3.8.34 \(\int \sqrt {1-x^2-y^4} \, dx\) [734]

Optimal. Leaf size=56 \[ \frac {1}{2} x \sqrt {1-x^2-y^4}-\frac {1}{2} i \left (-1+y^4\right ) \log \left (-i x+\sqrt {1-x^2-y^4}\right ) \]

[Out]

1/2*x*(-y^4-x^2+1)^(1/2)-1/2*I*(y^4-1)*ln(-I*x+(-y^4-x^2+1)^(1/2))

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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {201, 223, 209} \begin {gather*} \frac {1}{2} \left (1-y^4\right ) \text {ArcTan}\left (\frac {x}{\sqrt {-x^2-y^4+1}}\right )+\frac {1}{2} x \sqrt {-x^2-y^4+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2 - y^4],x]

[Out]

(x*Sqrt[1 - x^2 - y^4])/2 + ((1 - y^4)*ArcTan[x/Sqrt[1 - x^2 - y^4]])/2

Rule 201

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \sqrt {1-x^2-y^4} \, dx &=\frac {1}{2} x \sqrt {1-x^2-y^4}+\frac {1}{2} \left (1-y^4\right ) \int \frac {1}{\sqrt {1-x^2-y^4}} \, dx\\ &=\frac {1}{2} x \sqrt {1-x^2-y^4}+\frac {1}{2} \left (1-y^4\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1-x^2-y^4}}\right )\\ &=\frac {1}{2} x \sqrt {1-x^2-y^4}+\frac {1}{2} \left (1-y^4\right ) \tan ^{-1}\left (\frac {x}{\sqrt {1-x^2-y^4}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 49, normalized size = 0.88 \begin {gather*} \frac {1}{2} \left (x \sqrt {1-x^2-y^4}-\left (-1+y^4\right ) \text {ArcTan}\left (\frac {x}{\sqrt {1-x^2-y^4}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2 - y^4],x]

[Out]

(x*Sqrt[1 - x^2 - y^4] - (-1 + y^4)*ArcTan[x/Sqrt[1 - x^2 - y^4]])/2

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Maple [A]
time = 0.34, size = 45, normalized size = 0.80

method	result	size
default	\(\frac {x \sqrt {-y^{4}-x^{2}+1}}{2}-\frac {\left (4 y^{4}-4\right ) \arctan \left (\frac {x}{\sqrt {-y^{4}-x^{2}+1}}\right )}{8}\)	\(45\)
risch	\(-\frac {x \left (y^{4}+x^{2}-1\right )}{2 \sqrt {-y^{4}-x^{2}+1}}-\frac {\arctan \left (\frac {x}{\sqrt {-y^{4}-x^{2}+1}}\right ) y^{4}}{2}+\frac {\arctan \left (\frac {x}{\sqrt {-y^{4}-x^{2}+1}}\right )}{2}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*(-y^4-x^2+1)^(1/2)-1/8*(4*y^4-4)*arctan(1/(-y^4-x^2+1)^(1/2)*x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(y-1>0)', see `assume?` for mor
e details)Is

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Fricas [A]
time = 0.38, size = 44, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, {\left (y^{4} - 1\right )} \arctan \left (\frac {\sqrt {-y^{4} - x^{2} + 1}}{x}\right ) + \frac {1}{2} \, \sqrt {-y^{4} - x^{2} + 1} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(y^4 - 1)*arctan(sqrt(-y^4 - x^2 + 1)/x) + 1/2*sqrt(-y^4 - x^2 + 1)*x

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (39) = 78\).
time = 1.01, size = 745, normalized size = 13.30 \begin {gather*} \begin {cases} - \frac {2 i x^{3} \sqrt {\operatorname {polar\_lift}{\left (1 - y^{4} \right )}}}{4 y^{4} \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1} - 4 \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1}} - \frac {2 i x y^{4} \sqrt {\operatorname {polar\_lift}{\left (1 - y^{4} \right )}}}{4 y^{4} \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1} - 4 \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1}} + \frac {2 i x \sqrt {\operatorname {polar\_lift}{\left (1 - y^{4} \right )}}}{4 y^{4} \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1} - 4 \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1}} - \frac {2 i y^{4} \cdot \left (1 - y^{4}\right ) \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1} \operatorname {acosh}{\left (\frac {x}{\sqrt {\operatorname {polar\_lift}{\left (1 - y^{4} \right )}}} \right )}}{4 y^{4} \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1} - 4 \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1}} + \frac {\pi y^{4} \cdot \left (1 - y^{4}\right ) \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1}}{4 y^{4} \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1} - 4 \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1}} + \frac {2 i \left (1 - y^{4}\right ) \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1} \operatorname {acosh}{\left (\frac {x}{\sqrt {\operatorname {polar\_lift}{\left (1 - y^{4} \right )}}} \right )}}{4 y^{4} \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1} - 4 \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1}} - \frac {\pi \left (1 - y^{4}\right ) \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1}}{4 y^{4} \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1} - 4 \sqrt {\frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} - 1}} & \text {for}\: \left |{\frac {x^{2}}{y^{4} - 1}}\right | > 1 \\\frac {x^{3} \sqrt {\operatorname {polar\_lift}{\left (1 - y^{4} \right )}}}{2 y^{4} \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1} - 2 \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1}} + \frac {x y^{4} \sqrt {\operatorname {polar\_lift}{\left (1 - y^{4} \right )}}}{2 y^{4} \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1} - 2 \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1}} - \frac {x \sqrt {\operatorname {polar\_lift}{\left (1 - y^{4} \right )}}}{2 y^{4} \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1} - 2 \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1}} + \frac {y^{4} \cdot \left (1 - y^{4}\right ) \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1} \operatorname {asin}{\left (\frac {x}{\sqrt {\operatorname {polar\_lift}{\left (1 - y^{4} \right )}}} \right )}}{2 y^{4} \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1} - 2 \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1}} - \frac {\left (1 - y^{4}\right ) \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1} \operatorname {asin}{\left (\frac {x}{\sqrt {\operatorname {polar\_lift}{\left (1 - y^{4} \right )}}} \right )}}{2 y^{4} \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1} - 2 \sqrt {- \frac {x^{2}}{\operatorname {polar\_lift}{\left (1 - y^{4} \right )}} + 1}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-y**4-x**2+1)**(1/2),x)

