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

Optimal. Leaf size=146 \[ \sqrt {x^2+\sqrt {x^2-1} x}-\sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {x^2+\sqrt {x^2-1} x}}{\sqrt {1+\sqrt {2}}}\right )-\frac {\tanh ^{-1}\left (\sqrt {2} \sqrt {x^2+\sqrt {x^2-1} x}\right )}{\sqrt {2}}+\sqrt {2 \left (\sqrt {2}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {x^2+\sqrt {x^2-1} x}}{\sqrt {\sqrt {2}-1}}\right ) \]

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Rubi [F]  time = 1.08, 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^2} \, 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^2),x]

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 2.72, size = 808, normalized size = 5.53 \begin {gather*} \frac {\sqrt {x^2-1} \left (x+\sqrt {x^2-1}\right ) \left (4 \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} \left (\sqrt {2} \left (x+\sqrt {x^2-1}\right )^2+3 \sqrt {2}+4\right )}{\sqrt {\left (x+\sqrt {x^2-1}\right )^2+1} \left (\left (x+\sqrt {x^2-1}\right )^2+2 \sqrt {2}+3\right )}\right )+2 \log \left (x+\sqrt {x^2-1}\right )-2 \sqrt {-1+\sqrt {2}} \log \left (\left (x+\sqrt {x^2-1}\right )^4+\left (6-4 \sqrt {2}\right ) \left (x+\sqrt {x^2-1}\right )^2-12 \sqrt {2}+17\right )-2 i \sqrt {1+\sqrt {2}} \log \left (\left (x+\sqrt {x^2-1}\right )^4+\left (6+4 \sqrt {2}\right ) \left (x+\sqrt {x^2-1}\right )^2+12 \sqrt {2}+17\right )+i \sqrt {1+\sqrt {2}} \log \left (\left (5+4 \sqrt {2}\right ) \left (x+\sqrt {x^2-1}\right )^4-2 \sqrt {2 \left (1+\sqrt {2}\right )} \left (2+\sqrt {2}\right ) \sqrt {x \left (x+\sqrt {x^2-1}\right )} \left (x+\sqrt {x^2-1}\right )^3+\left (34+24 \sqrt {2}\right ) \left (x+\sqrt {x^2-1}\right )^2-\frac {2 \sqrt {2 \left (1+\sqrt {2}\right )} \left (10+7 \sqrt {2}\right ) \left (x \left (x+\sqrt {x^2-1}\right )\right )^{3/2}}{x}+12 \sqrt {2}+17\right )+i \sqrt {1+\sqrt {2}} \log \left (\left (5+4 \sqrt {2}\right ) \left (x+\sqrt {x^2-1}\right )^4+2 \sqrt {2 \left (1+\sqrt {2}\right )} \left (2+\sqrt {2}\right ) \sqrt {x \left (x+\sqrt {x^2-1}\right )} \left (x+\sqrt {x^2-1}\right )^3+\left (34+24 \sqrt {2}\right ) \left (x+\sqrt {x^2-1}\right )^2+\frac {2 \sqrt {2 \left (1+\sqrt {2}\right )} \left (10+7 \sqrt {2}\right ) \left (x \left (x+\sqrt {x^2-1}\right )\right )^{3/2}}{x}+12 \sqrt {2}+17\right )-2 \log \left (\sqrt {\left (x+\sqrt {x^2-1}\right )^2+1}+1\right )+2 \sqrt {-1+\sqrt {2}} \log \left (-\left (x+\sqrt {x^2-1}\right )^4-2 \left (2 \sqrt {-1+\sqrt {2}} \sqrt {x \left (x+\sqrt {x^2-1}\right )}+1\right ) \left (x+\sqrt {x^2-1}\right )^2+8 \sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {x \left (x+\sqrt {x^2-1}\right )}-12 \sqrt {-1+\sqrt {2}} \sqrt {x \left (x+\sqrt {x^2-1}\right )}-8 \sqrt {2}+11\right )+2 \sqrt {2} \sqrt {x \left (x+\sqrt {x^2-1}\right )}\right )}{2 \sqrt {2} \left (x^2+\sqrt {x^2-1} x-1\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-1 + x^2]*(x + Sqrt[-1 + x^2])*(2*Sqrt[2]*Sqrt[x*(x + Sqrt[-1 + x^2])] + 4*Sqrt[1 + Sqrt[2]]*ArcTan[(Sqr
t[1 + Sqrt[2]]*(4 + 3*Sqrt[2] + Sqrt[2]*(x + Sqrt[-1 + x^2])^2))/(Sqrt[1 + (x + Sqrt[-1 + x^2])^2]*(3 + 2*Sqrt
[2] + (x + Sqrt[-1 + x^2])^2))] + 2*Log[x + Sqrt[-1 + x^2]] - 2*Sqrt[-1 + Sqrt[2]]*Log[17 - 12*Sqrt[2] + (6 -
4*Sqrt[2])*(x + Sqrt[-1 + x^2])^2 + (x + Sqrt[-1 + x^2])^4] - (2*I)*Sqrt[1 + Sqrt[2]]*Log[17 + 12*Sqrt[2] + (6
+ 4*Sqrt[2])*(x + Sqrt[-1 + x^2])^2 + (x + Sqrt[-1 + x^2])^4] + I*Sqrt[1 + Sqrt[2]]*Log[17 + 12*Sqrt[2] + (34
+ 24*Sqrt[2])*(x + Sqrt[-1 + x^2])^2 + (5 + 4*Sqrt[2])*(x + Sqrt[-1 + x^2])^4 - 2*Sqrt[2*(1 + Sqrt[2])]*(2 +
Sqrt[2])*(x + Sqrt[-1 + x^2])^3*Sqrt[x*(x + Sqrt[-1 + x^2])] - (2*Sqrt[2*(1 + Sqrt[2])]*(10 + 7*Sqrt[2])*(x*(x
+ Sqrt[-1 + x^2]))^(3/2))/x] + I*Sqrt[1 + Sqrt[2]]*Log[17 + 12*Sqrt[2] + (34 + 24*Sqrt[2])*(x + Sqrt[-1 + x^2
])^2 + (5 + 4*Sqrt[2])*(x + Sqrt[-1 + x^2])^4 + 2*Sqrt[2*(1 + Sqrt[2])]*(2 + Sqrt[2])*(x + Sqrt[-1 + x^2])^3*S
qrt[x*(x + Sqrt[-1 + x^2])] + (2*Sqrt[2*(1 + Sqrt[2])]*(10 + 7*Sqrt[2])*(x*(x + Sqrt[-1 + x^2]))^(3/2))/x] - 2
*Log[1 + Sqrt[1 + (x + Sqrt[-1 + x^2])^2]] + 2*Sqrt[-1 + Sqrt[2]]*Log[11 - 8*Sqrt[2] - (x + Sqrt[-1 + x^2])^4
- 12*Sqrt[-1 + Sqrt[2]]*Sqrt[x*(x + Sqrt[-1 + x^2])] + 8*Sqrt[2*(-1 + Sqrt[2])]*Sqrt[x*(x + Sqrt[-1 + x^2])] -
2*(x + Sqrt[-1 + x^2])^2*(1 + 2*Sqrt[-1 + Sqrt[2]]*Sqrt[x*(x + Sqrt[-1 + x^2])])]))/(2*Sqrt[2]*(-1 + x^2 + x*
Sqrt[-1 + x^2]))

