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 = 0.0908678, 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 \[ -\frac{1}{4} \csc ^4(x)+6 \csc ^2(x)+16 \log (\sin (x)) \]

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]]

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Rubi in Sympy [A]  time = 23.4735, size = 29, normalized size = 1.45 \[ 8 \log{\left (- \cos ^{2}{\left (x \right )} + 1 \right )} + \frac{6}{- \cos ^{2}{\left (x \right )} + 1} - \frac{1}{4 \left (- \cos ^{2}{\left (x \right )} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(5*x)/sin(x)**5,x)

[Out]

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

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Mathematica [A]  time = 0.0125984, 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 = 0.066, size = 35, normalized size = 1.8 \[ -{\frac{5}{4\, \left ( \sin \left ( x \right ) \right ) ^{4}}}+5\,{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{ \left ( \sin \left ( x \right ) \right ) ^{4}}}-4\, \left ( \cot \left ( x \right ) \right ) ^{4}+8\, \left ( \cot \left ( x \right ) \right ) ^{2}+16\,\ln \left ( \sin \left ( x \right ) \right ) \]

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 = 1.34561, size = 45, normalized size = 2.25 \[ \frac{5}{\sin \left (x\right )^{2}} + \frac{4 \, \sin \left (x\right )^{2} - 1}{4 \, \sin \left (x\right )^{4}} + \frac{11}{2} \, \log \left (\sin \left (x\right )^{2}\right ) + 5 \, \log \left (\sin \left (x\right )\right ) \]

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 [A]  time = 0.230294, size = 58, normalized size = 2.9 \[ -\frac{24 \, \cos \left (x\right )^{2} - 64 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) - 23}{4 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(5*x)/sin(x)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.204975, size = 136, normalized size = 6.8 \[ -\frac{{\left (\frac{92 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac{768 \,{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{\left (\cos \left (x\right ) + 1\right )}^{2}}{64 \,{\left (\cos \left (x\right ) - 1\right )}^{2}} - \frac{23 \,{\left (\cos \left (x\right ) - 1\right )}}{16 \,{\left (\cos \left (x\right ) + 1\right )}} - \frac{{\left (\cos \left (x\right ) - 1\right )}^{2}}{64 \,{\left (\cos \left (x\right ) + 1\right )}^{2}} - 16 \,{\rm ln}\left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right ) + 8 \,{\rm ln}\left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \]

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)*(co
s(x) + 1)^2/(cos(x) - 1)^2 - 23/16*(cos(x) - 1)/(cos(x) + 1) - 1/64*(cos(x) - 1)
^2/(cos(x) + 1)^2 - 16*ln(-(cos(x) - 1)/(cos(x) + 1) + 1) + 8*ln(-(cos(x) - 1)/(
cos(x) + 1))