3.7.87 \(\int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx\)

Optimal. Leaf size=63 \[ \frac {1}{4} \sqrt {-x^2+\sqrt {x^2-1} x+1} \left (\sqrt {x^2-1}+3 x\right )+\frac {3 \sin ^{-1}\left (x-\sqrt {x^2-1}\right )}{4 \sqrt {2}} \]

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Rubi [F]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]],x]

[Out]

Defer[Int][Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]], x]

Rubi steps

\begin {align*} \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx &=\int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]],x]

[Out]

Integrate[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]], x]

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IntegrateAlgebraic [B]  time = 0.94, size = 133, normalized size = 2.11 \begin {gather*} \frac {3 x^3+\left (x^2-1\right )^{3/2}-3 x}{4 \sqrt {x^2-1} \sqrt {-x^2+\sqrt {x^2-1} x+1} x+4 \left (x^2-1\right ) \sqrt {-x^2+\sqrt {x^2-1} x+1}}-\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x^2-1}}{\sqrt {-x^2+\sqrt {x^2-1} x+1}}\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]],x]

[Out]

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

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fricas [A]  time = 0.98, size = 68, normalized size = 1.08 \begin {gather*} \frac {1}{4} \, \sqrt {-x^{2} + \sqrt {x^{2} - 1} x + 1} {\left (3 \, x + \sqrt {x^{2} - 1}\right )} + \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + \sqrt {x^{2} - 1} x + 1}}{2 \, \sqrt {x^{2} - 1}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(-x^2 + sqrt(x^2 - 1)*x + 1)*(3*x + sqrt(x^2 - 1)) + 3/8*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-x^2 + sqrt(x
^2 - 1)*x + 1)/sqrt(x^2 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {-x^{2} + \sqrt {x^{2} - 1} x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^2 + sqrt(x^2 - 1)*x + 1), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {-x^{2}+\sqrt {x^{2}-1}\, x +1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x^2+(x^2-1)^(1/2)*x)^(1/2),x)

[Out]

int((1-x^2+(x^2-1)^(1/2)*x)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {-x^{2} + \sqrt {x^{2} - 1} x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-x^2 + sqrt(x^2 - 1)*x + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {x\,\sqrt {x^2-1}-x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(x^2 - 1)^(1/2) - x^2 + 1)^(1/2),x)

[Out]

int((x*(x^2 - 1)^(1/2) - x^2 + 1)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- x^{2} + x \sqrt {x^{2} - 1} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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