3.10.50 \(\int \frac {1}{(1+x^4)^{3/2}} \, dx\) [950]

Optimal. Leaf size=58 \[ \frac {x}{2 \sqrt {1+x^4}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}} \]

[Out]

1/2*x/(x^4+1)^(1/2)+1/4*(x^2+1)*(cos(2*arctan(x))^2)^(1/2)/cos(2*arctan(x))*EllipticF(sin(2*arctan(x)),1/2*2^(
1/2))*((x^4+1)/(x^2+1)^2)^(1/2)/(x^4+1)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {205, 226} \begin {gather*} \frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{4 \sqrt {x^4+1}}+\frac {x}{2 \sqrt {x^4+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)^(-3/2),x]

[Out]

x/(2*Sqrt[1 + x^4]) + ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(4*Sqrt[1 + x^4])

Rule 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin {align*} \int \frac {1}{\left (1+x^4\right )^{3/2}} \, dx &=\frac {x}{2 \sqrt {1+x^4}}+\frac {1}{2} \int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=\frac {x}{2 \sqrt {1+x^4}}+\frac {\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 )}{4 \sqrt {1+x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 3.11, size = 30, normalized size = 0.52 \begin {gather*} \frac {1}{2} x \left (\frac {1}{\sqrt {1+x^4}}+\, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)^(-3/2),x]

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.14, size = 72, normalized size = 1.24

method	result	size
meijerg	\(x \hypergeom \left (\left [\frac {1}{4}, \frac {3}{2}\right ], \left [\frac {5}{4}\right ], -x^{4}\right )\)	\(14\)
default	\(\frac {x}{2 \sqrt {x^{4}+1}}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(72\)
risch	\(\frac {x}{2 \sqrt {x^{4}+1}}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(72\)
elliptic	\(\frac {x}{2 \sqrt {x^{4}+1}}+\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*x/(x^4+1)^(1/2)+1/2/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 + 1)^(-3/2), x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.09, size = 38, normalized size = 0.66 \begin {gather*} \frac {\sqrt {i} {\left (-i \, x^{4} - i\right )} F(\arcsin \left (\sqrt {i} x\right )\,|\,-1) + \sqrt {x^{4} + 1} x}{2 \, {\left (x^{4} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(I)*(-I*x^4 - I)*elliptic_f(arcsin(sqrt(I)*x), -1) + sqrt(x^4 + 1)*x)/(x^4 + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.32, size = 27, normalized size = 0.47 \begin {gather*} \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**4+1)**(3/2),x)

[Out]

x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), x**4*exp_polar(I*pi))/(4*gamma(5/4))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 + 1)^(-3/2), x)

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Mupad [B]
time = 0.08, size = 12, normalized size = 0.21 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{2};\ \frac {5}{4};\ -x^4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4 + 1)^(3/2),x)

[Out]

x*hypergeom([1/4, 3/2], 5/4, -x^4)

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