\(\int \frac {x}{\sqrt {1+x^2} \sqrt {1+\sqrt {1+x^2}}} \, dx\) [20]

Optimal result

Integrand size = 26, antiderivative size = 17 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {1+\sqrt {1+x^2}}} \, dx=2 \sqrt {1+\sqrt {1+x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6818} \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {1+\sqrt {1+x^2}}} \, dx=2 \sqrt {\sqrt {x^2+1}+1} \]

[In]

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

[Out]

2*Sqrt[1 + Sqrt[1 + x^2]]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = 2 \sqrt {1+\sqrt {1+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {1+\sqrt {1+x^2}}} \, dx=2 \sqrt {1+\sqrt {1+x^2}} \]

[In]

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

[Out]

2*Sqrt[1 + Sqrt[1 + x^2]]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(2 \sqrt {1+\sqrt {x^{2}+1}}\)	\(14\)
default	\(2 \sqrt {1+\sqrt {x^{2}+1}}\)	\(14\)

[In]

int(x/(x^2+1)^(1/2)/(1+(x^2+1)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {1+\sqrt {1+x^2}}} \, dx=2 \, \sqrt {\sqrt {x^{2} + 1} + 1} \]

[In]

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

[Out]

2*sqrt(sqrt(x^2 + 1) + 1)

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {1+\sqrt {1+x^2}}} \, dx=2 \sqrt {\sqrt {x^{2} + 1} + 1} \]

[In]

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

[Out]

2*sqrt(sqrt(x**2 + 1) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {1+\sqrt {1+x^2}}} \, dx=2 \, \sqrt {\sqrt {x^{2} + 1} + 1} \]

[In]

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

[Out]

2*sqrt(sqrt(x^2 + 1) + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {1+\sqrt {1+x^2}}} \, dx=2 \, \sqrt {\sqrt {x^{2} + 1} + 1} \]

[In]

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

[Out]

2*sqrt(sqrt(x^2 + 1) + 1)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {1+\sqrt {1+x^2}}} \, dx=2\,\sqrt {\sqrt {x^2+1}+1} \]

[In]

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

[Out]

2*((x^2 + 1)^(1/2) + 1)^(1/2)