[Out]

Piecewise((-2*I*x**3*sqrt(polar_lift(1 - y**4))/(4*y**4*sqrt(x**2/polar_lift(1 - y**4) - 1) - 4*sqrt(x**2/pola
r_lift(1 - y**4) - 1)) - 2*I*x*y**4*sqrt(polar_lift(1 - y**4))/(4*y**4*sqrt(x**2/polar_lift(1 - y**4) - 1) - 4
*sqrt(x**2/polar_lift(1 - y**4) - 1)) + 2*I*x*sqrt(polar_lift(1 - y**4))/(4*y**4*sqrt(x**2/polar_lift(1 - y**4
) - 1) - 4*sqrt(x**2/polar_lift(1 - y**4) - 1)) - 2*I*y**4*(1 - y**4)*sqrt(x**2/polar_lift(1 - y**4) - 1)*acos
h(x/sqrt(polar_lift(1 - y**4)))/(4*y**4*sqrt(x**2/polar_lift(1 - y**4) - 1) - 4*sqrt(x**2/polar_lift(1 - y**4)
- 1)) + pi*y**4*(1 - y**4)*sqrt(x**2/polar_lift(1 - y**4) - 1)/(4*y**4*sqrt(x**2/polar_lift(1 - y**4) - 1) -
4*sqrt(x**2/polar_lift(1 - y**4) - 1)) + 2*I*(1 - y**4)*sqrt(x**2/polar_lift(1 - y**4) - 1)*acosh(x/sqrt(polar
_lift(1 - y**4)))/(4*y**4*sqrt(x**2/polar_lift(1 - y**4) - 1) - 4*sqrt(x**2/polar_lift(1 - y**4) - 1)) - pi*(1
- y**4)*sqrt(x**2/polar_lift(1 - y**4) - 1)/(4*y**4*sqrt(x**2/polar_lift(1 - y**4) - 1) - 4*sqrt(x**2/polar_l
ift(1 - y**4) - 1)), Abs(x**2/(y**4 - 1)) > 1), (x**3*sqrt(polar_lift(1 - y**4))/(2*y**4*sqrt(-x**2/polar_lift
(1 - y**4) + 1) - 2*sqrt(-x**2/polar_lift(1 - y**4) + 1)) + x*y**4*sqrt(polar_lift(1 - y**4))/(2*y**4*sqrt(-x*
*2/polar_lift(1 - y**4) + 1) - 2*sqrt(-x**2/polar_lift(1 - y**4) + 1)) - x*sqrt(polar_lift(1 - y**4))/(2*y**4*
sqrt(-x**2/polar_lift(1 - y**4) + 1) - 2*sqrt(-x**2/polar_lift(1 - y**4) + 1)) + y**4*(1 - y**4)*sqrt(-x**2/po
lar_lift(1 - y**4) + 1)*asin(x/sqrt(polar_lift(1 - y**4)))/(2*y**4*sqrt(-x**2/polar_lift(1 - y**4) + 1) - 2*sq
rt(-x**2/polar_lift(1 - y**4) + 1)) - (1 - y**4)*sqrt(-x**2/polar_lift(1 - y**4) + 1)*asin(x/sqrt(polar_lift(1
- y**4)))/(2*y**4*sqrt(-x**2/polar_lift(1 - y**4) + 1) - 2*sqrt(-x**2/polar_lift(1 - y**4) + 1)), True))

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Giac [A]
time = 0.40, size = 37, normalized size = 0.66 \begin {gather*} -\frac {1}{2} \, {\left (y^{4} - 1\right )} \arcsin \left (\frac {x}{\sqrt {-y^{4} + 1}}\right ) + \frac {1}{2} \, \sqrt {-y^{4} - x^{2} + 1} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(y^4 - 1)*arcsin(x/sqrt(-y^4 + 1)) + 1/2*sqrt(-y^4 - x^2 + 1)*x

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Mupad [B]
time = 0.10, size = 48, normalized size = 0.86 \begin {gather*} \frac {x\,\sqrt {-x^2-y^4+1}}{2}+\ln \left (\sqrt {-x^2-y^4+1}+x\,1{}\mathrm {i}\right )\,\left (\frac {y^4\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - y^4 - x^2)^(1/2),x)

[Out]

(x*(1 - y^4 - x^2)^(1/2))/2 + log(x*1i + (1 - y^4 - x^2)^(1/2))*((y^4*1i)/2 - 1i/2)

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