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IntegrateAlgebraic [A]  time = 2.07, size = 146, normalized size = 1.00 \begin {gather*} \sqrt {x^2+x \sqrt {-1+x^2}}-\sqrt {2 \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\sqrt {-1+\sqrt {2}} \sqrt {x^2+x \sqrt {-1+x^2}}\right )-\frac {\tanh ^{-1}\left (\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right )}{\sqrt {2}}+\sqrt {2 \left (-1+\sqrt {2}\right )} \tanh ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {x^2+x \sqrt {-1+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 23.86, size = 516, normalized size = 3.53 \begin {gather*} -\sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {45602 \, {\left (4 \, x^{4} - 6 \, x^{2} - \sqrt {2} {\left (x^{4} - 1\right )} - {\left (4 \, x^{3} - \sqrt {2} {\left (x^{3} - 3 \, x\right )} - 4 \, x\right )} \sqrt {x^{2} - 1} - 2\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} \sqrt {2 \, \sqrt {2} + 2} + {\left (1902 \, x^{4} - 3056 \, x^{2} - \sqrt {2} {\left (1403 \, x^{4} - 2494 \, x^{2} + 343\right )} + 2 \, {\left (904 \, x^{3} - \sqrt {2} {\left (499 \, x^{3} - 873 \, x\right )} - 1216 \, x\right )} \sqrt {x^{2} - 1} + 530\right )} \sqrt {76309 \, \sqrt {2} + 105481} \sqrt {2 \, \sqrt {2} + 2}}{45602 \, {\left (7 \, x^{4} - 10 \, x^{2} - 1\right )}}\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 {1}{4} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (-\frac {4 \, {\left (31 \, x^{2} + \sqrt {2} {\left (109 \, x^{2} + 78\right )} - \sqrt {x^{2} - 1} {\left (109 \, \sqrt {2} x + 31 \, x\right )} - 187\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} + {\left (280 \, x^{2} + \sqrt {2} {\left (249 \, x^{2} - 187\right )} - 2 \, \sqrt {x^{2} - 1} {\left (31 \, \sqrt {2} x + 218 \, x\right )} + 156\right )} \sqrt {2 \, \sqrt {2} - 2}}{x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (-\frac {4 \, {\left (31 \, x^{2} + \sqrt {2} {\left (109 \, x^{2} + 78\right )} - \sqrt {x^{2} - 1} {\left (109 \, \sqrt {2} x + 31 \, x\right )} - 187\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} - {\left (280 \, x^{2} + \sqrt {2} {\left (249 \, x^{2} - 187\right )} - 2 \, \sqrt {x^{2} - 1} {\left (31 \, \sqrt {2} x + 218 \, x\right )} + 156\right )} \sqrt {2 \, \sqrt {2} - 2}}{x^{2} + 1}\right ) + \sqrt {x^{2} + \sqrt {x^{2} - 1} x} \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)/(x^2+1),x, algorithm="fricas")

