3.389 \(\int \csc (4 x) \sin ^3(x) \, dx\)

Optimal. Leaf size=26 \[ \frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{4 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\sin (x)) \]

[Out]

-ArcTanh[Sin[x]]/4 + ArcTanh[Sqrt[2]*Sin[x]]/(4*Sqrt[2])

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Rubi [A]  time = 0.0627791, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{4 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\sin (x)) \]

Antiderivative was successfully verified.

[In]  Int[Csc[4*x]*Sin[x]^3,x]

[Out]

-ArcTanh[Sin[x]]/4 + ArcTanh[Sqrt[2]*Sin[x]]/(4*Sqrt[2])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sin ^{3}{\left (x \right )}}{\sin{\left (4 x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)**3/sin(4*x),x)

[Out]

Integral(sin(x)**3/sin(4*x), x)

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Mathematica [C]  time = 0.493763, size = 218, normalized size = 8.38 \[ \frac{2 \log \left (2 \sin (x)+\sqrt{2}\right )+4 \sqrt{2} \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-4 \sqrt{2} \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (-\sqrt{2} \sin (x)-\sqrt{2} \cos (x)+2\right )-\log \left (-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+2\right )-2 i \tan ^{-1}\left (\frac{\cos \left (\frac{x}{2}\right )-\left (\sqrt{2}-1\right ) \sin \left (\frac{x}{2}\right )}{\left (1+\sqrt{2}\right ) \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )}\right )-2 i \tan ^{-1}\left (\frac{\cos \left (\frac{x}{2}\right )-\left (1+\sqrt{2}\right ) \sin \left (\frac{x}{2}\right )}{\left (\sqrt{2}-1\right ) \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )}\right )}{16 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[4*x]*Sin[x]^3,x]

[Out]

((-2*I)*ArcTan[(Cos[x/2] - (-1 + Sqrt[2])*Sin[x/2])/((1 + Sqrt[2])*Cos[x/2] - Si
n[x/2])] - (2*I)*ArcTan[(Cos[x/2] - (1 + Sqrt[2])*Sin[x/2])/((-1 + Sqrt[2])*Cos[
x/2] - Sin[x/2])] + 4*Sqrt[2]*Log[Cos[x/2] - Sin[x/2]] - 4*Sqrt[2]*Log[Cos[x/2]
+ Sin[x/2]] + 2*Log[Sqrt[2] + 2*Sin[x]] - Log[2 - Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x
]] - Log[2 + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]])/(16*Sqrt[2])

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Maple [A]  time = 0.095, size = 28, normalized size = 1.1 \[ -{\frac{\ln \left ( 1+\sin \left ( x \right ) \right ) }{8}}+{\frac{\ln \left ( -1+\sin \left ( x \right ) \right ) }{8}}+{\frac{{\it Artanh} \left ( \sin \left ( x \right ) \sqrt{2} \right ) \sqrt{2}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)^3/sin(4*x),x)

[Out]

-1/8*ln(1+sin(x))+1/8*ln(-1+sin(x))+1/8*arctanh(sin(x)*2^(1/2))*2^(1/2)

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Maxima [A]  time = 1.59756, size = 231, normalized size = 8.88 \[ \frac{1}{32} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt{2} \cos \left (x\right ) + 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) - \frac{1}{32} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt{2} \cos \left (x\right ) - 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) + \frac{1}{32} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) + 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) - \frac{1}{32} \, \sqrt{2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt{2} \cos \left (x\right ) - 2 \, \sqrt{2} \sin \left (x\right ) + 2\right ) - \frac{1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) + \frac{1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^3/sin(4*x),x, algorithm="maxima")

[Out]

1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) + 2*sqrt(2)*sin(x) +
2) - 1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 + 2*sqrt(2)*cos(x) - 2*sqrt(2)*si
n(x) + 2) + 1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 - 2*sqrt(2)*cos(x) + 2*sqrt
(2)*sin(x) + 2) - 1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 - 2*sqrt(2)*cos(x) -
2*sqrt(2)*sin(x) + 2) - 1/8*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) + 1/8*log(co
s(x)^2 + sin(x)^2 - 2*sin(x) + 1)

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Fricas [A]  time = 0.233936, size = 82, normalized size = 3.15 \[ -\frac{1}{16} \, \sqrt{2}{\left (\sqrt{2} \log \left (\sin \left (x\right ) + 1\right ) - \sqrt{2} \log \left (-\sin \left (x\right ) + 1\right ) - \log \left (-\frac{2 \, \sqrt{2} \cos \left (x\right )^{2} - 3 \, \sqrt{2} - 4 \, \sin \left (x\right )}{2 \, \cos \left (x\right )^{2} - 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^3/sin(4*x),x, algorithm="fricas")

[Out]

-1/16*sqrt(2)*(sqrt(2)*log(sin(x) + 1) - sqrt(2)*log(-sin(x) + 1) - log(-(2*sqrt
(2)*cos(x)^2 - 3*sqrt(2) - 4*sin(x))/(2*cos(x)^2 - 1)))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)**3/sin(4*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219861, size = 65, normalized size = 2.5 \[ -\frac{1}{16} \, \sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt{2} + 4 \, \sin \left (x\right ) \right |}}\right ) - \frac{1}{8} \,{\rm ln}\left (\sin \left (x\right ) + 1\right ) + \frac{1}{8} \,{\rm ln}\left (-\sin \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^3/sin(4*x),x, algorithm="giac")

[Out]

-1/16*sqrt(2)*ln(abs(-2*sqrt(2) + 4*sin(x))/abs(2*sqrt(2) + 4*sin(x))) - 1/8*ln(
sin(x) + 1) + 1/8*ln(-sin(x) + 1)