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

Optimal. Leaf size=134 \[ \frac {2 \sqrt {x^2+1} x}{3 \sqrt {\sqrt {x^2+1}+1}}+\frac {4 x}{3 \sqrt {\sqrt {x^2+1}+1}}+2 \sqrt {\sqrt {2}-1} \tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^2+1}+1}}\right )-2 \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^2+1}+1}}\right ) \]

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Rubi [F]  time = 0.46, 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 {\left (1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{-1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*x^3)/(3*(1 + Sqrt[1 + x^2])^(3/2)) + (2*x)/Sqrt[1 + Sqrt[1 + x^2]] - Defer[Int][Sqrt[1 + Sqrt[1 + x^2]]/(1
- x), x] - Defer[Int][Sqrt[1 + Sqrt[1 + x^2]]/(1 + x), x]

Rubi steps

\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{-1+x^2} \, dx &=\int \left (\sqrt {1+\sqrt {1+x^2}}+\frac {2 \sqrt {1+\sqrt {1+x^2}}}{-1+x^2}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+\sqrt {1+x^2}}}{-1+x^2} \, dx+\int \sqrt {1+\sqrt {1+x^2}} \, dx\\ &=\frac {2 x^3}{3 \left (1+\sqrt {1+x^2}\right )^{3/2}}+\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}+2 \int \left (-\frac {\sqrt {1+\sqrt {1+x^2}}}{2 (1-x)}-\frac {\sqrt {1+\sqrt {1+x^2}}}{2 (1+x)}\right ) \, dx\\ &=\frac {2 x^3}{3 \left (1+\sqrt {1+x^2}\right )^{3/2}}+\frac {2 x}{\sqrt {1+\sqrt {1+x^2}}}-\int \frac {\sqrt {1+\sqrt {1+x^2}}}{1-x} \, dx-\int \frac {\sqrt {1+\sqrt {1+x^2}}}{1+x} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^2\right ) \sqrt {1+\sqrt {1+x^2}}}{-1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.34, size = 134, normalized size = 1.00 \begin {gather*} \frac {4 x}{3 \sqrt {1+\sqrt {1+x^2}}}+\frac {2 x \sqrt {1+x^2}}{3 \sqrt {1+\sqrt {1+x^2}}}+2 \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {-1+\sqrt {2}} x}{\sqrt {1+\sqrt {1+x^2}}}\right )-2 \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} x}{\sqrt {1+\sqrt {1+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x)/(3*Sqrt[1 + Sqrt[1 + x^2]]) + (2*x*Sqrt[1 + x^2])/(3*Sqrt[1 + Sqrt[1 + x^2]]) + 2*Sqrt[-1 + Sqrt[2]]*Arc
Tan[(Sqrt[-1 + Sqrt[2]]*x)/Sqrt[1 + Sqrt[1 + x^2]]] - 2*Sqrt[1 + Sqrt[2]]*ArcTanh[(Sqrt[1 + Sqrt[2]]*x)/Sqrt[1
+ Sqrt[1 + x^2]]]

