3.37 \(\int \frac{\sin (x)}{1+\sin ^2(x)} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\tanh ^{-1}\left (\frac{\cos (x)}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

-(ArcTanh[Cos[x]/Sqrt[2]]/Sqrt[2])

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Rubi [A]  time = 0.0327096, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{\tanh ^{-1}\left (\frac{\cos (x)}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Sin[x]/(1 + Sin[x]^2),x]

[Out]

-(ArcTanh[Cos[x]/Sqrt[2]]/Sqrt[2])

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Rubi in Sympy [A]  time = 3.37331, size = 19, normalized size = 1.19 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \cos{\left (x \right )}}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)/(1+sin(x)**2),x)

[Out]

-sqrt(2)*atanh(sqrt(2)*cos(x)/2)/2

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Mathematica [C]  time = 0.0580657, size = 46, normalized size = 2.88 \[ -\frac{i \left (\tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )-i}{\sqrt{2}}\right )-\tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )+i}{\sqrt{2}}\right )\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[x]/(1 + Sin[x]^2),x]

[Out]

((-I)*(ArcTan[(-I + Tan[x/2])/Sqrt[2]] - ArcTan[(I + Tan[x/2])/Sqrt[2]]))/Sqrt[2
]

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Maple [A]  time = 0.016, size = 14, normalized size = 0.9 \[ -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{\cos \left ( x \right ) \sqrt{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)/(1+sin(x)^2),x)

[Out]

-1/2*arctanh(1/2*cos(x)*2^(1/2))*2^(1/2)

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Maxima [A]  time = 1.6312, size = 35, normalized size = 2.19 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \cos \left (x\right )\right )}}{2 \, \sqrt{2} + 2 \, \cos \left (x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)/(sin(x)^2 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.239245, size = 46, normalized size = 2.88 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{\sqrt{2} \cos \left (x\right )^{2} + 2 \, \sqrt{2} - 4 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)/(sin(x)^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 52.2298, size = 46, normalized size = 2.88 \[ \frac{\sqrt{2} \log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} - 2 \sqrt{2} + 3 \right )}}{4} - \frac{\sqrt{2} \log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 2 \sqrt{2} + 3 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)/(1+sin(x)**2),x)

[Out]

sqrt(2)*log(tan(x/2)**2 - 2*sqrt(2) + 3)/4 - sqrt(2)*log(tan(x/2)**2 + 2*sqrt(2)
+ 3)/4

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GIAC/XCAS [A]  time = 0.202009, size = 36, normalized size = 2.25 \[ -\frac{1}{4} \, \sqrt{2}{\rm ln}\left (\sqrt{2} + \cos \left (x\right )\right ) + \frac{1}{4} \, \sqrt{2}{\rm ln}\left (\sqrt{2} - \cos \left (x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)/(sin(x)^2 + 1),x, algorithm="giac")

[Out]

-1/4*sqrt(2)*ln(sqrt(2) + cos(x)) + 1/4*sqrt(2)*ln(sqrt(2) - cos(x))