3.940 \(\int \frac{x^9}{\left (1+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{3}{4} \sinh ^{-1}\left (x^2\right )-\frac{x^6}{2 \sqrt{x^4+1}}+\frac{3}{4} \sqrt{x^4+1} x^2 \]

[Out]

-x^6/(2*Sqrt[1 + x^4]) + (3*x^2*Sqrt[1 + x^4])/4 - (3*ArcSinh[x^2])/4

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Rubi [A]  time = 0.0475946, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{3}{4} \sinh ^{-1}\left (x^2\right )-\frac{x^6}{2 \sqrt{x^4+1}}+\frac{3}{4} \sqrt{x^4+1} x^2 \]

Antiderivative was successfully verified.

[In]  Int[x^9/(1 + x^4)^(3/2),x]

[Out]

-x^6/(2*Sqrt[1 + x^4]) + (3*x^2*Sqrt[1 + x^4])/4 - (3*ArcSinh[x^2])/4

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Rubi in Sympy [A]  time = 6.23299, size = 36, normalized size = 0.88 \[ - \frac{x^{6}}{2 \sqrt{x^{4} + 1}} + \frac{3 x^{2} \sqrt{x^{4} + 1}}{4} - \frac{3 \operatorname{asinh}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(x**4+1)**(3/2),x)

[Out]

-x**6/(2*sqrt(x**4 + 1)) + 3*x**2*sqrt(x**4 + 1)/4 - 3*asinh(x**2)/4

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Mathematica [A]  time = 0.0310787, size = 37, normalized size = 0.9 \[ \frac{x^6+3 x^2-3 \sqrt{x^4+1} \sinh ^{-1}\left (x^2\right )}{4 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(1 + x^4)^(3/2),x]

[Out]

(3*x^2 + x^6 - 3*Sqrt[1 + x^4]*ArcSinh[x^2])/(4*Sqrt[1 + x^4])

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Maple [A]  time = 0.015, size = 32, normalized size = 0.8 \[{\frac{{x}^{6}}{4}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{3\,{x}^{2}}{4}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{3\,{\it Arcsinh} \left ({x}^{2} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(x^4+1)^(3/2),x)

[Out]

1/4*x^6/(x^4+1)^(1/2)+3/4*x^2/(x^4+1)^(1/2)-3/4*arcsinh(x^2)

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Maxima [A]  time = 1.42306, size = 99, normalized size = 2.41 \[ -\frac{\frac{3 \,{\left (x^{4} + 1\right )}}{x^{4}} - 2}{4 \,{\left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{x^{6}}\right )}} - \frac{3}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) + \frac{3}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

-1/4*(3*(x^4 + 1)/x^4 - 2)/(sqrt(x^4 + 1)/x^2 - (x^4 + 1)^(3/2)/x^6) - 3/8*log(s
qrt(x^4 + 1)/x^2 + 1) + 3/8*log(sqrt(x^4 + 1)/x^2 - 1)

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Fricas [A]  time = 0.252759, size = 170, normalized size = 4.15 \[ -\frac{4 \, x^{12} + 7 \, x^{8} - x^{4} - 3 \,{\left (4 \, x^{8} + 5 \, x^{4} -{\left (4 \, x^{6} + 3 \, x^{2}\right )} \sqrt{x^{4} + 1} + 1\right )} \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) -{\left (4 \, x^{10} + 5 \, x^{6} - 3 \, x^{2}\right )} \sqrt{x^{4} + 1} - 2}{4 \,{\left (4 \, x^{8} + 5 \, x^{4} -{\left (4 \, x^{6} + 3 \, x^{2}\right )} \sqrt{x^{4} + 1} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/4*(4*x^12 + 7*x^8 - x^4 - 3*(4*x^8 + 5*x^4 - (4*x^6 + 3*x^2)*sqrt(x^4 + 1) +
1)*log(-x^2 + sqrt(x^4 + 1)) - (4*x^10 + 5*x^6 - 3*x^2)*sqrt(x^4 + 1) - 2)/(4*x^
8 + 5*x^4 - (4*x^6 + 3*x^2)*sqrt(x^4 + 1) + 1)

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Sympy [A]  time = 9.2043, size = 36, normalized size = 0.88 \[ \frac{x^{6}}{4 \sqrt{x^{4} + 1}} + \frac{3 x^{2}}{4 \sqrt{x^{4} + 1}} - \frac{3 \operatorname{asinh}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(x**4+1)**(3/2),x)

[Out]

x**6/(4*sqrt(x**4 + 1)) + 3*x**2/(4*sqrt(x**4 + 1)) - 3*asinh(x**2)/4

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GIAC/XCAS [A]  time = 0.226761, size = 46, normalized size = 1.12 \[ \frac{{\left (x^{4} + 3\right )} x^{2}}{4 \, \sqrt{x^{4} + 1}} + \frac{3}{4} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

1/4*(x^4 + 3)*x^2/sqrt(x^4 + 1) + 3/4*ln(-x^2 + sqrt(x^4 + 1))