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

Optimal. Leaf size=3 \[ \sinh ^{-1}(\sin (x)) \]

[Out]

ArcSinh[Sin[x]]

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Rubi [A]  time = 0.0227818, antiderivative size = 3, 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} \[ \sinh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/Sqrt[1 + Sin[x]^2],x]

[Out]

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

Mathematica [A]  time = 0.0078914, size = 3, normalized size = 1. \[ \sinh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[Sin[x]]

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Maple [A]  time = 0.016, size = 4, normalized size = 1.3 \begin{align*}{\it Arcsinh} \left ( \sin \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(sin(x))

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Maxima [A]  time = 1.4372, size = 4, normalized size = 1.33 \begin{align*} \operatorname{arsinh}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(sin(x))

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Fricas [B]  time = 2.17918, size = 119, normalized size = 39.67 \begin{align*} \frac{1}{4} \, \log \left (8 \, \cos \left (x\right )^{4} - 4 \,{\left (2 \, \cos \left (x\right )^{2} - 3\right )} \sqrt{-\cos \left (x\right )^{2} + 2} \sin \left (x\right ) - 24 \, \cos \left (x\right )^{2} + 17\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(8*cos(x)^4 - 4*(2*cos(x)^2 - 3)*sqrt(-cos(x)^2 + 2)*sin(x) - 24*cos(x)^2 + 17)

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

[Out]

Timed out

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Giac [B]  time = 1.07965, size = 22, normalized size = 7.33 \begin{align*} -\log \left (\sqrt{\sin \left (x\right )^{2} + 1} - \sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(sin(x)^2 + 1) - sin(x))