3.5.70 \(\int \frac {x}{\text {ArcSin}(\sin (x))} \, dx\) [470]

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

[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 = 0.03, 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 \frac {x}{\text {ArcSin}(\sin (x))} \, dx \end {gather*}

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 = 0.36, size = 28, normalized size = 1.04 \begin {gather*} -\text {ArcSin}(\sin (x)) (-1+\log (\text {ArcSin}(\sin (x))))+x \sqrt {\cos ^2(x)} \log (\text {ArcSin}(\sin (x))) \sec (x) \end {gather*}

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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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {x}{\arcsin \left (\sin \left (x \right )\right )}\, dx\]

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 = 0.48, size = 1, normalized size = 0.04 \begin {gather*} x \end {gather*}

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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Fricas [A]
time = 3.30, size = 3, normalized size = 0.11 \begin {gather*} -x \end {gather*}

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

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

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x}{\mathrm {asin}\left (\sin \left (x\right )\right )} \,d x \end {gather*}

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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