3.372 \(\int \csc ^4(x) \sin (4 x) \, dx\)

Optimal. Leaf size=12 \[ -2 \csc ^2(x)-8 \log (\sin (x)) \]

[Out]

-2*Csc[x]^2 - 8*Log[Sin[x]]

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Rubi [A]  time = 0.0214947, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {14} \[ -2 \csc ^2(x)-8 \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^4*Sin[4*x],x]

[Out]

-2*Csc[x]^2 - 8*Log[Sin[x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0091205, size = 12, normalized size = 1. \[ -2 \csc ^2(x)-8 \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^4*Sin[4*x],x]

[Out]

-2*Csc[x]^2 - 8*Log[Sin[x]]

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Maple [A]  time = 0.05, size = 19, normalized size = 1.6 \begin{align*} 2\, \left ( \sin \left ( x \right ) \right ) ^{-2}-4\, \left ( \cot \left ( x \right ) \right ) ^{2}-8\,\ln \left ( \sin \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/sin(x)^2-4*cot(x)^2-8*ln(sin(x))

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Maxima [A]  time = 0.936607, size = 26, normalized size = 2.17 \begin{align*} -\frac{2}{\sin \left (x\right )^{2}} - 2 \, \log \left (\sin \left (x\right )^{2}\right ) - 4 \, \log \left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(4*x)/sin(x)^4,x, algorithm="maxima")

[Out]

-2/sin(x)^2 - 2*log(sin(x)^2) - 4*log(sin(x))

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Fricas [B]  time = 2.24487, size = 78, normalized size = 6.5 \begin{align*} -\frac{2 \,{\left (4 \,{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) - 1\right )}}{\cos \left (x\right )^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(4*x)/sin(x)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 15.3909, size = 14, normalized size = 1.17 \begin{align*} - 8 \log{\left (\sin{\left (x \right )} \right )} - \frac{2}{\sin ^{2}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(4*x)/sin(x)**4,x)

[Out]

-8*log(sin(x)) - 2/sin(x)**2

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Giac [B]  time = 1.08126, size = 96, normalized size = 8. \begin{align*} \frac{{\left (\frac{8 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + 1\right )}{\left (\cos \left (x\right ) + 1\right )}}{2 \,{\left (\cos \left (x\right ) - 1\right )}} + \frac{\cos \left (x\right ) - 1}{2 \,{\left (\cos \left (x\right ) + 1\right )}} + 8 \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right ) - 4 \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(4*x)/sin(x)^4,x, algorithm="giac")

[Out]

1/2*(8*(cos(x) - 1)/(cos(x) + 1) + 1)*(cos(x) + 1)/(cos(x) - 1) + 1/2*(cos(x) - 1)/(cos(x) + 1) + 8*log(-(cos(
x) - 1)/(cos(x) + 1) + 1) - 4*log(-(cos(x) - 1)/(cos(x) + 1))