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

Optimal. Leaf size=44 \[ \frac{1}{3} \left (x^2-1\right )^{3/2}+\frac{1}{2} x \sqrt{x^2-1}-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.028681, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{3} \left (x^2-1\right )^{3/2}+\frac{1}{2} x \sqrt{x^2-1}-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)*Sqrt[-1 + x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.99358, size = 34, normalized size = 0.77 \[ \frac{x \sqrt{x^{2} - 1}}{2} + \frac{\left (x^{2} - 1\right )^{\frac{3}{2}}}{3} - \frac{\operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} - 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)*(x**2-1)**(1/2),x)

[Out]

x*sqrt(x**2 - 1)/2 + (x**2 - 1)**(3/2)/3 - atanh(x/sqrt(x**2 - 1))/2

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Mathematica [A]  time = 0.0273182, size = 39, normalized size = 0.89 \[ \frac{1}{6} \left (\sqrt{x^2-1} \left (2 x^2+3 x-2\right )-3 \log \left (\sqrt{x^2-1}+x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-1 + x^2]*(-2 + 3*x + 2*x^2) - 3*Log[x + Sqrt[-1 + x^2]])/6

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Maple [A]  time = 0.006, size = 33, normalized size = 0.8 \[{\frac{x}{2}\sqrt{{x}^{2}-1}}-{\frac{1}{2}\ln \left ( x+\sqrt{{x}^{2}-1} \right ) }+{\frac{1}{3} \left ({x}^{2}-1 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)*(x^2-1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.707162, size = 49, normalized size = 1.11 \[ \frac{1}{3} \,{\left (x^{2} - 1\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{x^{2} - 1} x - \frac{1}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 - 1)*(x + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.215104, size = 177, normalized size = 4.02 \[ -\frac{8 \, x^{6} + 12 \, x^{5} - 18 \, x^{4} - 15 \, x^{3} + 12 \, x^{2} - 3 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} - 1\right )} \sqrt{x^{2} - 1} - 3 \, x\right )} \log \left (-x + \sqrt{x^{2} - 1}\right ) -{\left (8 \, x^{5} + 12 \, x^{4} - 14 \, x^{3} - 9 \, x^{2} + 6 \, x\right )} \sqrt{x^{2} - 1} + 3 \, x - 2}{6 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} - 1\right )} \sqrt{x^{2} - 1} - 3 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 - 1)*(x + 1),x, algorithm="fricas")

[Out]

-1/6*(8*x^6 + 12*x^5 - 18*x^4 - 15*x^3 + 12*x^2 - 3*(4*x^3 - (4*x^2 - 1)*sqrt(x^
2 - 1) - 3*x)*log(-x + sqrt(x^2 - 1)) - (8*x^5 + 12*x^4 - 14*x^3 - 9*x^2 + 6*x)*
sqrt(x^2 - 1) + 3*x - 2)/(4*x^3 - (4*x^2 - 1)*sqrt(x^2 - 1) - 3*x)

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Sympy [A]  time = 0.522439, size = 39, normalized size = 0.89 \[ \frac{x^{2} \sqrt{x^{2} - 1}}{3} + \frac{x \sqrt{x^{2} - 1}}{2} - \frac{\sqrt{x^{2} - 1}}{3} - \frac{\operatorname{acosh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)*(x**2-1)**(1/2),x)

[Out]

x**2*sqrt(x**2 - 1)/3 + x*sqrt(x**2 - 1)/2 - sqrt(x**2 - 1)/3 - acosh(x)/2

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GIAC/XCAS [A]  time = 0.22906, size = 46, normalized size = 1.05 \[ \frac{1}{6} \,{\left ({\left (2 \, x + 3\right )} x - 2\right )} \sqrt{x^{2} - 1} + \frac{1}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 - 1)*(x + 1),x, algorithm="giac")

[Out]

1/6*((2*x + 3)*x - 2)*sqrt(x^2 - 1) + 1/2*ln(abs(-x + sqrt(x^2 - 1)))