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 = 0.0346002, 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, Rules used = {1130, 207} \[ \frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{4 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\sin (x)) \]

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])

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \csc (4 x) \sin ^3(x) \, dx &=\operatorname{Subst}\left (\int \frac{x^2}{4-12 x^2+8 x^4} \, dx,x,\sin (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1}{-8+8 x^2} \, dx,x,\sin (x)\right )-\operatorname{Subst}\left (\int \frac{1}{-4+8 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac{1}{4} \tanh ^{-1}(\sin (x))+\frac{\tanh ^{-1}\left (\sqrt{2} \sin (x)\right )}{4 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.315892, 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] - Sin[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*S
qrt[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 = 0.066, size = 28, normalized size = 1.1 \begin{align*}{\frac{{\it Artanh} \left ( \sin \left ( x \right ) \sqrt{2} \right ) \sqrt{2}}{8}}-{\frac{\ln \left ( 1+\sin \left ( x \right ) \right ) }{8}}+{\frac{\ln \left ( -1+\sin \left ( x \right ) \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/sin(4*x),x)

[Out]

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

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Maxima [B]  time = 1.47293, size = 231, normalized size = 8.88 \begin{align*} \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 ) \end{align*}

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)*sin(x) + 2) + 1/32*sqrt(2)*log(2*cos(x)^2 + 2*sin(x)^2 - 2*sqr
t(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(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1)

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Fricas [B]  time = 2.03274, size = 159, normalized size = 6.12 \begin{align*} \frac{1}{16} \, \sqrt{2} \log \left (-\frac{2 \, \cos \left (x\right )^{2} - 2 \, \sqrt{2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) - \frac{1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac{1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}

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)*log(-(2*cos(x)^2 - 2*sqrt(2)*sin(x) - 3)/(2*cos(x)^2 - 1)) - 1/8*log(sin(x) + 1) + 1/8*log(-sin(x
) + 1)

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

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 [B]  time = 1.12643, size = 65, normalized size = 2.5 \begin{align*} -\frac{1}{16} \, \sqrt{2} \log \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} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac{1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}

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