3.76 \(\int \csc ^5(x) \, dx\)

Optimal. Leaf size=26 \[ -\frac{3}{8} \tanh ^{-1}(\cos (x))-\frac{1}{4} \cot (x) \csc ^3(x)-\frac{3}{8} \cot (x) \csc (x) \]

[Out]

(-3*ArcTanh[Cos[x]])/8 - (3*Cot[x]*Csc[x])/8 - (Cot[x]*Csc[x]^3)/4

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Rubi [A]  time = 0.0124967, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3768, 3770} \[ -\frac{3}{8} \tanh ^{-1}(\cos (x))-\frac{1}{4} \cot (x) \csc ^3(x)-\frac{3}{8} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^5,x]

[Out]

(-3*ArcTanh[Cos[x]])/8 - (3*Cot[x]*Csc[x])/8 - (Cot[x]*Csc[x]^3)/4

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \csc ^5(x) \, dx &=-\frac{1}{4} \cot (x) \csc ^3(x)+\frac{3}{4} \int \csc ^3(x) \, dx\\ &=-\frac{3}{8} \cot (x) \csc (x)-\frac{1}{4} \cot (x) \csc ^3(x)+\frac{3}{8} \int \csc (x) \, dx\\ &=-\frac{3}{8} \tanh ^{-1}(\cos (x))-\frac{3}{8} \cot (x) \csc (x)-\frac{1}{4} \cot (x) \csc ^3(x)\\ \end{align*}

Mathematica [B]  time = 0.0058338, size = 71, normalized size = 2.73 \[ -\frac{1}{64} \csc ^4\left (\frac{x}{2}\right )-\frac{3}{32} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{64} \sec ^4\left (\frac{x}{2}\right )+\frac{3}{32} \sec ^2\left (\frac{x}{2}\right )+\frac{3}{8} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{3}{8} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^5,x]

[Out]

(-3*Csc[x/2]^2)/32 - Csc[x/2]^4/64 - (3*Log[Cos[x/2]])/8 + (3*Log[Sin[x/2]])/8 + (3*Sec[x/2]^2)/32 + Sec[x/2]^
4/64

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Maple [A]  time = 0.031, size = 26, normalized size = 1. \begin{align*} \left ( -{\frac{ \left ( \csc \left ( x \right ) \right ) ^{3}}{4}}-{\frac{3\,\csc \left ( x \right ) }{8}} \right ) \cot \left ( x \right ) +{\frac{3\,\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(x)^5,x)

[Out]

(-1/4*csc(x)^3-3/8*csc(x))*cot(x)+3/8*ln(csc(x)-cot(x))

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Maxima [B]  time = 0.932535, size = 57, normalized size = 2.19 \begin{align*} \frac{3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )}{8 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} - \frac{3}{16} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{3}{16} \, \log \left (\cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^5,x, algorithm="maxima")

[Out]

1/8*(3*cos(x)^3 - 5*cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1) - 3/16*log(cos(x) + 1) + 3/16*log(cos(x) - 1)

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Fricas [B]  time = 1.92129, size = 227, normalized size = 8.73 \begin{align*} \frac{6 \, \cos \left (x\right )^{3} - 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 10 \, \cos \left (x\right )}{16 \,{\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(1/sin(x)^5,x, algorithm="fricas")

[Out]

1/16*(6*cos(x)^3 - 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(-1/
2*cos(x) + 1/2) - 10*cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1)

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Sympy [A]  time = 0.135067, size = 46, normalized size = 1.77 \begin{align*} \frac{3 \cos ^{3}{\left (x \right )} - 5 \cos{\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} + \frac{3 \log{\left (\cos{\left (x \right )} - 1 \right )}}{16} - \frac{3 \log{\left (\cos{\left (x \right )} + 1 \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)**5,x)

[Out]

(3*cos(x)**3 - 5*cos(x))/(8*cos(x)**4 - 16*cos(x)**2 + 8) + 3*log(cos(x) - 1)/16 - 3*log(cos(x) + 1)/16

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Giac [A]  time = 1.05432, size = 51, normalized size = 1.96 \begin{align*} \frac{3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )}{8 \,{\left (\cos \left (x\right )^{2} - 1\right )}^{2}} - \frac{3}{16} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{3}{16} \, \log \left (-\cos \left (x\right ) + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^5,x, algorithm="giac")

[Out]

1/8*(3*cos(x)^3 - 5*cos(x))/(cos(x)^2 - 1)^2 - 3/16*log(cos(x) + 1) + 3/16*log(-cos(x) + 1)