3.9.75 \(\int \frac {\sqrt {1-x^4}}{\sqrt {1-x^2}} \, dx\) [875]

Optimal. Leaf size=21 \[ \frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \sinh ^{-1}(x) \]

[Out]

1/2*arcsinh(x)+1/2*x*(x^2+1)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {26, 201, 221} \begin {gather*} \frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \sinh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^4]/Sqrt[1 - x^2],x]

[Out]

(x*Sqrt[1 + x^2])/2 + ArcSinh[x]/2

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-b^2/d)^m, Int[u/
(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d, 0]
&& GtQ[a, 0] && LtQ[d, 0]

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 221

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1-x^4}}{\sqrt {1-x^2}} \, dx &=\int \sqrt {1+x^2} \, dx\\ &=\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \sinh ^{-1}(x)\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(21)=42\).
time = 0.62, size = 70, normalized size = 3.33 \begin {gather*} \frac {1}{2} \left (\frac {x \sqrt {1-x^4}}{\sqrt {1-x^2}}+\log \left (1-x^2\right )-\log \left (-x+x^3+\sqrt {1-x^2} \sqrt {1-x^4}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^4]/Sqrt[1 - x^2],x]

[Out]

((x*Sqrt[1 - x^4])/Sqrt[1 - x^2] + Log[1 - x^2] - Log[-x + x^3 + Sqrt[1 - x^2]*Sqrt[1 - x^4]])/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(15)=30\).
time = 0.70, size = 47, normalized size = 2.24

method	result	size
default	\(-\frac {\sqrt {-x^{4}+1}\, \sqrt {-x^{2}+1}\, \left (x \sqrt {x^{2}+1}+\arcsinh \left (x \right )\right )}{2 \left (x^{2}-1\right ) \sqrt {x^{2}+1}}\)	\(47\)
risch	\(-\frac {x \sqrt {x^{2}+1}\, \sqrt {\frac {\left (-x^{2}+1\right ) \left (-x^{4}+1\right )}{\left (x^{2}-1\right )^{2}}}\, \left (x^{2}-1\right )}{2 \sqrt {-x^{4}+1}\, \sqrt {-x^{2}+1}}-\frac {\arcsinh \left (x \right ) \sqrt {\frac {\left (-x^{2}+1\right ) \left (-x^{4}+1\right )}{\left (x^{2}-1\right )^{2}}}\, \left (x^{2}-1\right )}{2 \sqrt {-x^{4}+1}\, \sqrt {-x^{2}+1}}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-x^4+1)^(1/2)*(-x^2+1)^(1/2)*(x*(x^2+1)^(1/2)+arcsinh(x))/(x^2-1)/(x^2+1)^(1/2)

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

[Out]

integrate(sqrt(-x^4 + 1)/sqrt(-x^2 + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (15) = 30\).
time = 0.33, size = 120, normalized size = 5.71 \begin {gather*} -\frac {2 \, \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} x + {\left (x^{2} - 1\right )} \log \left (\frac {x^{3} + \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} - x}{x^{3} - x}\right ) - {\left (x^{2} - 1\right )} \log \left (-\frac {x^{3} - \sqrt {-x^{4} + 1} \sqrt {-x^{2} + 1} - x}{x^{3} - x}\right )}{4 \, {\left (x^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(-x^4 + 1)*sqrt(-x^2 + 1)*x + (x^2 - 1)*log((x^3 + sqrt(-x^4 + 1)*sqrt(-x^2 + 1) - x)/(x^3 - x)) -
(x^2 - 1)*log(-(x^3 - sqrt(-x^4 + 1)*sqrt(-x^2 + 1) - x)/(x^3 - x)))/(x^2 - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(x - 1)*(x + 1)*(x**2 + 1))/sqrt(-(x - 1)*(x + 1)), x)

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

[Out]

integrate(sqrt(-x^4 + 1)/sqrt(-x^2 + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\sqrt {1-x^4}}{\sqrt {1-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^4)^(1/2)/(1 - x^2)^(1/2),x)

[Out]

int((1 - x^4)^(1/2)/(1 - x^2)^(1/2), x)

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