3.3.25 \(\int \frac {1}{x \sqrt {1+x}} \, dx\) [225]

Optimal. Leaf size=10 \[ -2 \tanh ^{-1}\left (\sqrt {1+x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {65, 213} \begin {gather*} -2 \tanh ^{-1}\left (\sqrt {x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[1 + x]),x]

[Out]

-2*ArcTanh[Sqrt[1 + x]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 213

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 10, normalized size = 1.00 \begin {gather*} -2 \tanh ^{-1}\left (\sqrt {1+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[1 + x]),x]

[Out]

-2*ArcTanh[Sqrt[1 + x]]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.91, size = 25, normalized size = 2.50 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-2 \text {ArcCoth}\left [\sqrt {1+x}\right ],\text {Abs}\left [1+x\right ]>1\right \}\right \},-2 \text {ArcTanh}\left [\sqrt {1+x}\right ]\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x*Sqrt[1 + x]),x]')

[Out]

Piecewise[{{-2 ArcCoth[Sqrt[1 + x]], Abs[1 + x] > 1}}, -2 ArcTanh[Sqrt[1 + x]]]

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Maple [A]
time = 0.06, size = 9, normalized size = 0.90

method	result	size
derivativedivides	\(-2 \arctanh \left (\sqrt {1+x}\right )\)	\(9\)
default	\(-2 \arctanh \left (\sqrt {1+x}\right )\)	\(9\)
trager	\(-\ln \left (\frac {2 \sqrt {1+x}+2+x}{x}\right )\)	\(18\)
meijerg	\(\frac {\left (\ln \left (x \right )-2 \ln \left (2\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+x}}{2}\right )}{\sqrt {\pi }}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (8) = 16\).
time = 0.25, size = 19, normalized size = 1.90 \begin {gather*} -\log \left (\sqrt {x + 1} + 1\right ) + \log \left (\sqrt {x + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (8) = 16\).
time = 0.33, size = 19, normalized size = 1.90 \begin {gather*} -\log \left (\sqrt {x + 1} + 1\right ) + \log \left (\sqrt {x + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.29, size = 26, normalized size = 2.60 \begin {gather*} \begin {cases} - 2 \operatorname {acoth}{\left (\sqrt {x + 1} \right )} & \text {for}\: \left |{x + 1}\right | > 1 \\- 2 \operatorname {atanh}{\left (\sqrt {x + 1} \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*acoth(sqrt(x + 1)), Abs(x + 1) > 1), (-2*atanh(sqrt(x + 1)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).
time = 0.00, size = 31, normalized size = 3.10 \begin {gather*} 2 \left (\frac {\ln \left |\sqrt {x+1}-1\right |}{2}-\frac {\ln \left (\sqrt {x+1}+1\right )}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(x + 1) + 1) + log(abs(sqrt(x + 1) - 1))

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Mupad [B]
time = 0.16, size = 8, normalized size = 0.80 \begin {gather*} -2\,\mathrm {atanh}\left (\sqrt {x+1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x + 1)^(1/2)),x)

[Out]

-2*atanh((x + 1)^(1/2))

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