3.945 \(\int \frac{x^{12}}{\left (1+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{15}{14} \sqrt{x^4+1} x-\frac{x^9}{2 \sqrt{x^4+1}}+\frac{9}{14} \sqrt{x^4+1} x^5+\frac{15 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{x^4+1}} \]

[Out]

-x^9/(2*Sqrt[1 + x^4]) - (15*x*Sqrt[1 + x^4])/14 + (9*x^5*Sqrt[1 + x^4])/14 + (1
5*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(28*Sqrt[1
+ x^4])

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Rubi [A]  time = 0.0619925, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{15}{14} \sqrt{x^4+1} x-\frac{x^9}{2 \sqrt{x^4+1}}+\frac{9}{14} \sqrt{x^4+1} x^5+\frac{15 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^12/(1 + x^4)^(3/2),x]

[Out]

-x^9/(2*Sqrt[1 + x^4]) - (15*x*Sqrt[1 + x^4])/14 + (9*x^5*Sqrt[1 + x^4])/14 + (1
5*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(28*Sqrt[1
+ x^4])

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Rubi in Sympy [A]  time = 5.68319, size = 82, normalized size = 0.91 \[ - \frac{x^{9}}{2 \sqrt{x^{4} + 1}} + \frac{9 x^{5} \sqrt{x^{4} + 1}}{14} - \frac{15 x \sqrt{x^{4} + 1}}{14} + \frac{15 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{28 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12/(x**4+1)**(3/2),x)

[Out]

-x**9/(2*sqrt(x**4 + 1)) + 9*x**5*sqrt(x**4 + 1)/14 - 15*x*sqrt(x**4 + 1)/14 + 1
5*sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_f(2*atan(x), 1/2)/(28*sqrt(
x**4 + 1))

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Mathematica [C]  time = 0.0507083, size = 57, normalized size = 0.63 \[ -\frac{-2 x^9+6 x^5+15 \sqrt [4]{-1} \sqrt{x^4+1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+15 x}{14 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12/(1 + x^4)^(3/2),x]

[Out]

-(15*x + 6*x^5 - 2*x^9 + 15*(-1)^(1/4)*Sqrt[1 + x^4]*EllipticF[I*ArcSinh[(-1)^(1
/4)*x], -1])/(14*Sqrt[1 + x^4])

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Maple [C]  time = 0.013, size = 94, normalized size = 1. \[ -{\frac{x}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{{x}^{5}}{7}\sqrt{{x}^{4}+1}}-{\frac{4\,x}{7}\sqrt{{x}^{4}+1}}+{\frac{15\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{7\,\sqrt{2}+7\,i\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12/(x^4+1)^(3/2),x)

[Out]

-1/2*x/(x^4+1)^(1/2)+1/7*x^5*(x^4+1)^(1/2)-4/7*x*(x^4+1)^(1/2)+15/14/(1/2*2^(1/2
)+1/2*I*2^(1/2))*(1-I*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticF(x*(1/2*
2^(1/2)+1/2*I*2^(1/2)),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{12}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^12/(x^4 + 1)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{12}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

integral(x^12/(x^4 + 1)^(3/2), x)

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Sympy [A]  time = 5.7449, size = 29, normalized size = 0.32 \[ \frac{x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{17}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12/(x**4+1)**(3/2),x)

[Out]

x**13*gamma(13/4)*hyper((3/2, 13/4), (17/4,), x**4*exp_polar(I*pi))/(4*gamma(17/
4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{12}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^12/(x^4 + 1)^(3/2), x)