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fricas [B]  time = 4.16, size = 444, normalized size = 3.31 \begin {gather*} -\frac {12 \, x \sqrt {\sqrt {2} - 1} \arctan \left (\frac {{\left (51 \, x^{5} - 222 \, x^{3} - 2 \, \sqrt {2} {\left (5 \, x^{5} - 163 \, x^{3} - 46 \, x\right )} + 2 \, {\left (31 \, x^{3} + \sqrt {2} {\left (41 \, x^{3} - 81 \, x\right )} + 173 \, x\right )} \sqrt {x^{2} + 1} + 11 \, x\right )} \sqrt {3821 \, \sqrt {2} + 4841} \sqrt {\sqrt {2} - 1} + 4802 \, {\left (x^{4} + \sqrt {2} {\left (x^{4} - 3 \, x^{2} - 2\right )} + {\left (3 \, x^{2} + 2 \, \sqrt {2} {\left (x^{2} + 1\right )} + 1\right )} \sqrt {x^{2} + 1} - 1\right )} \sqrt {\sqrt {2} - 1} \sqrt {\sqrt {x^{2} + 1} + 1}}{2401 \, {\left (x^{5} - 10 \, x^{3} - 7 \, x\right )}}\right ) + 3 \, x \sqrt {\sqrt {2} + 1} \log \left (-\frac {2 \, {\left ({\left (71 \, x^{3} - \sqrt {2} {\left (61 \, x^{3} + 325 \, x\right )} + 2 \, \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} x - 193 \, x\right )} + 457 \, x\right )} \sqrt {\sqrt {2} + 1} + 2 \, {\left (71 \, x^{2} - \sqrt {2} {\left (61 \, x^{2} + 132\right )} + \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} - 193\right )} + 193\right )} \sqrt {\sqrt {x^{2} + 1} + 1}\right )}}{x^{3} - x}\right ) - 3 \, x \sqrt {\sqrt {2} + 1} \log \left (\frac {2 \, {\left ({\left (71 \, x^{3} - \sqrt {2} {\left (61 \, x^{3} + 325 \, x\right )} + 2 \, \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} x - 193 \, x\right )} + 457 \, x\right )} \sqrt {\sqrt {2} + 1} - 2 \, {\left (71 \, x^{2} - \sqrt {2} {\left (61 \, x^{2} + 132\right )} + \sqrt {x^{2} + 1} {\left (132 \, \sqrt {2} - 193\right )} + 193\right )} \sqrt {\sqrt {x^{2} + 1} + 1}\right )}}{x^{3} - x}\right ) - 4 \, {\left (x^{2} + \sqrt {x^{2} + 1} - 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{6 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(12*x*sqrt(sqrt(2) - 1)*arctan(1/2401*((51*x^5 - 222*x^3 - 2*sqrt(2)*(5*x^5 - 163*x^3 - 46*x) + 2*(31*x^3
+ sqrt(2)*(41*x^3 - 81*x) + 173*x)*sqrt(x^2 + 1) + 11*x)*sqrt(3821*sqrt(2) + 4841)*sqrt(sqrt(2) - 1) + 4802*(
x^4 + sqrt(2)*(x^4 - 3*x^2 - 2) + (3*x^2 + 2*sqrt(2)*(x^2 + 1) + 1)*sqrt(x^2 + 1) - 1)*sqrt(sqrt(2) - 1)*sqrt(
sqrt(x^2 + 1) + 1))/(x^5 - 10*x^3 - 7*x)) + 3*x*sqrt(sqrt(2) + 1)*log(-2*((71*x^3 - sqrt(2)*(61*x^3 + 325*x) +
2*sqrt(x^2 + 1)*(132*sqrt(2)*x - 193*x) + 457*x)*sqrt(sqrt(2) + 1) + 2*(71*x^2 - sqrt(2)*(61*x^2 + 132) + sqr
t(x^2 + 1)*(132*sqrt(2) - 193) + 193)*sqrt(sqrt(x^2 + 1) + 1))/(x^3 - x)) - 3*x*sqrt(sqrt(2) + 1)*log(2*((71*x
^3 - sqrt(2)*(61*x^3 + 325*x) + 2*sqrt(x^2 + 1)*(132*sqrt(2)*x - 193*x) + 457*x)*sqrt(sqrt(2) + 1) - 2*(71*x^2
- sqrt(2)*(61*x^2 + 132) + sqrt(x^2 + 1)*(132*sqrt(2) - 193) + 193)*sqrt(sqrt(x^2 + 1) + 1))/(x^3 - x)) - 4*(
x^2 + sqrt(x^2 + 1) - 1)*sqrt(sqrt(x^2 + 1) + 1))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}+1\right ) \sqrt {1+\sqrt {x^{2}+1}}}{x^{2}-1}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^2+1)*(1+(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 {{\left (x^{2} + 1\right )} \sqrt {\sqrt {x^{2} + 1} + 1}}{x^{2} - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((x^2 + 1)*((x^2 + 1)^(1/2) + 1)^(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 {\left (x^{2} + 1\right ) \sqrt {\sqrt {x^{2} + 1} + 1}}{\left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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