3.3.65 \(\int \arccos (x) \arcsin (x) \, dx\) [265]

Optimal. Leaf size=38 \[ 2 x-\sqrt {1-x^2} \arcsin (x)+\arccos (x) \left (\sqrt {1-x^2}+x \arcsin (x)\right ) \]

[Out]

2*x-(-x^2+1)^(1/2)*arcsin(x)+arccos(x)*((-x^2+1)^(1/2)+x*arcsin(x))

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Rubi [F]
time = 0.01, 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 \arccos (x) \arcsin (x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[ArcCos[x]*ArcSin[x],x]

[Out]

Defer[Int][ArcCos[x]*ArcSin[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \arccos (x) \arcsin (x) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.00 \begin {gather*} 2 x-\sqrt {1-x^2} \arcsin (x)+\arccos (x) \left (\sqrt {1-x^2}+x \arcsin (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[x]*ArcSin[x],x]

[Out]

2*x - Sqrt[1 - x^2]*ArcSin[x] + ArcCos[x]*(Sqrt[1 - x^2] + x*ArcSin[x])

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \arcsin \left (x \right ) \arccos \left (x \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(x)*arccos(x),x)

[Out]

int(arcsin(x)*arccos(x),x)

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Maxima [A]
time = 0.42, size = 34, normalized size = 0.89 \begin {gather*} {\left (x \arcsin \left (x\right ) + \sqrt {-x^{2} + 1}\right )} \arccos \left (x\right ) - \sqrt {-x^{2} + 1} \arcsin \left (x\right ) + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)*arccos(x),x, algorithm="maxima")

[Out]

(x*arcsin(x) + sqrt(-x^2 + 1))*arccos(x) - sqrt(-x^2 + 1)*arcsin(x) + 2*x

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Fricas [A]
time = 0.63, size = 34, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, \pi x \arccos \left (x\right ) - x \arccos \left (x\right )^{2} - \frac {1}{2} \, {\left (\pi - 4 \, \arccos \left (x\right )\right )} \sqrt {-x^{2} + 1} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)*arccos(x),x, algorithm="fricas")

[Out]

1/2*pi*x*arccos(x) - x*arccos(x)^2 - 1/2*(pi - 4*arccos(x))*sqrt(-x^2 + 1) + 2*x

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Sympy [A]
time = 0.10, size = 34, normalized size = 0.89 \begin {gather*} x \operatorname {acos}{\left (x \right )} \operatorname {asin}{\left (x \right )} + 2 x + \sqrt {1 - x^{2}} \operatorname {acos}{\left (x \right )} - \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(x)*acos(x),x)

[Out]

x*acos(x)*asin(x) + 2*x + sqrt(1 - x**2)*acos(x) - sqrt(1 - x**2)*asin(x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (34) = 68\).
time = 0.46, size = 163, normalized size = 4.29 \begin {gather*} -\pi \left (-1\right )^{\left \lfloor -\frac {\arccos \left (x\right )}{\pi } + 1 \right \rfloor } x \arccos \left (x\right ) \left \lfloor -\frac {\arccos \left (x\right )}{\pi } + 1 \right \rfloor + \frac {1}{2} \, \pi \left (-1\right )^{\left \lfloor -\frac {\arccos \left (x\right )}{\pi } + 1 \right \rfloor } x \arccos \left (x\right ) - \left (-1\right )^{\left \lfloor -\frac {\arccos \left (x\right )}{\pi } + 1 \right \rfloor } x \arccos \left (x\right )^{2} + \pi \sqrt {-x^{2} + 1} \left (-1\right )^{\left \lfloor -\frac {\arccos \left (x\right )}{\pi } + 1 \right \rfloor } \left \lfloor -\frac {\arccos \left (x\right )}{\pi } + 1 \right \rfloor - \frac {1}{2} \, \pi \sqrt {-x^{2} + 1} \left (-1\right )^{\left \lfloor -\frac {\arccos \left (x\right )}{\pi } + 1 \right \rfloor } + 2 \, \sqrt {-x^{2} + 1} \left (-1\right )^{\left \lfloor -\frac {\arccos \left (x\right )}{\pi } + 1 \right \rfloor } \arccos \left (x\right ) + 2 \, \left (-1\right )^{\left \lfloor -\frac {\arccos \left (x\right )}{\pi } + 1 \right \rfloor } x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(x)*arccos(x),x, algorithm="giac")

[Out]

-pi*(-1)^floor(-arccos(x)/pi + 1)*x*arccos(x)*floor(-arccos(x)/pi + 1) + 1/2*pi*(-1)^floor(-arccos(x)/pi + 1)*
x*arccos(x) - (-1)^floor(-arccos(x)/pi + 1)*x*arccos(x)^2 + pi*sqrt(-x^2 + 1)*(-1)^floor(-arccos(x)/pi + 1)*fl
oor(-arccos(x)/pi + 1) - 1/2*pi*sqrt(-x^2 + 1)*(-1)^floor(-arccos(x)/pi + 1) + 2*sqrt(-x^2 + 1)*(-1)^floor(-ar
ccos(x)/pi + 1)*arccos(x) + 2*(-1)^floor(-arccos(x)/pi + 1)*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \mathrm {acos}\left (x\right )\,\mathrm {asin}\left (x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(x)*asin(x),x)

[Out]

int(acos(x)*asin(x), x)

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Chatgpt [F] Failed to verify
time = 1.00, size = 11, normalized size = 0.29 \begin {gather*} x \arcsin \left (x \right ) \arccos \left (x \right )+\frac {\arcsin \left (x \right )}{2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(arcsin(x)*arccos(x),x)

[Out]

x*arcsin(x)*arccos(x)+1/2*arcsin(x)

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