Optimal. Leaf size=55 \[ (a+x) \sin ^{-1}\left (\sqrt {\frac {x-a}{a+x}}\right )-\frac {\sqrt {2} a \sqrt {\frac {x-a}{a+x}}}{\sqrt {\frac {a}{a+x}}} \]
________________________________________________________________________________________
Rubi [B] time = 0.84, antiderivative size = 118, normalized size of antiderivative = 2.15, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4840, 12, 6677, 6720, 385, 217, 206} \[ -\sqrt {2} \sqrt {\frac {a}{a+x}} \sqrt {-\frac {a-x}{a+x}} (a+x)+x \sin ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right )-\frac {a \sqrt {\frac {a}{a+x}} \tanh ^{-1}\left (\frac {\sqrt {-\frac {a-x}{a+x}}}{\sqrt {2} \sqrt {-\frac {a}{a+x}}}\right )}{\sqrt {-\frac {a}{a+x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 206
Rule 217
Rule 385
Rule 4840
Rule 6677
Rule 6720
Rubi steps
\begin {align*} \int \sin ^{-1}\left (\sqrt {\frac {-a+x}{a+x}}\right ) \, dx &=x \sin ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right )-\int \frac {x \left (\frac {a}{a+x}\right )^{3/2}}{\sqrt {2} a \sqrt {\frac {-a+x}{a+x}}} \, dx\\ &=x \sin ^{-1}\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}\\ &=x \sin ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right )-\frac {\left (\sqrt {\frac {a}{a+x}} \sqrt {a+x}\right ) \int \frac {x}{\sqrt {\frac {-a+x}{a+x}} (a+x)^{3/2}} \, dx}{\sqrt {2}}\\ &=x \sin ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right )-\left (a \sqrt {\frac {a}{a+x}} \sqrt {a+x}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {-\frac {a}{-1+x^2}} \left (-1+x^2\right )^2} \, dx,x,\sqrt {\frac {-a+x}{a+x}}\right )\\ &=x \sin ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right )-\frac {\left (a \sqrt {\frac {a}{a+x}}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\left (-1+x^2\right )^{3/2}} \, dx,x,\sqrt {\frac {-a+x}{a+x}}\right )}{\sqrt {-\frac {a}{a+x}}}\\ &=-\sqrt {2} \sqrt {\frac {a}{a+x}} \sqrt {-\frac {a-x}{a+x}} (a+x)+x \sin ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right )-\frac {\left (a \sqrt {\frac {a}{a+x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,\sqrt {\frac {-a+x}{a+x}}\right )}{\sqrt {-\frac {a}{a+x}}}\\ &=-\sqrt {2} \sqrt {\frac {a}{a+x}} \sqrt {-\frac {a-x}{a+x}} (a+x)+x \sin ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right )-\frac {\left (a \sqrt {\frac {a}{a+x}}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {\frac {-a+x}{a+x}}}{\sqrt {2} \sqrt {-\frac {a}{a+x}}}\right )}{\sqrt {-\frac {a}{a+x}}}\\ &=-\sqrt {2} \sqrt {\frac {a}{a+x}} \sqrt {-\frac {a-x}{a+x}} (a+x)+x \sin ^{-1}\left (\sqrt {-\frac {a-x}{a+x}}\right )-\frac {a \sqrt {\frac {a}{a+x}} \tanh ^{-1}\left (\frac {\sqrt {-\frac {a-x}{a+x}}}{\sqrt {2} \sqrt {-\frac {a}{a+x}}}\right )}{\sqrt {-\frac {a}{a+x}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 99, normalized size = 1.80 \[ x \sin ^{-1}\left (\sqrt {\frac {x-a}{a+x}}\right )+\frac {\sqrt {\frac {a}{a+x}} \left (\sqrt {2} \sqrt {a} \sqrt {x-a} \tan ^{-1}\left (\frac {\sqrt {x-a}}{\sqrt {2} \sqrt {a}}\right )+2 a-2 x\right )}{\sqrt {2} \sqrt {\frac {x-a}{a+x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin ^{-1}\left (\sqrt {\frac {-a+x}{a+x}}\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.14, size = 51, normalized size = 0.93 \[ -\sqrt {2} {\left (a + x\right )} \sqrt {-\frac {a - x}{a + x}} \sqrt {\frac {a}{a + x}} + {\left (a + x\right )} \arcsin \left (\sqrt {-\frac {a - x}{a + x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \arcsin \left (\sqrt {-\frac {a - x}{a + x}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 87, normalized size = 1.58
method | result | size |
default | \(x \arcsin \left (\sqrt {\frac {-a +x}{a +x}}\right )-\frac {\sqrt {-a +x}\, \sqrt {2}\, \sqrt {\frac {a}{a +x}}\, \left (-\sqrt {a}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-a +x}\, \sqrt {2}}{2 \sqrt {a}}\right )+2 \sqrt {-a +x}\right )}{2 \sqrt {-\frac {a -x}{a +x}}}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.01, size = 103, normalized size = 1.87 \[ a {\left (\frac {2 \, \arcsin \left (\sqrt {-\frac {a - x}{a + x}}\right )}{\frac {a - x}{a + x} + 1} + \frac {\sqrt {\frac {a - x}{a + x} + 1}}{\sqrt {-\frac {a - x}{a + x}} + 1} + \frac {\sqrt {\frac {a - x}{a + x} + 1}}{\sqrt {-\frac {a - x}{a + x}} - 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {asin}\left (\sqrt {-\frac {a-x}{a+x}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {asin}{\left (\sqrt {\frac {- a + x}{a + x}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________