]

x^2/12 + (x*ArcTan[x])/2 - (x^3*ArcTan[x])/6 - ArcTan[x]^2/4 + (x^4*ArcTan[x]^2)
/4 - Log[1 + x^2]/3

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Rubi in Sympy [A]  time = 11.0979, size = 44, normalized size = 0.83 \[ \frac{x^{4} \operatorname{atan}^{2}{\left (x \right )}}{4} - \frac{x^{3} \operatorname{atan}{\left (x \right )}}{6} + \frac{x^{2}}{12} + \frac{x \operatorname{atan}{\left (x \right )}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{3} - \frac{\operatorname{atan}^{2}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*atan(x)**2,x)

[Out]

x**4*atan(x)**2/4 - x**3*atan(x)/6 + x**2/12 + x*atan(x)/2 - log(x**2 + 1)/3 - a
tan(x)**2/4

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Mathematica [A]  time = 0.0110817, size = 37, normalized size = 0.7 \[ \frac{1}{12} \left (3 \left (x^4-1\right ) \tan ^{-1}(x)^2+x^2-4 \log \left (x^2+1\right )-2 \left (x^2-3\right ) x \tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*ArcTan[x]^2,x]

[Out]

(x^2 - 2*x*(-3 + x^2)*ArcTan[x] + 3*(-1 + x^4)*ArcTan[x]^2 - 4*Log[1 + x^2])/12

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Maple [A]  time = 0.016, size = 42, normalized size = 0.8 \[{\frac{{x}^{2}}{12}}+{\frac{x\arctan \left ( x \right ) }{2}}-{\frac{{x}^{3}\arctan \left ( x \right ) }{6}}-{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{4}}+{\frac{{x}^{4} \left ( \arctan \left ( x \right ) \right ) ^{2}}{4}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*arctan(x)^2,x)

[Out]

1/12*x^2+1/2*x*arctan(x)-1/6*x^3*arctan(x)-1/4*arctan(x)^2+1/4*x^4*arctan(x)^2-1
/3*ln(x^2+1)

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Maxima [A]  time = 1.53726, size = 59, normalized size = 1.11 \[ \frac{1}{4} \, x^{4} \arctan \left (x\right )^{2} + \frac{1}{12} \, x^{2} - \frac{1}{6} \,{\left (x^{3} - 3 \, x + 3 \, \arctan \left (x\right )\right )} \arctan \left (x\right ) + \frac{1}{4} \, \arctan \left (x\right )^{2} - \frac{1}{3} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*arctan(x)^2,x, algorithm="maxima")

[Out]

1/4*x^4*arctan(x)^2 + 1/12*x^2 - 1/6*(x^3 - 3*x + 3*arctan(x))*arctan(x) + 1/4*a
rctan(x)^2 - 1/3*log(x^2 + 1)

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Fricas [A]  time = 0.227866, size = 49, normalized size = 0.92 \[ \frac{1}{4} \,{\left (x^{4} - 1\right )} \arctan \left (x\right )^{2} + \frac{1}{12} \, x^{2} - \frac{1}{6} \,{\left (x^{3} - 3 \, x\right )} \arctan \left (x\right ) - \frac{1}{3} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*arctan(x)^2,x, algorithm="fricas")

[Out]

1/4*(x^4 - 1)*arctan(x)^2 + 1/12*x^2 - 1/6*(x^3 - 3*x)*arctan(x) - 1/3*log(x^2 +
1)

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Sympy [A]  time = 0.994043, size = 44, normalized size = 0.83 \[ \frac{x^{4} \operatorname{atan}^{2}{\left (x \right )}}{4} - \frac{x^{3} \operatorname{atan}{\left (x \right )}}{6} + \frac{x^{2}}{12} + \frac{x \operatorname{atan}{\left (x \right )}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{3} - \frac{\operatorname{atan}^{2}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*atan(x)**2,x)

[Out]

x**4*atan(x)**2/4 - x**3*atan(x)/6 + x**2/12 + x*atan(x)/2 - log(x**2 + 1)/3 - a
tan(x)**2/4

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GIAC/XCAS [A]  time = 0.212072, size = 58, normalized size = 1.09 \[ \frac{1}{4} \, x^{4} \arctan \left (x\right )^{2} - \frac{1}{6} \, x^{3} \arctan \left (x\right ) + \frac{1}{12} \, x^{2} + \frac{1}{2} \, x \arctan \left (x\right ) - \frac{1}{4} \, \arctan \left (x\right )^{2} - \frac{1}{3} \,{\rm ln}\left (i \, x^{2} + i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*arctan(x)^2,x, algorithm="giac")

[Out]

1/4*x^4*arctan(x)^2 - 1/6*x^3*arctan(x) + 1/12*x^2 + 1/2*x*arctan(x) - 1/4*arcta
n(x)^2 - 1/3*ln(I*x^2 + I)