3.41 \(\int \log (\sin (x)) \sqrt{1+\sin (x)} \, dx\)

Optimal. Leaf size=42 \[ \frac{4 \cos (x)}{\sqrt{\sin (x)+1}}-\frac{2 \cos (x) \log (\sin (x))}{\sqrt{\sin (x)+1}}-4 \tanh ^{-1}\left (\frac{\cos (x)}{\sqrt{\sin (x)+1}}\right ) \]

[Out]

-4*ArcTanh[Cos[x]/Sqrt[1 + Sin[x]]] + (4*Cos[x])/Sqrt[1 + Sin[x]] - (2*Cos[x]*Lo
g[Sin[x]])/Sqrt[1 + Sin[x]]

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Rubi [A]  time = 0.238017, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583 \[ \frac{4 \cos (x)}{\sqrt{\sin (x)+1}}-\frac{2 \cos (x) \log (\sin (x))}{\sqrt{\sin (x)+1}}-4 \tanh ^{-1}\left (\frac{\cos (x)}{\sqrt{\sin (x)+1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Log[Sin[x]]*Sqrt[1 + Sin[x]],x]

[Out]

-4*ArcTanh[Cos[x]/Sqrt[1 + Sin[x]]] + (4*Cos[x])/Sqrt[1 + Sin[x]] - (2*Cos[x]*Lo
g[Sin[x]])/Sqrt[1 + Sin[x]]

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Rubi in Sympy [A]  time = 10.8485, size = 46, normalized size = 1.1 \[ - 4 \operatorname{atanh}{\left (\frac{\cos{\left (x \right )}}{\sqrt{\sin{\left (x \right )} + 1}} \right )} - \frac{2 \log{\left (\sin{\left (x \right )} \right )} \cos{\left (x \right )}}{\sqrt{\sin{\left (x \right )} + 1}} + \frac{4 \cos{\left (x \right )}}{\sqrt{\sin{\left (x \right )} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*atanh(cos(x)/sqrt(sin(x) + 1)) - 2*log(sin(x))*cos(x)/sqrt(sin(x) + 1) + 4*co
s(x)/sqrt(sin(x) + 1)

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Mathematica [B]  time = 0.105702, size = 87, normalized size = 2.07 \[ \frac{2 \sqrt{\sin (x)+1} \left (\sin \left (\frac{x}{2}\right ) (\log (\sin (x))-2)-\log \left (-\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )+1\right )+\log \left (\sin \left (\frac{x}{2}\right )-\cos \left (\frac{x}{2}\right )+1\right )-\cos \left (\frac{x}{2}\right ) (\log (\sin (x))-2)\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[Log[Sin[x]]*Sqrt[1 + Sin[x]],x]

[Out]

(2*(-Log[1 + Cos[x/2] - Sin[x/2]] + Log[1 - Cos[x/2] + Sin[x/2]] - Cos[x/2]*(-2
+ Log[Sin[x]]) + (-2 + Log[Sin[x]])*Sin[x/2])*Sqrt[1 + Sin[x]])/(Cos[x/2] + Sin[
x/2])

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Maple [F]  time = 0.207, size = 0, normalized size = 0. \[ \int \ln \left ( \sin \left ( x \right ) \right ) \sqrt{1+\sin \left ( x \right ) }\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(sin(x))*(1+sin(x))^(1/2),x)

[Out]

int(ln(sin(x))*(1+sin(x))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sin(x) + 1)*log(sin(x)),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.239538, size = 142, normalized size = 3.38 \[ \frac{{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )} \log \left (-\frac{\sqrt{\sin \left (x\right ) + 1}{\left (\sin \left (x\right ) - 2\right )} + 2 \, \cos \left (x\right )}{2 \, \sqrt{\sin \left (x\right ) + 1}}\right ) -{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )} \log \left (-\frac{\sqrt{\sin \left (x\right ) + 1}{\left (\sin \left (x\right ) - 2\right )} - 2 \, \cos \left (x\right )}{2 \, \sqrt{\sin \left (x\right ) + 1}}\right ) - 2 \,{\left ({\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )} \log \left (\sin \left (x\right )\right ) - 2 \, \cos \left (x\right ) + 2 \, \sin \left (x\right ) - 2\right )} \sqrt{\sin \left (x\right ) + 1}}{\cos \left (x\right ) + \sin \left (x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sin(x) + 1)*log(sin(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sin{\left (x \right )} + 1} \log{\left (\sin{\left (x \right )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(sin(x) + 1)*log(sin(x)), x)

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GIAC/XCAS [A]  time = 0.43222, size = 1, normalized size = 0.02 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sin(x) + 1)*log(sin(x)),x, algorithm="giac")

[Out]

Done