3.957 \(\int \frac{1}{\left (1+x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=72 \[ \frac{5 x}{12 \sqrt{x^4+1}}+\frac{x}{6 \left (x^4+1\right )^{3/2}}+\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{24 \sqrt{x^4+1}} \]

[Out]

x/(6*(1 + x^4)^(3/2)) + (5*x)/(12*Sqrt[1 + x^4]) + (5*(1 + x^2)*Sqrt[(1 + x^4)/(
1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(24*Sqrt[1 + x^4])

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Rubi [A]  time = 0.0312607, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{5 x}{12 \sqrt{x^4+1}}+\frac{x}{6 \left (x^4+1\right )^{3/2}}+\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{24 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^4)^(-5/2),x]

[Out]

x/(6*(1 + x^4)^(3/2)) + (5*x)/(12*Sqrt[1 + x^4]) + (5*(1 + x^2)*Sqrt[(1 + x^4)/(
1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(24*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 1.69381, size = 65, normalized size = 0.9 \[ \frac{5 x}{12 \sqrt{x^{4} + 1}} + \frac{x}{6 \left (x^{4} + 1\right )^{\frac{3}{2}}} + \frac{5 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{24 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**4+1)**(5/2),x)

[Out]

5*x/(12*sqrt(x**4 + 1)) + x/(6*(x**4 + 1)**(3/2)) + 5*sqrt((x**4 + 1)/(x**2 + 1)
**2)*(x**2 + 1)*elliptic_f(2*atan(x), 1/2)/(24*sqrt(x**4 + 1))

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Mathematica [C]  time = 0.0614723, size = 52, normalized size = 0.72 \[ \frac{5 x^5-5 \sqrt [4]{-1} \left (x^4+1\right )^{3/2} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+7 x}{12 \left (x^4+1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^4)^(-5/2),x]

[Out]

(7*x + 5*x^5 - 5*(-1)^(1/4)*(1 + x^4)^(3/2)*EllipticF[I*ArcSinh[(-1)^(1/4)*x], -
1])/(12*(1 + x^4)^(3/2))

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Maple [C]  time = 0.013, size = 82, normalized size = 1.1 \[{\frac{x}{6} \left ({x}^{4}+1 \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,x}{12}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{5\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{6\,\sqrt{2}+6\,i\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4+1)^(5/2),x)

[Out]

1/6*x/(x^4+1)^(3/2)+5/12*x/(x^4+1)^(1/2)+5/12/(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. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(-5/2),x, algorithm="maxima")

[Out]

integrate((x^4 + 1)^(-5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{8} + 2 \, x^{4} + 1\right )} \sqrt{x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(-5/2),x, algorithm="fricas")

[Out]

integral(1/((x^8 + 2*x^4 + 1)*sqrt(x^4 + 1)), x)

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Sympy [A]  time = 2.25683, size = 27, normalized size = 0.38 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4+1)**(5/2),x)

[Out]

x*gamma(1/4)*hyper((1/4, 5/2), (5/4,), x**4*exp_polar(I*pi))/(4*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(-5/2),x, algorithm="giac")

[Out]

integrate((x^4 + 1)^(-5/2), x)