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

________________________________________________________________________________________

Rubi [F]  time = 0.0387074, 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 = {} \[ \int \frac{x}{\sin ^{-1}(\sin (x))} \, dx \]

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*}

Mathematica [A]  time = 0.526656, 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]

________________________________________________________________________________________

Maple [F]  time = 0.073, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{\arcsin \left ( \sin \left ( x \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.51935, size = 1, normalized size = 0.04 \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

________________________________________________________________________________________

Fricas [A]  time = 1.7056, size = 5, normalized size = 0.19 \begin{align*} -x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/asin(sin(x)), x)

________________________________________________________________________________________

Giac [A]  time = 1.15805, size = 1, normalized size = 0.04 \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x