∫ cos−1(∘__x 1+x ) dx

Optimal. Leaf size=38 \[ (x+1) \left (\sqrt {\frac {1}{x+1}} \sqrt {\frac {x}{x+1}}+\cos ^{-1}\left (\sqrt {\frac {x}{x+1}}\right )\right ) \]

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Rubi [A] time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.50, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4841, 12, 6719, 50, 63, 203} \[ \sqrt {\frac {x}{(x+1)^2}} (x+1)+x \cos ^{-1}\left (\sqrt {\frac {x}{x+1}}\right )-\frac {\sqrt {\frac {x}{(x+1)^2}} (x+1) \tan ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]

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

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*

(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a

, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !IntegerQ[n] || (G

tQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;

FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Int[ArcCos[u_], x_Symbol] :> Simp[x*ArcCos[u], x] + Int[SimplifyIntegrand[(x*D[u, x])/Sqrt[1 - u^2], x], x] /;

InverseFunctionFreeQ[u, x] && !FunctionOfExponentialQ[u, x]

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F

racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !

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

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Mathematica [A] time = 0.06, size = 49, normalized size = 1.29 \[ x \cos ^{-1}\left (\sqrt {\frac {x}{x+1}}\right )+\frac {\sqrt {\frac {x}{(x+1)^2}} (x+1) \left (\sqrt {x}-\tan ^{-1}\left (\sqrt {x}\right )\right )}{\sqrt {x}} \]

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

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^{-1}\left (\sqrt {\frac {x}{1+x}}\right ) \, dx \]

IntegrateAlgebraic[ArcCos[Sqrt[x/(1 + x)]],x]

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fricas [A] time = 0.83, size = 30, normalized size = 0.79 \[ {\left (x + 1\right )} \arccos \left (\sqrt {\frac {x}{x + 1}}\right ) + \sqrt {x + 1} \sqrt {\frac {x}{x + 1}} \]

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

(x + 1)*arccos(sqrt(x/(x + 1))) + sqrt(x + 1)*sqrt(x/(x + 1))

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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn

ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab

s(x+1)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real

):Check [abs(t_nostep+1)]Warning, choosing root of [1,0,%%%{2,[2,2]%%%}+%%%{2,[2,1]%%%}+%%%{-4,[0,2]%%%}+%%%{-

6,[0,1]%%%}+%%%{-2,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{2,[4,3]%%%}+%%%{1,[4,2]%%%}+%%%{-2,[2,3]%%%}+%%%{-4,[2,2]%%

%}+%%%{-2,[2,1]%%%}+%%%{1,[0,2]%%%}+%%%{2,[0,1]%%%}+%%%{1,[0,0]%%%}] at parameters values [86,-97]Warning, cho

osing root of [1,0,%%%{2,[2,2]%%%}+%%%{-2,[2,1]%%%}+%%%{-4,[0,2]%%%}+%%%{6,[0,1]%%%}+%%%{-2,[0,0]%%%},0,%%%{1,

[4,4]%%%}+%%%{-2,[4,3]%%%}+%%%{1,[4,2]%%%}+%%%{2,[2,3]%%%}+%%%{-4,[2,2]%%%}+%%%{2,[2,1]%%%}+%%%{1,[0,2]%%%}+%%

%{-2,[0,1]%%%}+%%%{1,[0,0]%%%}] at parameters values [-82,7]Undef/Unsigned Inf encountered in limit

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method	result	size

default	\(x \arccos \left (\sqrt {\frac {x}{1+x}}\right )-\frac {\sqrt {x}\, \sqrt {\frac {1}{1+x}}\, \left (-\sqrt {x}+\arctan \left (\sqrt {x}\right )\right )}{\sqrt {\frac {x}{1+x}}}\)	\(45\)

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

x*arccos((x/(1+x))^(1/2))-1/(x/(1+x))^(1/2)*x^(1/2)*(1/(1+x))^(1/2)*(-x^(1/2)+arctan(x^(1/2)))

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maxima [B] time = 0.97, size = 78, normalized size = 2.05 \[ -\frac {\arccos \left (\sqrt {\frac {x}{x + 1}}\right )}{\frac {x}{x + 1} - 1} - \frac {\sqrt {-\frac {x}{x + 1} + 1}}{2 \, {\left (\sqrt {\frac {x}{x + 1}} + 1\right )}} - \frac {\sqrt {-\frac {x}{x + 1} + 1}}{2 \, {\left (\sqrt {\frac {x}{x + 1}} - 1\right )}} \]

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

-arccos(sqrt(x/(x + 1)))/(x/(x + 1) - 1) - 1/2*sqrt(-x/(x + 1) + 1)/(sqrt(x/(x + 1)) + 1) - 1/2*sqrt(-x/(x + 1

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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {acos}\left (\sqrt {\frac {x}{x+1}}\right ) \,d x \]

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acos}{\left (\sqrt {\frac {x}{x + 1}} \right )}\, dx \]

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3.86 \(\int \cos ^{-1}(\sqrt {\frac {x}{1+x}}) \, dx\)