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 = 0.02, antiderivative size = 5, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 5, normalized size = 1.00 \[ -\sinh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcSinh[Cos[x]]

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fricas [B]  time = 0.96, size = 36, normalized size = 7.20 \[ \frac {1}{4} \, \log \left (8 \, \cos \relax (x)^{4} + 8 \, \cos \relax (x)^{2} - 4 \, {\left (2 \, \cos \relax (x)^{3} + \cos \relax (x)\right )} \sqrt {\cos \relax (x)^{2} + 1} + 1\right ) \]

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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giac [B]  time = 0.13, size = 14, normalized size = 2.80 \[ \log \left (\sqrt {\cos \relax (x)^{2} + 1} - \cos \relax (x)\right ) \]

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))

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maple [A]  time = 0.06, size = 6, normalized size = 1.20 \[ -\arcsinh \left (\cos \relax (x )\right ) \]

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 = 1.41, size = 5, normalized size = 1.00 \[ -\operatorname {arsinh}\left (\cos \relax (x)\right ) \]

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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mupad [B]  time = 3.08, size = 5, normalized size = 1.00 \[ -\mathrm {asinh}\left (\cos \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(cos(x)^2 + 1)^(1/2),x)

[Out]

-asinh(cos(x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\relax (x )}}{\sqrt {\cos ^{2}{\relax (x )} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(1+cos(x)**2)**(1/2),x)

[Out]

Integral(sin(x)/sqrt(cos(x)**2 + 1), x)

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