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

Optimal. Leaf size=119 \[ \frac {1}{192} \sqrt {x^2-1} \sqrt {x^2+\sqrt {x^2-1} x} \left (56 x^2-39\right )+\frac {13 \log \left (\sqrt {x^2-1}-\sqrt {2} \sqrt {x^2+\sqrt {x^2-1} x}+x\right )}{64 \sqrt {2}}+\frac {1}{192} \left (13 x-8 x^3\right ) \sqrt {x^2+\sqrt {x^2-1} x} \]

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Rubi [F]  time = 0.33, 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 x \sqrt {-1+x^2} \sqrt {x^2+x \sqrt {-1+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.12, size = 238, normalized size = 2.00 \begin {gather*} \frac {\sqrt {x^2-1} \left (\sqrt {x^2-1}+x\right )^4 \left (\sqrt {2} \sqrt {x \left (\sqrt {x^2-1}+x\right )} \left (192 x^6-360 x^4+212 x^2+104 \sqrt {x^2-1} x+192 \sqrt {x^2-1} x^5-264 \sqrt {x^2-1} x^3-39\right )-39 \left (4 x^3+4 \sqrt {x^2-1} x^2-\sqrt {x^2-1}-3 x\right ) \sinh ^{-1}\left (\sqrt {x^2-1}+x\right )\right )}{192 \sqrt {2} \left (64 x^8-144 x^6+104 x^4-25 x^2-7 \sqrt {x^2-1} x+64 \sqrt {x^2-1} x^7-112 \sqrt {x^2-1} x^5+56 \sqrt {x^2-1} x^3+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + x^2]*(x + Sqrt[-1 + x^2])^4*(Sqrt[2]*Sqrt[x*(x + Sqrt[-1 + x^2])]*(-39 + 212*x^2 - 360*x^4 + 192*x^
6 + 104*x*Sqrt[-1 + x^2] - 264*x^3*Sqrt[-1 + x^2] + 192*x^5*Sqrt[-1 + x^2]) - 39*(-3*x + 4*x^3 - Sqrt[-1 + x^2
] + 4*x^2*Sqrt[-1 + x^2])*ArcSinh[x + Sqrt[-1 + x^2]]))/(192*Sqrt[2]*(1 - 25*x^2 + 104*x^4 - 144*x^6 + 64*x^8
- 7*x*Sqrt[-1 + x^2] + 56*x^3*Sqrt[-1 + x^2] - 112*x^5*Sqrt[-1 + x^2] + 64*x^7*Sqrt[-1 + x^2]))

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IntegrateAlgebraic [A]  time = 1.58, size = 119, normalized size = 1.00 \begin {gather*} \frac {1}{192} \sqrt {-1+x^2} \left (-39+56 x^2\right ) \sqrt {x^2+x \sqrt {-1+x^2}}+\frac {1}{192} \left (13 x-8 x^3\right ) \sqrt {x^2+x \sqrt {-1+x^2}}+\frac {13 \log \left (x+\sqrt {-1+x^2}-\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right )}{64 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-1 + x^2]*(-39 + 56*x^2)*Sqrt[x^2 + x*Sqrt[-1 + x^2]])/192 + ((13*x - 8*x^3)*Sqrt[x^2 + x*Sqrt[-1 + x^2]
])/192 + (13*Log[x + Sqrt[-1 + x^2] - Sqrt[2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]]])/(64*Sqrt[2])

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fricas [A]  time = 1.03, size = 100, normalized size = 0.84 \begin {gather*} -\frac {1}{192} \, {\left (8 \, x^{3} - {\left (56 \, x^{2} - 39\right )} \sqrt {x^{2} - 1} - 13 \, x\right )} \sqrt {x^{2} + \sqrt {x^{2} - 1} x} + \frac {13}{256} \, \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*(x^2-1)^(1/2)*(x^2+x*(x^2-1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/192*(8*x^3 - (56*x^2 - 39)*sqrt(x^2 - 1) - 13*x)*sqrt(x^2 + sqrt(x^2 - 1)*x) + 13/256*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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{2} + \sqrt {x^{2} - 1} x} \sqrt {x^{2} - 1} x\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int x \sqrt {x^{2}-1}\, \sqrt {x^{2}+x \sqrt {x^{2}-1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {x \left (x + \sqrt {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*(x**2-1)**(1/2)*(x**2+x*(x**2-1)**(1/2))**(1/2),x)

[Out]

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

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