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

Optimal. Leaf size=48 \[ \frac {\sqrt {x^2+1} \sqrt {x^2+2 x+1}}{x+1}+\frac {\sqrt {x^2+2 x+1} \sinh ^{-1}(x)}{x+1} \]

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Rubi [A]  time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {970, 641, 215} \begin {gather*} \frac {\sqrt {x^2+1} \sqrt {x^2+2 x+1}}{x+1}+\frac {\sqrt {x^2+2 x+1} \sinh ^{-1}(x)}{x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + x^2]*Sqrt[1 + 2*x + x^2])/(1 + x) + (Sqrt[1 + 2*x + x^2]*ArcSinh[x])/(1 + x)

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 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 970

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[(a + b*x + c*x^2)^F
racPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(b + 2*c*x)^(2*p)*(d + f*x^2)^q, x], x] /; Free
Q[{a, b, c, d, f, p, q}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 27, normalized size = 0.56 \begin {gather*} \frac {\sqrt {(x+1)^2} \left (\sqrt {x^2+1}+\sinh ^{-1}(x)\right )}{x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(1 + x)^2]*(Sqrt[1 + x^2] + ArcSinh[x]))/(1 + x)

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IntegrateAlgebraic [A]  time = 0.14, size = 41, normalized size = 0.85 \begin {gather*} \frac {\sqrt {(x+1)^2} \left (\sqrt {x^2+1}-\log \left (\sqrt {x^2+1}-x\right )\right )}{x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 22, normalized size = 0.46 \begin {gather*} \sqrt {x^{2} + 1} - \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 1) - log(-x + sqrt(x^2 + 1))

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giac [A]  time = 0.23, size = 49, normalized size = 1.02 \begin {gather*} -{\left (\sqrt {2} - \log \left (\sqrt {2} + 1\right )\right )} \mathrm {sgn}\left (x + 1\right ) - \log \left (-x + \sqrt {x^{2} + 1}\right ) \mathrm {sgn}\left (x + 1\right ) + \sqrt {x^{2} + 1} \mathrm {sgn}\left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(2) - log(sqrt(2) + 1))*sgn(x + 1) - log(-x + sqrt(x^2 + 1))*sgn(x + 1) + sqrt(x^2 + 1)*sgn(x + 1)

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maple [C]  time = 0.05, size = 16, normalized size = 0.33 \begin {gather*} \left (\arcsinh \relax (x )+\sqrt {x^{2}+1}\right ) \mathrm {csgn}\left (x +1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

csgn(1+x)*(arcsinh(x)+(x^2+1)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {{\left (x+1\right )}^2}}{\sqrt {x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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