Integrand size = 16, antiderivative size = 55 \[ \int \arcsin \left (\sqrt {\frac {-a+x}{a+x}}\right ) \, dx=-\frac {\sqrt {2} a \sqrt {\frac {-a+x}{a+x}}}{\sqrt {\frac {a}{a+x}}}+(a+x) \arcsin \left (\sqrt {\frac {-a+x}{a+x}}\right ) \] Output:
(a+x)*arcsin(((-a+x)/(a+x))^(1/2))-a*2^(1/2)*((-a+x)/(a+x))^(1/2)/(a/(a+x) )^(1/2)
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.80 \[ \int \arcsin \left (\sqrt {\frac {-a+x}{a+x}}\right ) \, dx=x \arcsin \left (\sqrt {\frac {-a+x}{a+x}}\right )+\frac {\sqrt {\frac {a}{a+x}} \left (2 a-2 x+\sqrt {2} \sqrt {a} \sqrt {-a+x} \arctan \left (\frac {\sqrt {-a+x}}{\sqrt {2} \sqrt {a}}\right )\right )}{\sqrt {2} \sqrt {\frac {-a+x}{a+x}}} \] Input:
Integrate[ArcSin[Sqrt[(-a + x)/(a + x)]],x]
Output:
x*ArcSin[Sqrt[(-a + x)/(a + x)]] + (Sqrt[a/(a + x)]*(2*a - 2*x + Sqrt[2]*S qrt[a]*Sqrt[-a + x]*ArcTan[Sqrt[-a + x]/(Sqrt[2]*Sqrt[a])]))/(Sqrt[2]*Sqrt [(-a + x)/(a + x)])
Leaf count is larger than twice the leaf count of optimal. \(136\) vs. \(2(55)=110\).
Time = 0.88 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.47, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5339, 27, 2045, 7268, 2044, 298, 223}
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.
\(\displaystyle \int \arcsin \left (\sqrt {\frac {x-a}{a+x}}\right ) \, dx\) |
\(\Big \downarrow \) 5339 |
\(\displaystyle x \arcsin \left (\sqrt {-\frac {a-x}{a+x}}\right )-\int \frac {x \left (\frac {a}{a+x}\right )^{3/2}}{\sqrt {2} a \sqrt {\frac {x-a}{a+x}}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle x \arcsin \left (\sqrt {-\frac {a-x}{a+x}}\right )-\frac {\int \frac {x \left (\frac {a}{a+x}\right )^{3/2}}{\sqrt {-\frac {a-x}{a+x}}}dx}{\sqrt {2} a}\) |
\(\Big \downarrow \) 2045 |
\(\displaystyle x \arcsin \left (\sqrt {-\frac {a-x}{a+x}}\right )-\frac {\sqrt {\frac {a}{a+x}} \sqrt {\frac {x}{a}+1} \int \frac {x}{\sqrt {-\frac {a-x}{a+x}} \left (\frac {x}{a}+1\right )^{3/2}}dx}{\sqrt {2} a}\) |
\(\Big \downarrow \) 7268 |
\(\displaystyle x \arcsin \left (\sqrt {-\frac {a-x}{a+x}}\right )-a \sqrt {\frac {a}{a+x}} \sqrt {\frac {x}{a}+1} \int \left (1-\frac {a-x}{a+x}\right ) \left (\frac {1}{\frac {a-x}{a+x}+1}\right )^{3/2}d\sqrt {-\frac {a-x}{a+x}}\) |
\(\Big \downarrow \) 2044 |
\(\displaystyle x \arcsin \left (\sqrt {-\frac {a-x}{a+x}}\right )-a \sqrt {\frac {a}{a+x}} \sqrt {\frac {x}{a}+1} \sqrt {\frac {1}{\frac {a-x}{a+x}+1}} \sqrt {\frac {a-x}{a+x}+1} \int \frac {1-\frac {a-x}{a+x}}{\left (\frac {a-x}{a+x}+1\right )^{3/2}}d\sqrt {-\frac {a-x}{a+x}}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle x \arcsin \left (\sqrt {-\frac {a-x}{a+x}}\right )-a \sqrt {\frac {a}{a+x}} \sqrt {\frac {x}{a}+1} \sqrt {\frac {1}{\frac {a-x}{a+x}+1}} \sqrt {\frac {a-x}{a+x}+1} \left (\frac {2 \sqrt {-\frac {a-x}{a+x}}}{\sqrt {\frac {a-x}{a+x}+1}}-\int \frac {1}{\sqrt {\frac {a-x}{a+x}+1}}d\sqrt {-\frac {a-x}{a+x}}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle x \arcsin \left (\sqrt {-\frac {a-x}{a+x}}\right )-a \sqrt {\frac {a}{a+x}} \sqrt {\frac {x}{a}+1} \sqrt {\frac {1}{\frac {a-x}{a+x}+1}} \sqrt {\frac {a-x}{a+x}+1} \left (\frac {2 \sqrt {-\frac {a-x}{a+x}}}{\sqrt {\frac {a-x}{a+x}+1}}-\arcsin \left (\sqrt {-\frac {a-x}{a+x}}\right )\right )\) |
