3.371 \(\int \cos (5 x) \csc ^5(x) \, dx\)

Optimal. Leaf size=20 \[ -\frac{1}{4} \csc ^4(x)+6 \csc ^2(x)+16 \log (\sin (x)) \]

[Out]

6*Csc[x]^2 - Csc[x]^4/4 + 16*Log[Sin[x]]

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Rubi [A]  time = 0.0461352, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4366, 1247, 698} \[ -\frac{1}{4} \csc ^4(x)+6 \csc ^2(x)+16 \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[5*x]*Csc[x]^5,x]

[Out]

6*Csc[x]^2 - Csc[x]^4/4 + 16*Log[Sin[x]]

Rule 4366

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dis
t[d/(b*c), Subst[Int[SubstFor[(1 - d^2*x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d]
, x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] &&
(EqQ[F, Sin] || EqQ[F, sin])

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \cos (5 x) \csc ^5(x) \, dx &=-\operatorname{Subst}\left (\int \frac{x \left (5-20 x^2+16 x^4\right )}{\left (1-x^2\right )^3} \, dx,x,\cos (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{5-20 x+16 x^2}{(1-x)^3} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{(-1+x)^3}-\frac{12}{(-1+x)^2}-\frac{16}{-1+x}\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=6 \csc ^2(x)-\frac{\csc ^4(x)}{4}+16 \log (\sin (x))\\ \end{align*}

Mathematica [A]  time = 0.01147, size = 20, normalized size = 1. \[ -\frac{1}{4} \csc ^4(x)+6 \csc ^2(x)+16 \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[5*x]*Csc[x]^5,x]

[Out]

6*Csc[x]^2 - Csc[x]^4/4 + 16*Log[Sin[x]]

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Maple [A]  time = 0.048, size = 35, normalized size = 1.8 \begin{align*} -{\frac{5}{4\, \left ( \sin \left ( x \right ) \right ) ^{4}}}+5\,{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{ \left ( \sin \left ( x \right ) \right ) ^{4}}}-4\, \left ( \cot \left ( x \right ) \right ) ^{4}+8\, \left ( \cot \left ( x \right ) \right ) ^{2}+16\,\ln \left ( \sin \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(5*x)/sin(x)^5,x)

[Out]

-5/4/sin(x)^4+5/sin(x)^4*cos(x)^4-4*cot(x)^4+8*cot(x)^2+16*ln(sin(x))

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Maxima [A]  time = 0.942205, size = 45, normalized size = 2.25 \begin{align*} \frac{5}{\sin \left (x\right )^{2}} + \frac{4 \, \sin \left (x\right )^{2} - 1}{4 \, \sin \left (x\right )^{4}} + \frac{11}{2} \, \log \left (\sin \left (x\right )^{2}\right ) + 5 \, \log \left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)/sin(x)^5,x, algorithm="maxima")

[Out]

5/sin(x)^2 + 1/4*(4*sin(x)^2 - 1)/sin(x)^4 + 11/2*log(sin(x)^2) + 5*log(sin(x))

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Fricas [B]  time = 2.36719, size = 138, normalized size = 6.9 \begin{align*} -\frac{24 \, \cos \left (x\right )^{2} - 64 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) - 23}{4 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)/sin(x)^5,x, algorithm="fricas")

[Out]

-1/4*(24*cos(x)^2 - 64*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*sin(x)) - 23)/(cos(x)^4 - 2*cos(x)^2 + 1)

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Sympy [A]  time = 105.548, size = 22, normalized size = 1.1 \begin{align*} 8 \log{\left (\sin ^{2}{\left (x \right )} \right )} + \frac{6}{\sin ^{2}{\left (x \right )}} - \frac{1}{4 \sin ^{4}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)/sin(x)**5,x)

[Out]

8*log(sin(x)**2) + 6/sin(x)**2 - 1/(4*sin(x)**4)

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Giac [B]  time = 1.10203, size = 136, normalized size = 6.8 \begin{align*} -\frac{{\left (\frac{92 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac{768 \,{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (x\right ) + 1\right )}^{2}}{64 \,{\left (\cos \left (x\right ) - 1\right )}^{2}} - \frac{23 \,{\left (\cos \left (x\right ) - 1\right )}}{16 \,{\left (\cos \left (x\right ) + 1\right )}} - \frac{{\left (\cos \left (x\right ) - 1\right )}^{2}}{64 \,{\left (\cos \left (x\right ) + 1\right )}^{2}} - 16 \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right ) + 8 \, \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(cos(5*x)/sin(x)^5,x, algorithm="giac")

[Out]

-1/64*(92*(cos(x) - 1)/(cos(x) + 1) + 768*(cos(x) - 1)^2/(cos(x) + 1)^2 + 1)*(cos(x) + 1)^2/(cos(x) - 1)^2 - 2
3/16*(cos(x) - 1)/(cos(x) + 1) - 1/64*(cos(x) - 1)^2/(cos(x) + 1)^2 - 16*log(-(cos(x) - 1)/(cos(x) + 1) + 1) +
8*log(-(cos(x) - 1)/(cos(x) + 1))