[Out]

-sqrt(2*sqrt(2) + 2)*arctan(1/45602*(45602*(4*x^4 - 6*x^2 - sqrt(2)*(x^4 - 1) - (4*x^3 - sqrt(2)*(x^3 - 3*x) -
4*x)*sqrt(x^2 - 1) - 2)*sqrt(x^2 + sqrt(x^2 - 1)*x)*sqrt(2*sqrt(2) + 2) + (1902*x^4 - 3056*x^2 - sqrt(2)*(140
3*x^4 - 2494*x^2 + 343) + 2*(904*x^3 - sqrt(2)*(499*x^3 - 873*x) - 1216*x)*sqrt(x^2 - 1) + 530)*sqrt(76309*sqr
t(2) + 105481)*sqrt(2*sqrt(2) + 2))/(7*x^4 - 10*x^2 - 1)) + 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) + 1/4*sqrt(2*sqrt(2) - 2)*log(-
(4*(31*x^2 + sqrt(2)*(109*x^2 + 78) - sqrt(x^2 - 1)*(109*sqrt(2)*x + 31*x) - 187)*sqrt(x^2 + sqrt(x^2 - 1)*x)
+ (280*x^2 + sqrt(2)*(249*x^2 - 187) - 2*sqrt(x^2 - 1)*(31*sqrt(2)*x + 218*x) + 156)*sqrt(2*sqrt(2) - 2))/(x^2
+ 1)) - 1/4*sqrt(2*sqrt(2) - 2)*log(-(4*(31*x^2 + sqrt(2)*(109*x^2 + 78) - sqrt(x^2 - 1)*(109*sqrt(2)*x + 31*
x) - 187)*sqrt(x^2 + sqrt(x^2 - 1)*x) - (280*x^2 + sqrt(2)*(249*x^2 - 187) - 2*sqrt(x^2 - 1)*(31*sqrt(2)*x + 2
18*x) + 156)*sqrt(2*sqrt(2) - 2))/(x^2 + 1)) + sqrt(x^2 + sqrt(x^2 - 1)*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + \sqrt {x^{2} - 1} x} \sqrt {x^{2} - 1}}{x^{2} + 1}\,{d x} \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)/(x^2+1),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}-1}\, \sqrt {x^{2}+x \sqrt {x^{2}-1}}}{x^{2}+1}\, dx\]

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)/(x^2+1),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + \sqrt {x^{2} - 1} x} \sqrt {x^{2} - 1}}{x^{2} + 1}\,{d x} \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)/(x^2+1),x, algorithm="maxima")

[Out]

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

[Out]

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

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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^{2} + 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)/(x**2+1),x)

[Out]

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

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