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]

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

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Rubi [F]  time = 0.15314, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [B]  time = 0.6045, 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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Maple [B]  time = 0.688, size = 251, normalized size = 3.6 \begin{align*} -{\frac{1}{16} \left ( \arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \left ( \tan \left ({\frac{1}{2}\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) } \right ) \right ) ^{8}+2\,\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \left ( \tan \left ( 1/2\,\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \right ) \right ) ^{6}-2\, \left ( \tan \left ( 1/2\,\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \right ) \right ) ^{7}+18\,\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \left ( \tan \left ( 1/2\,\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \right ) \right ) ^{4}-6\, \left ( \tan \left ( 1/2\,\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \right ) \right ) ^{5}+2\,\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \left ( \tan \left ( 1/2\,\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \right ) \right ) ^{2}+6\, \left ( \tan \left ( 1/2\,\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \right ) \right ) ^{3}+\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) +2\,\tan \left ( 1/2\,\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) \right ) \right ) \left ( \left ( \tan \left ({\frac{1}{2}\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) } \right ) \right ) ^{2}+1 \right ) ^{-2} \left ( \tan \left ({\frac{1}{2}\arcsin \left ( \sqrt{x}-\sqrt{1+x} \right ) } \right ) \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 4.05797, size = 5, normalized size = 0.07 \begin{align*} \frac{1}{2} \, \pi x \end{align*}

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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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

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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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out