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 = 0.02, antiderivative size = 3, 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} \[ \sinh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]/Sqrt[1 + Sin[x]^2],x]

[Out]

ArcSinh[Sin[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 {\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*}

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Mathematica [A]  time = 0.01, size = 3, normalized size = 1.00 \[ \sinh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[Sin[x]]

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fricas [B]  time = 1.19, size = 39, normalized size = 13.00 \[ \frac {1}{4} \, \log \left (8 \, \cos \relax (x)^{4} - 4 \, {\left (2 \, \cos \relax (x)^{2} - 3\right )} \sqrt {-\cos \relax (x)^{2} + 2} \sin \relax (x) - 24 \, \cos \relax (x)^{2} + 17\right ) \]

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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giac [B]  time = 0.13, size = 16, normalized size = 5.33 \[ -\log \left (\sqrt {\sin \relax (x)^{2} + 1} - \sin \relax (x)\right ) \]

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))

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maple [A]  time = 0.06, size = 4, normalized size = 1.33 \[ \arcsinh \left (\sin \relax (x )\right ) \]

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 = 0.41, size = 3, normalized size = 1.00 \[ \operatorname {arsinh}\left (\sin \relax (x)\right ) \]

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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mupad [B]  time = 0.02, size = 9, normalized size = 3.00 \[ -\mathrm {asin}\left (\sin \relax (x)\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)/(sin(x)^2 + 1)^(1/2),x)

[Out]

-asin(sin(x)*1i)*1i

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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