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

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

________________________________________________________________________________________

Rubi [A]  time = 0.0069667, 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, Rules used = {641, 195, 217, 206} \[ \frac{1}{3} \left (x^2-1\right )^{3/2}+\frac{1}{2} x \sqrt{x^2-1}-\frac{1}{2} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

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

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0313316, size = 49, normalized size = 1.11 \[ \frac{\left (x^2-1\right ) \left (\sqrt{1-x^2} \left (2 x^2+3 x-2\right )+3 \sin ^{-1}(x)\right )}{6 \sqrt{-\left (x^2-1\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)*Sqrt[-1 + x^2],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.04, size = 33, normalized size = 0.8 \begin{align*}{\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}}}} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.02171, size = 49, normalized size = 1.11 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Fricas [A]  time = 2.13397, size = 90, normalized size = 2.05 \begin{align*} \frac{1}{6} \,{\left (2 \, x^{2} + 3 \, x - 2\right )} \sqrt{x^{2} - 1} + \frac{1}{2} \, \log \left (-x + \sqrt{x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.202426, size = 39, normalized size = 0.89 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.33293, size = 46, normalized size = 1.05 \begin{align*} \frac{1}{6} \,{\left ({\left (2 \, x + 3\right )} x - 2\right )} \sqrt{x^{2} - 1} + \frac{1}{2} \, \log \left ({\left | -x + \sqrt{x^{2} - 1} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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