3.113 \(\int \csc (2 x) (\cos (x)+\sin (x)) \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{2} \tanh ^{-1}(\sin (x))-\frac{1}{2} \tanh ^{-1}(\cos (x)) \]

[Out]

-ArcTanh[Cos[x]]/2 + ArcTanh[Sin[x]]/2

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Rubi [A]  time = 0.0733967, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{1}{2} \tanh ^{-1}(\sin (x))-\frac{1}{2} \tanh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]  Int[Csc[2*x]*(Cos[x] + Sin[x]),x]

[Out]

-ArcTanh[Cos[x]]/2 + ArcTanh[Sin[x]]/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sin{\left (x \right )} + \cos{\left (x \right )}}{\sin{\left (2 x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.014988, size = 61, normalized size = 4.07 \[ \frac{1}{2} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{1}{2} \log \left (\cos \left (\frac{x}{2}\right )\right )-\frac{1}{2} \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\frac{1}{2} \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[2*x]*(Cos[x] + Sin[x]),x]

[Out]

-Log[Cos[x/2]]/2 - Log[Cos[x/2] - Sin[x/2]]/2 + Log[Sin[x/2]]/2 + Log[Cos[x/2] +
Sin[x/2]]/2

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Maple [A]  time = 0.063, size = 20, normalized size = 1.3 \[{\frac{\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) }{2}}+{\frac{\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((cos(x)+sin(x))/sin(2*x),x)

[Out]

1/2*ln(sec(x)+tan(x))+1/2*ln(csc(x)-cot(x))

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Maxima [A]  time = 1.65356, size = 93, normalized size = 6.2 \[ -\frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*log(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1) + 1/4*log(cos(x)^2 + sin(x)^2 - 2*c
os(x) + 1) + 1/4*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) - 1/4*log(cos(x)^2 + si
n(x)^2 - 2*sin(x) + 1)

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Fricas [A]  time = 0.232271, size = 47, normalized size = 3.13 \[ -\frac{1}{4} \, \log \left (-\frac{1}{2} \,{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) + \frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{4} \, \log \left (-\frac{1}{2} \,{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right ) - \frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.875583, size = 32, normalized size = 2.13 \[ - \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{4} + \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{4} + \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{4} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(sin(x) - 1)/4 + log(sin(x) + 1)/4 + log(cos(x) - 1)/4 - log(cos(x) + 1)/4

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GIAC/XCAS [A]  time = 0.222145, size = 39, normalized size = 2.6 \[ \frac{1}{2} \,{\rm ln}\left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(abs(tan(1/2*x) + 1)) - 1/2*ln(abs(tan(1/2*x) - 1)) + 1/2*ln(abs(tan(1/2*x
)))