3.2.90 \(\int \frac {1}{(1+x^2)^{3/2}} \, dx\) [190]

Optimal. Leaf size=11 \[ \frac {x}{\sqrt {1+x^2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {197} \begin {gather*} \frac {x}{\sqrt {x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)^(-3/2),x]

[Out]

x/Sqrt[1 + x^2]

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\frac {x}{\sqrt {1+x^2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 11, normalized size = 1.00 \begin {gather*} \frac {x}{\sqrt {1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)^(-3/2),x]

[Out]

x/Sqrt[1 + x^2]

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Maple [A]
time = 0.06, size = 10, normalized size = 0.91

method	result	size
gosper	\(\frac {x}{\sqrt {x^{2}+1}}\)	\(10\)
default	\(\frac {x}{\sqrt {x^{2}+1}}\)	\(10\)
trager	\(\frac {x}{\sqrt {x^{2}+1}}\)	\(10\)
meijerg	\(\frac {x}{\sqrt {x^{2}+1}}\)	\(10\)
risch	\(\frac {x}{\sqrt {x^{2}+1}}\)	\(10\)
pseudoelliptic	\(\frac {x}{\sqrt {x^{2}+1}}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.42, size = 9, normalized size = 0.82 \begin {gather*} \frac {x}{\sqrt {x^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^(3/2),x, algorithm="maxima")

[Out]

x/sqrt(x^2 + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (9) = 18\).
time = 0.56, size = 22, normalized size = 2.00 \begin {gather*} \frac {x^{2} + \sqrt {x^{2} + 1} x + 1}{x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^(3/2),x, algorithm="fricas")

[Out]

(x^2 + sqrt(x^2 + 1)*x + 1)/(x^2 + 1)

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Sympy [A]
time = 0.45, size = 8, normalized size = 0.73 \begin {gather*} \frac {x}{\sqrt {x^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+1)**(3/2),x)

[Out]

x/sqrt(x**2 + 1)

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Giac [A]
time = 0.51, size = 9, normalized size = 0.82 \begin {gather*} \frac {x}{\sqrt {x^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^(3/2),x, algorithm="giac")

[Out]

x/sqrt(x^2 + 1)

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Mupad [B]
time = 0.03, size = 9, normalized size = 0.82 \begin {gather*} \frac {x}{\sqrt {x^2+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 + 1)^(3/2),x)

[Out]

x/(x^2 + 1)^(1/2)

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Chatgpt [F] Failed to verify
time = 1.00, size = 10, normalized size = 0.91 \begin {gather*} \ln \left (x +\sqrt {x^{2}+1}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(1/(x^2+1)^(3/2),x)

[Out]

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

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