3.1.3 \(\int -\sin ^{-1}(\sqrt {x}-\sqrt {1+x}) \, dx\) [3]

Optimal. Leaf size=69 \[ \frac {\left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x+\sqrt {x} \sqrt {1+x}}}{4 \sqrt {2}}-\left (\frac {3}{8}+x\right ) \sin ^{-1}\left (\sqrt {x}-\sqrt {1+x}\right ) \]

[Out]

-(3/8+x)*arcsin(x^(1/2)-(1+x)^(1/2))+1/8*(x^(1/2)+3*(1+x)^(1/2))*(-x+x^(1/2)*(1+x)^(1/2))^(1/2)*2^(1/2)

________________________________________________________________________________________

Rubi [F]
time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int -\text {ArcSin}\left (\sqrt {x}-\sqrt {1+x}\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[-ArcSin[Sqrt[x] - Sqrt[1 + x]],x]

[Out]

-(x*ArcSin[Sqrt[x] - Sqrt[1 + x]]) + Defer[Subst][Defer[Int][Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]], x], x, Sqrt[1 +
x]]/Sqrt[2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.71, size = 106, normalized size = 1.54 \begin {gather*} \frac {1}{8} \left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-2 x+2 \sqrt {x} \sqrt {1+x}}-x \sin ^{-1}\left (\sqrt {x}-\sqrt {1+x}\right )-\frac {3}{8} \tan ^{-1}\left (\frac {\sqrt {-2 x+2 \sqrt {x} \sqrt {1+x}}}{-\sqrt {x}+\sqrt {1+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-ArcSin[Sqrt[x] - Sqrt[1 + x]],x]

[Out]

((Sqrt[x] + 3*Sqrt[1 + x])*Sqrt[-2*x + 2*Sqrt[x]*Sqrt[1 + x]])/8 - x*ArcSin[Sqrt[x] - Sqrt[1 + x]] - (3*ArcTan
[Sqrt[-2*x + 2*Sqrt[x]*Sqrt[1 + x]]/(-Sqrt[x] + Sqrt[1 + x])])/8

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(250\) vs. \(2(49)=98\).
time = 0.58, size = 251, normalized size = 3.64

method	result	size
default	\(-\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right ) \left (\tan ^{8}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2}\right )\right )-2 \left (\tan ^{7}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2}\right )\right )+2 \arcsin \left (\sqrt {x}-\sqrt {1+x}\right ) \left (\tan ^{6}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2}\right )\right )-6 \left (\tan ^{5}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2}\right )\right )+18 \arcsin \left (\sqrt {x}-\sqrt {1+x}\right ) \left (\tan ^{4}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2}\right )\right )+6 \left (\tan ^{3}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2}\right )\right )+2 \arcsin \left (\sqrt {x}-\sqrt {1+x}\right ) \left (\tan ^{2}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2}\right )\right )+2 \tan \left (\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2}\right )+\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{16 \left (1+\tan ^{2}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2}\right )\right )^{2} \tan \left (\frac {\arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2}\right )^{2}}\)	\(251\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(arcsin(x^(1/2)-(1+x)^(1/2))*tan(1/2*arcsin(x^(1/2)-(1+x)^(1/2)))^8-2*tan(1/2*arcsin(x^(1/2)-(1+x)^(1/2)
))^7+2*arcsin(x^(1/2)-(1+x)^(1/2))*tan(1/2*arcsin(x^(1/2)-(1+x)^(1/2)))^6-6*tan(1/2*arcsin(x^(1/2)-(1+x)^(1/2)
))^5+18*arcsin(x^(1/2)-(1+x)^(1/2))*tan(1/2*arcsin(x^(1/2)-(1+x)^(1/2)))^4+6*tan(1/2*arcsin(x^(1/2)-(1+x)^(1/2
)))^3+2*arcsin(x^(1/2)-(1+x)^(1/2))*tan(1/2*arcsin(x^(1/2)-(1+x)^(1/2)))^2+2*tan(1/2*arcsin(x^(1/2)-(1+x)^(1/2
)))+arcsin(x^(1/2)-(1+x)^(1/2)))/(1+tan(1/2*arcsin(x^(1/2)-(1+x)^(1/2)))^2)^2/tan(1/2*arcsin(x^(1/2)-(1+x)^(1/
2)))^2

________________________________________________________________________________________

Maxima [A]
time = 4.16, size = 4, normalized size = 0.06 \begin {gather*} \frac {1}{2} \, \pi x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*pi*x

________________________________________________________________________________________

Fricas [A]
time = 22.13, size = 49, normalized size = 0.71 \begin {gather*} \frac {1}{8} \, {\left (8 \, x + 3\right )} \arcsin \left (\sqrt {x + 1} - \sqrt {x}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x} {\left (3 \, \sqrt {x + 1} + \sqrt {x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(8*x + 3)*arcsin(sqrt(x + 1) - sqrt(x)) + 1/8*sqrt(2*sqrt(x + 1)*sqrt(x) - 2*x)*(3*sqrt(x + 1) + sqrt(x))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \operatorname {asin}{\left (\sqrt {x} - \sqrt {x + 1} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-asin(x**(1/2)-(1+x)**(1/2)),x)

[Out]

-Integral(asin(sqrt(x) - sqrt(x + 1)), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-arcsin(-sqrt(x + 1) + sqrt(x)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {asin}\left (\sqrt {x+1}-\sqrt {x}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin((x + 1)^(1/2) - x^(1/2)),x)

[Out]

int(asin((x + 1)^(1/2) - x^(1/2)), x)

________________________________________________________________________________________