3.3 \(\int -\sin ^{-1}(\sqrt {x}-\sqrt {1+x}) \, dx\)

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

[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)

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Rubi [F]  time = 0.15, 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 = {} \[ \int -\sin ^{-1}\left (\sqrt {x}-\sqrt {1+x}\right ) \, dx \]

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 {\operatorname {Subst}\left (\int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx,x,\sqrt {1+x}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [B]  time = 0.67, size = 205, normalized size = 2.97 \[ -\frac {(x+1) \left (2 x-2 \sqrt {x+1} \sqrt {x}+1\right )^2 \left (2 \sqrt {\sqrt {x} \sqrt {x+1}-x} \left (-2 x+2 \sqrt {x+1} \sqrt {x}-3\right )+3 \sqrt {-4 x+4 \sqrt {x+1} \sqrt {x}-2} \log \left (2 \sqrt {\sqrt {x} \sqrt {x+1}-x}+\sqrt {-4 x+4 \sqrt {x+1} \sqrt {x}-2}\right )\right )}{8 \sqrt {2} \left (\sqrt {x+1}-\sqrt {x}\right )^3 \left (x-\sqrt {x+1} \sqrt {x}+1\right )^2}-x \sin ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*ArcSin[Sqrt[x] - Sqrt[1 + x]]) - ((1 + x)*(1 + 2*x - 2*Sqrt[x]*Sqrt[1 + x])^2*(2*Sqrt[-x + Sqrt[x]*Sqrt[1
+ x]]*(-3 - 2*x + 2*Sqrt[x]*Sqrt[1 + x]) + 3*Sqrt[-2 - 4*x + 4*Sqrt[x]*Sqrt[1 + x]]*Log[2*Sqrt[-x + Sqrt[x]*Sq
rt[1 + x]] + Sqrt[-2 - 4*x + 4*Sqrt[x]*Sqrt[1 + x]]]))/(8*Sqrt[2]*(-Sqrt[x] + Sqrt[1 + x])^3*(1 + x - Sqrt[x]*
Sqrt[1 + x])^2)

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fricas [A]  time = 43.87, size = 49, normalized size = 0.71 \[ \frac {1}{8} \, {\left (8 \, x + 3\right )} \arcsin \left (\sqrt {x + 1} - \sqrt {x}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x} {\left (3 \, \sqrt {x + 1} + \sqrt {x}\right )} \]

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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\arcsin \left (-\sqrt {x + 1} + \sqrt {x}\right )\,{d x} \]

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)

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maple [B]  time = 0.97, size = 251, normalized size = 3.64 \[ -\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right ) \left (\tan ^{8}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )}{2}\right )\right )+2 \arcsin \left (\sqrt {x}-\sqrt {x +1}\right ) \left (\tan ^{6}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )}{2}\right )\right )-2 \left (\tan ^{7}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )}{2}\right )\right )+18 \arcsin \left (\sqrt {x}-\sqrt {x +1}\right ) \left (\tan ^{4}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )}{2}\right )\right )-6 \left (\tan ^{5}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )}{2}\right )\right )+2 \arcsin \left (\sqrt {x}-\sqrt {x +1}\right ) \left (\tan ^{2}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )}{2}\right )\right )+6 \left (\tan ^{3}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )}{2}\right )\right )+\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )+2 \tan \left (\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )}{2}\right )}{16 \left (\tan ^{2}\left (\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )}{2}\right )+1\right )^{2} \tan \left (\frac {\arcsin \left (\sqrt {x}-\sqrt {x +1}\right )}{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.85, size = 4, normalized size = 0.06 \[ \frac {1}{2} \, \pi x \]

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

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

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)

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

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)

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