Input:
Int[ArcSin[Sqrt[(-a + x)/(a + x)]],x]
Output:
-(a*Sqrt[a/(a + x)]*Sqrt[1 + x/a]*Sqrt[(1 + (a - x)/(a + x))^(-1)]*Sqrt[1 + (a - x)/(a + x)]*((2*Sqrt[-((a - x)/(a + x))])/Sqrt[1 + (a - x)/(a + x)] - ArcSin[Sqrt[-((a - x)/(a + x))]])) + x*ArcSin[Sqrt[-((a - x)/(a + x))]]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[(u_.)*((c_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[S imp[(c*(a + b*x^n)^q)^p/(a + b*x^n)^(p*q)] Int[u*(a + b*x^n)^(p*q), x], x ] /; FreeQ[{a, b, c, n, p, q}, x] && GeQ[a, 0]
Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[Si mp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/a))^(p*q)] Int[u*(1 + b*(x^n/a))^(p*q) , x], x] /; FreeQ[{a, b, c, n, p, q}, x] && !GeQ[a, 0]
Int[ArcSin[u_], x_Symbol] :> Simp[x*ArcSin[u], x] - Int[SimplifyIntegrand[x *(D[u, x]/Sqrt[1 - u^2]), x], x] /; InverseFunctionFreeQ[u, x] && !Functio nOfExponentialQ[u, x]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears [u, x]}, Simp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls t[[2]])], x] /; !FalseQ[lst]]
Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.56
method | result | size |
default | \(x \arcsin \left (\sqrt {\frac {-a +x}{a +x}}\right )+\frac {\sqrt {-a +x}\, \sqrt {2}\, \sqrt {\frac {a}{a +x}}\, \left (-2 \sqrt {-a +x}+\sqrt {a}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-a +x}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{2 \sqrt {-\frac {a -x}{a +x}}}\) | \(86\) |
parts | \(x \arcsin \left (\sqrt {\frac {-a +x}{a +x}}\right )+\frac {\sqrt {-a +x}\, \sqrt {2}\, \sqrt {\frac {a}{a +x}}\, \left (-2 \sqrt {-a +x}+\sqrt {a}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-a +x}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{2 \sqrt {-\frac {a -x}{a +x}}}\) | \(86\) |
Input:
int(arcsin(((-a+x)/(a+x))^(1/2)),x,method=_RETURNVERBOSE)
Output:
x*arcsin(((-a+x)/(a+x))^(1/2))+1/2/(-(a-x)/(a+x))^(1/2)*(-a+x)^(1/2)*2^(1/ 2)*(a/(a+x))^(1/2)*(-2*(-a+x)^(1/2)+a^(1/2)*2^(1/2)*arctan(1/2*(-a+x)^(1/2 )*2^(1/2)/a^(1/2)))
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \arcsin \left (\sqrt {\frac {-a+x}{a+x}}\right ) \, dx=-\sqrt {2} {\left (a + x\right )} \sqrt {-\frac {a - x}{a + x}} \sqrt {\frac {a}{a + x}} + {\left (a + x\right )} \arcsin \left (\sqrt {-\frac {a - x}{a + x}}\right ) \] Input:
integrate(arcsin(((-a+x)/(a+x))^(1/2)),x, algorithm="fricas")
Output:
-sqrt(2)*(a + x)*sqrt(-(a - x)/(a + x))*sqrt(a/(a + x)) + (a + x)*arcsin(s qrt(-(a - x)/(a + x)))
\[ \int \arcsin \left (\sqrt {\frac {-a+x}{a+x}}\right ) \, dx=\int \operatorname {asin}{\left (\sqrt {\frac {- a + x}{a + x}} \right )}\, dx \] Input:
integrate(asin(((-a+x)/(a+x))**(1/2)),x)
Output:
Integral(asin(sqrt((-a + x)/(a + x))), x)
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (49) = 98\).
Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.87 \[ \int \arcsin \left (\sqrt {\frac {-a+x}{a+x}}\right ) \, dx=a {\left (\frac {2 \, \arcsin \left (\sqrt {-\frac {a - x}{a + x}}\right )}{\frac {a - x}{a + x} + 1} + \frac {\sqrt {\frac {a - x}{a + x} + 1}}{\sqrt {-\frac {a - x}{a + x}} + 1} + \frac {\sqrt {\frac {a - x}{a + x} + 1}}{\sqrt {-\frac {a - x}{a + x}} - 1}\right )} \] Input:
integrate(arcsin(((-a+x)/(a+x))^(1/2)),x, algorithm="maxima")
Output:
a*(2*arcsin(sqrt(-(a - x)/(a + x)))/((a - x)/(a + x) + 1) + sqrt((a - x)/( a + x) + 1)/(sqrt(-(a - x)/(a + x)) + 1) + sqrt((a - x)/(a + x) + 1)/(sqrt (-(a - x)/(a + x)) - 1))
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.71 \[ \int \arcsin \left (\sqrt {\frac {-a+x}{a+x}}\right ) \, dx=x \arcsin \left (\frac {\sqrt {-a^{2} + x^{2}} \mathrm {sgn}\left (a + x\right )}{a + x}\right ) + \frac {\sqrt {2} {\left (\sqrt {2} a \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} + a x}}{2 \, a}\right ) - \sqrt {2} {\left (a \arctan \left (\frac {i \, {\left | a \right |}}{a}\right ) - 2 i \, {\left | a \right |}\right )} - 2 \, \sqrt {-a^{2} + a x}\right )} a}{2 \, {\left | a \right |}} \] Input:
integrate(arcsin(((-a+x)/(a+x))^(1/2)),x, algorithm="giac")
Output:
x*arcsin(sqrt(-a^2 + x^2)*sgn(a + x)/(a + x)) + 1/2*sqrt(2)*(sqrt(2)*a*arc tan(1/2*sqrt(2)*sqrt(-a^2 + a*x)/a) - sqrt(2)*(a*arctan(I*abs(a)/a) - 2*I* abs(a)) - 2*sqrt(-a^2 + a*x))*a/abs(a)
Timed out. \[ \int \arcsin \left (\sqrt {\frac {-a+x}{a+x}}\right ) \, dx=\int \mathrm {asin}\left (\sqrt {-\frac {a-x}{a+x}}\right ) \,d x \] Input:
int(asin((-(a - x)/(a + x))^(1/2)),x)
Output:
int(asin((-(a - x)/(a + x))^(1/2)), x)
Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.55 \[ \int \arcsin \left (\sqrt {\frac {-a+x}{a+x}}\right ) \, dx=\mathit {asin} \left (\frac {\sqrt {a +x}\, \sqrt {-a +x}-a +x}{\sqrt {a +x}\, \sqrt {-a +x}+a +x}\right ) a +\mathit {asin} \left (\frac {\sqrt {a +x}\, \sqrt {-a +x}-a +x}{\sqrt {a +x}\, \sqrt {-a +x}+a +x}\right ) x -\sqrt {a}\, \sqrt {-a +x}\, \sqrt {2} \] Input:
int(asin(((-a+x)/(a+x))^(1/2)),x)
Output:
asin((sqrt(a + x)*sqrt( - a + x) - a + x)/(sqrt(a + x)*sqrt( - a + x) + a + x))*a + asin((sqrt(a + x)*sqrt( - a + x) - a + x)/(sqrt(a + x)*sqrt( - a + x) + a + x))*x - sqrt(a)*sqrt( - a + x)*sqrt(2)