\(\int \frac {\arccos (\sqrt {x})}{x^3} \, dx\) [65]

Optimal result

Integrand size = 10, antiderivative size = 50 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^3} \, dx=\frac {\sqrt {1-x}}{6 x^{3/2}}+\frac {\sqrt {1-x}}{3 \sqrt {x}}-\frac {\arccos \left (\sqrt {x}\right )}{2 x^2} \]

[Out]

-1/2*arccos(x^(1/2))/x^2+1/6*(1-x)^(1/2)/x^(3/2)+1/3*(1-x)^(1/2)/x^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4927, 12, 47, 37} \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^3} \, dx=-\frac {\arccos \left (\sqrt {x}\right )}{2 x^2}+\frac {\sqrt {1-x}}{6 x^{3/2}}+\frac {\sqrt {1-x}}{3 \sqrt {x}} \]

[In]

Int[ArcCos[Sqrt[x]]/x^3,x]

[Out]

Sqrt[1 - x]/(6*x^(3/2)) + Sqrt[1 - x]/(3*Sqrt[x]) - ArcCos[Sqrt[x]]/(2*x^2)

Rule 12

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 4927

Int[((a_.) + ArcCos[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcCos[
u])/(d*(m + 1))), x] + Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/Sqrt[1 - u^2]), x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos \left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2} \int \frac {1}{2 \sqrt {1-x} x^{5/2}} \, dx \\ & = -\frac {\arccos \left (\sqrt {x}\right )}{2 x^2}-\frac {1}{4} \int \frac {1}{\sqrt {1-x} x^{5/2}} \, dx \\ & = \frac {\sqrt {1-x}}{6 x^{3/2}}-\frac {\arccos \left (\sqrt {x}\right )}{2 x^2}-\frac {1}{6} \int \frac {1}{\sqrt {1-x} x^{3/2}} \, dx \\ & = \frac {\sqrt {1-x}}{6 x^{3/2}}+\frac {\sqrt {1-x}}{3 \sqrt {x}}-\frac {\arccos \left (\sqrt {x}\right )}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^3} \, dx=\left (\frac {1}{6 x^{3/2}}+\frac {1}{3 \sqrt {x}}\right ) \sqrt {1-x}-\frac {\arccos \left (\sqrt {x}\right )}{2 x^2} \]

[In]

Integrate[ArcCos[Sqrt[x]]/x^3,x]

[Out]

(1/(6*x^(3/2)) + 1/(3*Sqrt[x]))*Sqrt[1 - x] - ArcCos[Sqrt[x]]/(2*x^2)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(-\frac {\arccos \left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {1-x}}{6 x^{\frac {3}{2}}}+\frac {\sqrt {1-x}}{3 \sqrt {x}}\)	\(35\)
default	\(-\frac {\arccos \left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {1-x}}{6 x^{\frac {3}{2}}}+\frac {\sqrt {1-x}}{3 \sqrt {x}}\)	\(35\)
parts	\(-\frac {\arccos \left (\sqrt {x}\right )}{2 x^{2}}+\frac {\sqrt {1-x}}{6 x^{\frac {3}{2}}}+\frac {\sqrt {1-x}}{3 \sqrt {x}}\)	\(35\)

[In]

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

[Out]

-1/2*arccos(x^(1/2))/x^2+1/6*(1-x)^(1/2)/x^(3/2)+1/3*(1-x)^(1/2)/x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.56 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^3} \, dx=\frac {{\left (2 \, x + 1\right )} \sqrt {x} \sqrt {-x + 1} - 3 \, \arccos \left (\sqrt {x}\right )}{6 \, x^{2}} \]

[In]

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

[Out]

1/6*((2*x + 1)*sqrt(x)*sqrt(-x + 1) - 3*arccos(sqrt(x)))/x^2

Sympy [A] (verification not implemented)

Time = 2.79 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.06 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^3} \, dx=- \frac {\begin {cases} - \frac {\sqrt {1 - x}}{\sqrt {x}} - \frac {\left (1 - x\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}}} & \text {for}\: \sqrt {x} > -1 \wedge \sqrt {x} < 1 \end {cases}}{2} - \frac {\operatorname {acos}{\left (\sqrt {x} \right )}}{2 x^{2}} \]

[In]

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

[Out]

-Piecewise((-sqrt(1 - x)/sqrt(x) - (1 - x)**(3/2)/(3*x**(3/2)), (sqrt(x) > -1) & (sqrt(x) < 1)))/2 - acos(sqrt
(x))/(2*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^3} \, dx=\frac {\sqrt {-x + 1}}{3 \, \sqrt {x}} + \frac {\sqrt {-x + 1}}{6 \, x^{\frac {3}{2}}} - \frac {\arccos \left (\sqrt {x}\right )}{2 \, x^{2}} \]

[In]

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

[Out]

1/3*sqrt(-x + 1)/sqrt(x) + 1/6*sqrt(-x + 1)/x^(3/2) - 1/2*arccos(sqrt(x))/x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (34) = 68\).

Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.48 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^3} \, dx=\frac {{\left (\sqrt {-x + 1} - 1\right )}^{3}}{48 \, x^{\frac {3}{2}}} + \frac {3 \, {\left (\sqrt {-x + 1} - 1\right )}}{16 \, \sqrt {x}} - \frac {x^{\frac {3}{2}} {\left (\frac {9 \, {\left (\sqrt {-x + 1} - 1\right )}^{2}}{x} + 1\right )}}{48 \, {\left (\sqrt {-x + 1} - 1\right )}^{3}} - \frac {\arccos \left (\sqrt {x}\right )}{2 \, x^{2}} \]

[In]

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

[Out]

1/48*(sqrt(-x + 1) - 1)^3/x^(3/2) + 3/16*(sqrt(-x + 1) - 1)/sqrt(x) - 1/48*x^(3/2)*(9*(sqrt(-x + 1) - 1)^2/x +
1)/(sqrt(-x + 1) - 1)^3 - 1/2*arccos(sqrt(x))/x^2

Mupad [F(-1)]

Timed out. \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^3} \, dx=\int \frac {\mathrm {acos}\left (\sqrt {x}\right )}{x^3} \,d x \]

[In]

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

[Out]

int(acos(x^(1/2))/x^3, x)