Integrand size = 10, antiderivative size = 86 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^5} \, dx=\frac {\sqrt {1-x}}{28 x^{7/2}}+\frac {3 \sqrt {1-x}}{70 x^{5/2}}+\frac {2 \sqrt {1-x}}{35 x^{3/2}}+\frac {4 \sqrt {1-x}}{35 \sqrt {x}}-\frac {\arccos \left (\sqrt {x}\right )}{4 x^4} \] Output:
1/28*(1-x)^(1/2)/x^(7/2)+3/70*(1-x)^(1/2)/x^(5/2)+2/35*(1-x)^(1/2)/x^(3/2) +4/35*(1-x)^(1/2)/x^(1/2)-1/4*arccos(x^(1/2))/x^4
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.49 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^5} \, dx=\frac {\sqrt {-((-1+x) x)} \left (5+6 x+8 x^2+16 x^3\right )-35 \arccos \left (\sqrt {x}\right )}{140 x^4} \] Input:
Integrate[ArcCos[Sqrt[x]]/x^5,x]
Output:
(Sqrt[-((-1 + x)*x)]*(5 + 6*x + 8*x^2 + 16*x^3) - 35*ArcCos[Sqrt[x]])/(140 *x^4)
Time = 0.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5342, 27, 55, 55, 55, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\arccos \left (\sqrt {x}\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 5342 |
\(\displaystyle -\frac {1}{4} \int \frac {1}{2 \sqrt {1-x} x^{9/2}}dx-\frac {\arccos \left (\sqrt {x}\right )}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{8} \int \frac {1}{\sqrt {1-x} x^{9/2}}dx-\frac {\arccos \left (\sqrt {x}\right )}{4 x^4}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{8} \left (\frac {2 \sqrt {1-x}}{7 x^{7/2}}-\frac {6}{7} \int \frac {1}{\sqrt {1-x} x^{7/2}}dx\right )-\frac {\arccos \left (\sqrt {x}\right )}{4 x^4}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{8} \left (\frac {2 \sqrt {1-x}}{7 x^{7/2}}-\frac {6}{7} \left (\frac {4}{5} \int \frac {1}{\sqrt {1-x} x^{5/2}}dx-\frac {2 \sqrt {1-x}}{5 x^{5/2}}\right )\right )-\frac {\arccos \left (\sqrt {x}\right )}{4 x^4}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{8} \left (\frac {2 \sqrt {1-x}}{7 x^{7/2}}-\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{\sqrt {1-x} x^{3/2}}dx-\frac {2 \sqrt {1-x}}{3 x^{3/2}}\right )-\frac {2 \sqrt {1-x}}{5 x^{5/2}}\right )\right )-\frac {\arccos \left (\sqrt {x}\right )}{4 x^4}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {1}{8} \left (\frac {2 \sqrt {1-x}}{7 x^{7/2}}-\frac {6}{7} \left (\frac {4}{5} \left (-\frac {2 \sqrt {1-x}}{3 x^{3/2}}-\frac {4 \sqrt {1-x}}{3 \sqrt {x}}\right )-\frac {2 \sqrt {1-x}}{5 x^{5/2}}\right )\right )-\frac {\arccos \left (\sqrt {x}\right )}{4 x^4}\) |
Input:
Int[ArcCos[Sqrt[x]]/x^5,x]
Output:
((-6*((4*((-2*Sqrt[1 - x])/(3*x^(3/2)) - (4*Sqrt[1 - x])/(3*Sqrt[x])))/5 - (2*Sqrt[1 - x])/(5*x^(5/2))))/7 + (2*Sqrt[1 - x])/(7*x^(7/2)))/8 - ArcCos [Sqrt[x]]/(4*x^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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] - Simp[d*(S implify[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] && 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] || !SumSimp lerQ[n, 1])
Int[((a_.) + ArcCos[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim p[(c + d*x)^(m + 1)*((a + b*ArcCos[u])/(d*(m + 1))), x] + Simp[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]
Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {\sqrt {1-x}}{28 x^{\frac {7}{2}}}+\frac {3 \sqrt {1-x}}{70 x^{\frac {5}{2}}}+\frac {2 \sqrt {1-x}}{35 x^{\frac {3}{2}}}+\frac {4 \sqrt {1-x}}{35 \sqrt {x}}-\frac {\arccos \left (\sqrt {x}\right )}{4 x^{4}}\) | \(59\) |
default | \(\frac {\sqrt {1-x}}{28 x^{\frac {7}{2}}}+\frac {3 \sqrt {1-x}}{70 x^{\frac {5}{2}}}+\frac {2 \sqrt {1-x}}{35 x^{\frac {3}{2}}}+\frac {4 \sqrt {1-x}}{35 \sqrt {x}}-\frac {\arccos \left (\sqrt {x}\right )}{4 x^{4}}\) | \(59\) |
parts | \(\frac {\sqrt {1-x}}{28 x^{\frac {7}{2}}}+\frac {3 \sqrt {1-x}}{70 x^{\frac {5}{2}}}+\frac {2 \sqrt {1-x}}{35 x^{\frac {3}{2}}}+\frac {4 \sqrt {1-x}}{35 \sqrt {x}}-\frac {\arccos \left (\sqrt {x}\right )}{4 x^{4}}\) | \(59\) |
