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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.010459, 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, Rules used = {199, 220} \[ \frac{5 x}{12 \sqrt{x^4+1}}+\frac{x}{6 \left (x^4+1\right )^{3/2}}+\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{24 \sqrt{x^4+1}} \]

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])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\left (1+x^4\right )^{5/2}} \, dx &=\frac{x}{6 \left (1+x^4\right )^{3/2}}+\frac{5}{6} \int \frac{1}{\left (1+x^4\right )^{3/2}} \, dx\\ &=\frac{x}{6 \left (1+x^4\right )^{3/2}}+\frac{5 x}{12 \sqrt{1+x^4}}+\frac{5}{12} \int \frac{1}{\sqrt{1+x^4}} \, dx\\ &=\frac{x}{6 \left (1+x^4\right )^{3/2}}+\frac{5 x}{12 \sqrt{1+x^4}}+\frac{5 \left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{24 \sqrt{1+x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0155247, size = 49, normalized size = 0.68 \[ \frac{5}{12} x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^4\right )+\frac{5 x}{12 \sqrt{x^4+1}}+\frac{x}{6 \left (x^4+1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)^(-5/2),x]

[Out]

x/(6*(1 + x^4)^(3/2)) + (5*x)/(12*Sqrt[1 + x^4]) + (5*x*Hypergeometric2F1[1/4, 1/2, 5/4, -x^4])/12

________________________________________________________________________________________

Maple [C]  time = 0.008, size = 82, normalized size = 1.1 \begin{align*}{\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}}}} \end{align*}

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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 + 1)^(-5/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{4} + 1}}{x^{12} + 3 \, x^{8} + 3 \, x^{4} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x^4 + 1)/(x^12 + 3*x^8 + 3*x^4 + 1), x)

________________________________________________________________________________________

Sympy [C]  time = 0.92844, size = 27, normalized size = 0.38 \begin{align*} \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 )} \end{align*}

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))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{4} + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 + 1)^(-5/2), x)