3.22.14 \(\int \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x} \, dx\) [2114]

Optimal. Leaf size=153 \[ \frac {1}{4} (-4-x) \sqrt {x^2+x \sqrt {-1+x^2}}+\frac {3}{4} \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}-\sqrt {2} \tanh ^{-1}\left (x+\sqrt {-1+x^2}-\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right )-\frac {3 \log \left (x+\sqrt {-1+x^2}-\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right )}{4 \sqrt {2}} \]

[Out]

1/4*(-4-x)*(x^2+x*(x^2-1)^(1/2))^(1/2)+3/4*(x^2-1)^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2)-2^(1/2)*arctanh(x+(x^2-1)
^(1/2)-2^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2))-3/8*2^(1/2)*ln(x+(x^2-1)^(1/2)-2^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2)
)

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Rubi [F]
time = 0.24, 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 \frac {\sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}}}{1+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.46, size = 197, normalized size = 1.29 \begin {gather*} \frac {\sqrt {x \left (x+\sqrt {-1+x^2}\right )} \left (2 \sqrt {x} \sqrt {x+\sqrt {-1+x^2}} \left (-3+2 x^2-4 \sqrt {-1+x^2}+2 x \left (-2+\sqrt {-1+x^2}\right )\right )+3 \sqrt {2} \left (x+\sqrt {-1+x^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x+\sqrt {-1+x^2}}}\right )+8 \sqrt {2} \left (x+\sqrt {-1+x^2}\right ) \tanh ^{-1}\left (x+\sqrt {-1+x^2}+\sqrt {2} \sqrt {x} \sqrt {x+\sqrt {-1+x^2}}\right )\right )}{8 \sqrt {x} \left (x+\sqrt {-1+x^2}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(x + Sqrt[-1 + x^2])]*(2*Sqrt[x]*Sqrt[x + Sqrt[-1 + x^2]]*(-3 + 2*x^2 - 4*Sqrt[-1 + x^2] + 2*x*(-2 + S
qrt[-1 + x^2])) + 3*Sqrt[2]*(x + Sqrt[-1 + x^2])*ArcTanh[(Sqrt[2]*Sqrt[x])/Sqrt[x + Sqrt[-1 + x^2]]] + 8*Sqrt[
2]*(x + Sqrt[-1 + x^2])*ArcTanh[x + Sqrt[-1 + x^2] + Sqrt[2]*Sqrt[x]*Sqrt[x + Sqrt[-1 + x^2]]]))/(8*Sqrt[x]*(x
+ Sqrt[-1 + x^2])^(3/2))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}-1}\, \sqrt {x^{2}+x \sqrt {x^{2}-1}}}{1+x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]
time = 2.81, size = 153, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \, \sqrt {x^{2} + \sqrt {x^{2} - 1} x} {\left (x - 3 \, \sqrt {x^{2} - 1} + 4\right )} + \frac {1}{4} \, \sqrt {2} \log \left (4 \, x^{2} - 2 \, {\left (2 \, \sqrt {2} \sqrt {x^{2} - 1} x - \sqrt {2} {\left (2 \, x^{2} - 1\right )}\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} - 4 \, \sqrt {x^{2} - 1} x - 1\right ) + \frac {3}{16} \, \sqrt {2} \log \left (-4 \, x^{2} - 2 \, \sqrt {x^{2} + \sqrt {x^{2} - 1} x} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} - 1}\right )} - 4 \, \sqrt {x^{2} - 1} x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(x^2 + sqrt(x^2 - 1)*x)*(x - 3*sqrt(x^2 - 1) + 4) + 1/4*sqrt(2)*log(4*x^2 - 2*(2*sqrt(2)*sqrt(x^2 - 1
)*x - sqrt(2)*(2*x^2 - 1))*sqrt(x^2 + sqrt(x^2 - 1)*x) - 4*sqrt(x^2 - 1)*x - 1) + 3/16*sqrt(2)*log(-4*x^2 - 2*
sqrt(x^2 + sqrt(x^2 - 1)*x)*(sqrt(2)*x + sqrt(2)*sqrt(x^2 - 1)) - 4*sqrt(x^2 - 1)*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x^2-1}\,\sqrt {x\,\sqrt {x^2-1}+x^2}}{x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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