Input:
int(arccos(x^(1/2))/x^5,x,method=_RETURNVERBOSE)
Output:
1/28*(1-x)^(1/2)/x^(7/2)+3/70*(1-x)^(1/2)/x^(5/2)+2/35*(1-x)^(1/2)/x^(3/2) +4/35*(1-x)^(1/2)/x^(1/2)-1/4*arccos(x^(1/2))/x^4
Time = 0.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.44 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^5} \, dx=\frac {{\left (16 \, x^{3} + 8 \, x^{2} + 6 \, x + 5\right )} \sqrt {x} \sqrt {-x + 1} - 35 \, \arccos \left (\sqrt {x}\right )}{140 \, x^{4}} \] Input:
integrate(arccos(x^(1/2))/x^5,x, algorithm="fricas")
Output:
1/140*((16*x^3 + 8*x^2 + 6*x + 5)*sqrt(x)*sqrt(-x + 1) - 35*arccos(sqrt(x) ))/x^4
Time = 22.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^5} \, dx=- \frac {\begin {cases} - \frac {\sqrt {1 - x}}{\sqrt {x}} - \frac {\left (1 - x\right )^{\frac {3}{2}}}{x^{\frac {3}{2}}} - \frac {3 \left (1 - x\right )^{\frac {5}{2}}}{5 x^{\frac {5}{2}}} - \frac {\left (1 - x\right )^{\frac {7}{2}}}{7 x^{\frac {7}{2}}} & \text {for}\: \sqrt {x} > -1 \wedge \sqrt {x} < 1 \end {cases}}{4} - \frac {\operatorname {acos}{\left (\sqrt {x} \right )}}{4 x^{4}} \] Input:
integrate(acos(x**(1/2))/x**5,x)
Output:
-Piecewise((-sqrt(1 - x)/sqrt(x) - (1 - x)**(3/2)/x**(3/2) - 3*(1 - x)**(5 /2)/(5*x**(5/2)) - (1 - x)**(7/2)/(7*x**(7/2)), (sqrt(x) > -1) & (sqrt(x) < 1)))/4 - acos(sqrt(x))/(4*x**4)
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^5} \, dx=\frac {4 \, \sqrt {-x + 1}}{35 \, \sqrt {x}} + \frac {2 \, \sqrt {-x + 1}}{35 \, x^{\frac {3}{2}}} + \frac {3 \, \sqrt {-x + 1}}{70 \, x^{\frac {5}{2}}} + \frac {\sqrt {-x + 1}}{28 \, x^{\frac {7}{2}}} - \frac {\arccos \left (\sqrt {x}\right )}{4 \, x^{4}} \] Input:
integrate(arccos(x^(1/2))/x^5,x, algorithm="maxima")
Output:
4/35*sqrt(-x + 1)/sqrt(x) + 2/35*sqrt(-x + 1)/x^(3/2) + 3/70*sqrt(-x + 1)/ x^(5/2) + 1/28*sqrt(-x + 1)/x^(7/2) - 1/4*arccos(sqrt(x))/x^4
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (58) = 116\).
Time = 0.13 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.60 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^5} \, dx=\frac {{\left (\sqrt {-x + 1} - 1\right )}^{7}}{3584 \, x^{\frac {7}{2}}} + \frac {7 \, {\left (\sqrt {-x + 1} - 1\right )}^{5}}{2560 \, x^{\frac {5}{2}}} + \frac {7 \, {\left (\sqrt {-x + 1} - 1\right )}^{3}}{512 \, x^{\frac {3}{2}}} + \frac {35 \, {\left (\sqrt {-x + 1} - 1\right )}}{512 \, \sqrt {x}} - \frac {{\left (\frac {1225 \, {\left (\sqrt {-x + 1} - 1\right )}^{6}}{x^{3}} + \frac {245 \, {\left (\sqrt {-x + 1} - 1\right )}^{4}}{x^{2}} + \frac {49 \, {\left (\sqrt {-x + 1} - 1\right )}^{2}}{x} + 5\right )} x^{\frac {7}{2}}}{17920 \, {\left (\sqrt {-x + 1} - 1\right )}^{7}} - \frac {\arccos \left (\sqrt {x}\right )}{4 \, x^{4}} \] Input:
integrate(arccos(x^(1/2))/x^5,x, algorithm="giac")
Output:
1/3584*(sqrt(-x + 1) - 1)^7/x^(7/2) + 7/2560*(sqrt(-x + 1) - 1)^5/x^(5/2) + 7/512*(sqrt(-x + 1) - 1)^3/x^(3/2) + 35/512*(sqrt(-x + 1) - 1)/sqrt(x) - 1/17920*(1225*(sqrt(-x + 1) - 1)^6/x^3 + 245*(sqrt(-x + 1) - 1)^4/x^2 + 4 9*(sqrt(-x + 1) - 1)^2/x + 5)*x^(7/2)/(sqrt(-x + 1) - 1)^7 - 1/4*arccos(sq rt(x))/x^4
Timed out. \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^5} \, dx=\int \frac {\mathrm {acos}\left (\sqrt {x}\right )}{x^5} \,d x \] Input:
int(acos(x^(1/2))/x^5,x)
Output:
int(acos(x^(1/2))/x^5, x)
Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {\arccos \left (\sqrt {x}\right )}{x^5} \, dx=\frac {-35 \mathit {acos} \left (\sqrt {x}\right )+16 \sqrt {x}\, \sqrt {1-x}\, x^{3}+8 \sqrt {x}\, \sqrt {1-x}\, x^{2}+6 \sqrt {x}\, \sqrt {1-x}\, x +5 \sqrt {x}\, \sqrt {1-x}}{140 x^{4}} \] Input:
int(acos(x^(1/2))/x^5,x)
Output:
( - 35*acos(sqrt(x)) + 16*sqrt(x)*sqrt( - x + 1)*x**3 + 8*sqrt(x)*sqrt( - x + 1)*x**2 + 6*sqrt(x)*sqrt( - x + 1)*x + 5*sqrt(x)*sqrt( - x + 1))/(140* x**4)