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

Optimal. Leaf size=45 \[ -\sqrt {2} \log \left (\sqrt {x^2-1}-\sqrt {2} \sqrt {x^2+\sqrt {x^2-1} x}+x\right ) \]

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Rubi [A]  time = 0.18, antiderivative size = 18, normalized size of antiderivative = 0.40, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2130, 215} \begin {gather*} \sqrt {2} \sinh ^{-1}\left (\sqrt {x^2-1}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2]*ArcSinh[x + Sqrt[-1 + x^2]]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 2130

Int[Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)*Sqrt[(c_) + (d_.)*(x_)^2]]/((x_)*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> D
ist[(Sqrt[2]*b)/a, Subst[Int[1/Sqrt[1 + x^2/a], x], x, a*x + b*Sqrt[c + d*x^2]], x] /; FreeQ[{a, b, c, d}, x]
&& EqQ[a^2 - b^2*d, 0] && EqQ[b^2*c + a, 0]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 70, normalized size = 1.56 \begin {gather*} \frac {\sqrt {2} \sqrt {x \left (\sqrt {x^2-1}+x\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {\sqrt {x^2-1}+x}}\right )}{\sqrt {x} \sqrt {\sqrt {x^2-1}+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.31, size = 45, normalized size = 1.00 \begin {gather*} -\sqrt {2} \log \left (x+\sqrt {-1+x^2}-\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2]*Log[x + Sqrt[-1 + x^2] - Sqrt[2]*Sqrt[x^2 + x*Sqrt[-1 + x^2]]])

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fricas [A]  time = 0.78, size = 57, normalized size = 1.27 \begin {gather*} \frac {1}{2} \, \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+x*(x^2-1)^(1/2))^(1/2)/x/(x^2-1)^(1/2),x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x*(x^2-1)^(1/2))^(1/2)/x/(x^2-1)^(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 = -0.02 \begin {gather*} \int \frac {\sqrt {x\,\sqrt {x^2-1}+x^2}}{x\,\sqrt {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)^(1/2)/(x*(x^2 - 1)^(1/2)),x)

[Out]

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

[Out]

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

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