3.470 \(\int \frac {x}{\sin ^{-1}(\sin (x))} \, dx\)

Optimal. Leaf size=27 \[ \sin ^{-1}(\sin (x))+\log \left (\sin ^{-1}(\sin (x))\right ) \left (x \sqrt {\cos ^2(x)} \sec (x)-\sin ^{-1}(\sin (x))\right ) \]

[Out]

arcsin(sin(x))+ln(arcsin(sin(x)))*(-arcsin(sin(x))+x*sec(x)*(cos(x)^2)^(1/2))

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Rubi [F]  time = 0.04, 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 = {} \[ \int \frac {x}{\sin ^{-1}(\sin (x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x/ArcSin[Sin[x]],x]

[Out]

Defer[Int][x/ArcSin[Sin[x]], x]

Rubi steps

\begin {align*} \int \frac {x}{\sin ^{-1}(\sin (x))} \, dx &=\int \frac {x}{\sin ^{-1}(\sin (x))} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.65, size = 28, normalized size = 1.04 \[ x \sqrt {\cos ^2(x)} \sec (x) \log \left (\sin ^{-1}(\sin (x))\right )-\sin ^{-1}(\sin (x)) \left (\log \left (\sin ^{-1}(\sin (x))\right )-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/ArcSin[Sin[x]],x]

[Out]

-(ArcSin[Sin[x]]*(-1 + Log[ArcSin[Sin[x]]])) + x*Sqrt[Cos[x]^2]*Log[ArcSin[Sin[x]]]*Sec[x]

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fricas [A]  time = 0.43, size = 3, normalized size = 0.11 \[ -x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arcsin(sin(x)),x, algorithm="fricas")

[Out]

-x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arcsin(sin(x)),x, algorithm="giac")

[Out]

sage0*x

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maple [F]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {x}{\arcsin \left (\sin \relax (x )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/arcsin(sin(x)),x)

[Out]

int(x/arcsin(sin(x)),x)

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maxima [A]  time = 0.41, size = 1, normalized size = 0.04 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arcsin(sin(x)),x, algorithm="maxima")

[Out]

x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x}{\mathrm {asin}\left (\sin \relax (x)\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/asin(sin(x)),x)

[Out]

int(x/asin(sin(x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\operatorname {asin}{\left (\sin {\relax (x )} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/asin(sin(x)),x)

[Out]

Integral(x/asin(sin(x)), x)

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