3.338 \(\int \csc ^7(x) \, dx\)

Optimal. Leaf size=36 \[ -\frac{5}{16} \tanh ^{-1}(\cos (x))-\frac{1}{6} \cot (x) \csc ^5(x)-\frac{5}{24} \cot (x) \csc ^3(x)-\frac{5}{16} \cot (x) \csc (x) \]

[Out]

(-5*ArcTanh[Cos[x]])/16 - (5*Cot[x]*Csc[x])/16 - (5*Cot[x]*Csc[x]^3)/24 - (Cot[x
]*Csc[x]^5)/6

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Rubi [A]  time = 0.0329247, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{5}{16} \tanh ^{-1}(\cos (x))-\frac{1}{6} \cot (x) \csc ^5(x)-\frac{5}{24} \cot (x) \csc ^3(x)-\frac{5}{16} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]  Int[Csc[x]^7,x]

[Out]

(-5*ArcTanh[Cos[x]])/16 - (5*Cot[x]*Csc[x])/16 - (5*Cot[x]*Csc[x]^3)/24 - (Cot[x
]*Csc[x]^5)/6

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Rubi in Sympy [A]  time = 0.700672, size = 42, normalized size = 1.17 \[ - \frac{5 \operatorname{atanh}{\left (\cos{\left (x \right )} \right )}}{16} - \frac{5 \cos{\left (x \right )}}{16 \sin ^{2}{\left (x \right )}} - \frac{5 \cos{\left (x \right )}}{24 \sin ^{4}{\left (x \right )}} - \frac{\cos{\left (x \right )}}{6 \sin ^{6}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(csc(x)**7,x)

[Out]

-5*atanh(cos(x))/16 - 5*cos(x)/(16*sin(x)**2) - 5*cos(x)/(24*sin(x)**4) - cos(x)
/(6*sin(x)**6)

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Mathematica [B]  time = 0.0106596, size = 95, normalized size = 2.64 \[ -\frac{1}{384} \csc ^6\left (\frac{x}{2}\right )-\frac{1}{64} \csc ^4\left (\frac{x}{2}\right )-\frac{5}{64} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{384} \sec ^6\left (\frac{x}{2}\right )+\frac{1}{64} \sec ^4\left (\frac{x}{2}\right )+\frac{5}{64} \sec ^2\left (\frac{x}{2}\right )+\frac{5}{16} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{5}{16} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[x]^7,x]

[Out]

(-5*Csc[x/2]^2)/64 - Csc[x/2]^4/64 - Csc[x/2]^6/384 - (5*Log[Cos[x/2]])/16 + (5*
Log[Sin[x/2]])/16 + (5*Sec[x/2]^2)/64 + Sec[x/2]^4/64 + Sec[x/2]^6/384

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Maple [A]  time = 0.054, size = 32, normalized size = 0.9 \[ \left ( -{\frac{ \left ( \csc \left ( x \right ) \right ) ^{5}}{6}}-{\frac{5\, \left ( \csc \left ( x \right ) \right ) ^{3}}{24}}-{\frac{5\,\csc \left ( x \right ) }{16}} \right ) \cot \left ( x \right ) +{\frac{5\,\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(csc(x)^7,x)

[Out]

(-1/6*csc(x)^5-5/24*csc(x)^3-5/16*csc(x))*cot(x)+5/16*ln(csc(x)-cot(x))

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Maxima [A]  time = 1.55237, size = 73, normalized size = 2.03 \[ \frac{15 \, \cos \left (x\right )^{5} - 40 \, \cos \left (x\right )^{3} + 33 \, \cos \left (x\right )}{48 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} - \frac{5}{32} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{5}{32} \, \log \left (\cos \left (x\right ) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csc(x)^7,x, algorithm="maxima")

[Out]

1/48*(15*cos(x)^5 - 40*cos(x)^3 + 33*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2
- 1) - 5/32*log(cos(x) + 1) + 5/32*log(cos(x) - 1)

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Fricas [A]  time = 0.235632, size = 126, normalized size = 3.5 \[ \frac{30 \, \cos \left (x\right )^{5} - 80 \, \cos \left (x\right )^{3} - 15 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 15 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 66 \, \cos \left (x\right )}{96 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csc(x)^7,x, algorithm="fricas")

[Out]

1/96*(30*cos(x)^5 - 80*cos(x)^3 - 15*(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*lo
g(1/2*cos(x) + 1/2) + 15*(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*log(-1/2*cos(x
) + 1/2) + 66*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)

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Sympy [A]  time = 0.213763, size = 60, normalized size = 1.67 \[ \frac{15 \cos ^{5}{\left (x \right )} - 40 \cos ^{3}{\left (x \right )} + 33 \cos{\left (x \right )}}{48 \cos ^{6}{\left (x \right )} - 144 \cos ^{4}{\left (x \right )} + 144 \cos ^{2}{\left (x \right )} - 48} + \frac{5 \log{\left (\cos{\left (x \right )} - 1 \right )}}{32} - \frac{5 \log{\left (\cos{\left (x \right )} + 1 \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csc(x)**7,x)

[Out]

(15*cos(x)**5 - 40*cos(x)**3 + 33*cos(x))/(48*cos(x)**6 - 144*cos(x)**4 + 144*co
s(x)**2 - 48) + 5*log(cos(x) - 1)/32 - 5*log(cos(x) + 1)/32

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GIAC/XCAS [A]  time = 0.204668, size = 151, normalized size = 4.19 \[ -\frac{{\left (\frac{9 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac{45 \,{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{110 \,{\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - 1\right )}{\left (\cos \left (x\right ) + 1\right )}^{3}}{384 \,{\left (\cos \left (x\right ) - 1\right )}^{3}} - \frac{15 \,{\left (\cos \left (x\right ) - 1\right )}}{128 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{3 \,{\left (\cos \left (x\right ) - 1\right )}^{2}}{128 \,{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac{{\left (\cos \left (x\right ) - 1\right )}^{3}}{384 \,{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac{5}{32} \,{\rm ln}\left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csc(x)^7,x, algorithm="giac")

[Out]

-1/384*(9*(cos(x) - 1)/(cos(x) + 1) - 45*(cos(x) - 1)^2/(cos(x) + 1)^2 + 110*(co
s(x) - 1)^3/(cos(x) + 1)^3 - 1)*(cos(x) + 1)^3/(cos(x) - 1)^3 - 15/128*(cos(x) -
1)/(cos(x) + 1) + 3/128*(cos(x) - 1)^2/(cos(x) + 1)^2 - 1/384*(cos(x) - 1)^3/(c
os(x) + 1)^3 + 5/32*ln(-(cos(x) - 1)/(cos(x) + 1))