3.650 \(\int \frac{\sin (x)}{\sqrt{1+\cos ^2(x)}} \, dx\)

Optimal. Leaf size=5 \[ -\sinh ^{-1}(\cos (x)) \]

[Out]

-ArcSinh[Cos[x]]

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Rubi [A]  time = 0.022591, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3190, 215} \[ -\sinh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/Sqrt[1 + Cos[x]^2],x]

[Out]

-ArcSinh[Cos[x]]

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{\sin (x)}{\sqrt{1+\cos ^2(x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\cos (x)\right )\\ &=-\sinh ^{-1}(\cos (x))\\ \end{align*}

Mathematica [A]  time = 0.0208214, size = 5, normalized size = 1. \[ -\sinh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/Sqrt[1 + Cos[x]^2],x]

[Out]

-ArcSinh[Cos[x]]

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Maple [A]  time = 0.013, size = 6, normalized size = 1.2 \begin{align*} -{\it Arcsinh} \left ( \cos \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(1+cos(x)^2)^(1/2),x)

[Out]

-arcsinh(cos(x))

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Maxima [A]  time = 1.42716, size = 7, normalized size = 1.4 \begin{align*} -\operatorname{arsinh}\left (\cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x)^2)^(1/2),x, algorithm="maxima")

[Out]

-arcsinh(cos(x))

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Fricas [B]  time = 2.1824, size = 112, normalized size = 22.4 \begin{align*} \frac{1}{4} \, \log \left (8 \, \cos \left (x\right )^{4} + 8 \, \cos \left (x\right )^{2} - 4 \,{\left (2 \, \cos \left (x\right )^{3} + \cos \left (x\right )\right )} \sqrt{\cos \left (x\right )^{2} + 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/4*log(8*cos(x)^4 + 8*cos(x)^2 - 4*(2*cos(x)^3 + cos(x))*sqrt(cos(x)^2 + 1) + 1)

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Sympy [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(sin(x)/(1+cos(x)**2)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.10103, size = 19, normalized size = 3.8 \begin{align*} \log \left (\sqrt{\cos \left (x\right )^{2} + 1} - \cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x)^2)^(1/2),x, algorithm="giac")

[Out]

log(sqrt(cos(x)^2 + 1) - cos(x))