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

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

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Rubi [A]  time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \[ -2 \csc ^2(x)-8 \log (\sin (x)) \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^4(x) \sin (4 x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [B]  time = 0.67, size = 25, normalized size = 2.08 \[ -\frac {2 \, {\left (4 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \relax (x)\right ) - 1\right )}}{\cos \relax (x)^{2} - 1} \]

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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giac [A]  time = 0.60, size = 13, normalized size = 1.08 \[ -\frac {2}{\sin \relax (x)^{2}} - 8 \, \log \left ({\left | \sin \relax (x) \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/sin(x)^2 - 8*log(abs(sin(x)))

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maple [A]  time = 0.13, size = 19, normalized size = 1.58

method	result	size
default	\(\frac {2}{\sin \relax (x )^{2}}-4 \left (\cot ^{2}\relax (x )\right )-8 \ln \left (\sin \relax (x )\right )\)	\(19\)
risch	\(8 i x +\frac {8 \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}-8 \ln \left ({\mathrm e}^{2 i x}-1\right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(4*x)/sin(x)^4,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.43, size = 19, normalized size = 1.58 \[ -\frac {2}{\sin \relax (x)^{2}} - 2 \, \log \left (\sin \relax (x)^{2}\right ) - 4 \, \log \left (\sin \relax (x)\right ) \]

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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mupad [B]  time = 0.30, size = 35, normalized size = 2.92 \[ 8\,\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-8\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\frac {1}{2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*log(tan(x/2)^2 + 1) - 8*log(tan(x/2)) - 1/(2*tan(x/2)^2) - tan(x/2)^2/2

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sympy [A]  time = 5.89, size = 14, normalized size = 1.17 \[ - 8 \log {\left (\sin {\relax (x )} \right )} - \frac {2}{\sin ^{2}{\relax (x )}} \]

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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