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 = 0.0195031, 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, Rules used = {3768, 3770} \[ -\frac{5}{16} \tanh ^{-1}(\cos (x))-\frac{1}{6} \cot (x) \csc ^5(x)-\frac{5}{24} \cot (x) \csc ^3(x)-\frac{5}{16} \cot (x) \csc (x) \]

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

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 ^7(x) \, dx &=-\frac{1}{6} \cot (x) \csc ^5(x)+\frac{5}{6} \int \csc ^5(x) \, dx\\ &=-\frac{5}{24} \cot (x) \csc ^3(x)-\frac{1}{6} \cot (x) \csc ^5(x)+\frac{5}{8} \int \csc ^3(x) \, dx\\ &=-\frac{5}{16} \cot (x) \csc (x)-\frac{5}{24} \cot (x) \csc ^3(x)-\frac{1}{6} \cot (x) \csc ^5(x)+\frac{5}{16} \int \csc (x) \, dx\\ &=-\frac{5}{16} \tanh ^{-1}(\cos (x))-\frac{5}{16} \cot (x) \csc (x)-\frac{5}{24} \cot (x) \csc ^3(x)-\frac{1}{6} \cot (x) \csc ^5(x)\\ \end{align*}

Mathematica [B]  time = 0.0081332, 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 = 0.034, size = 32, normalized size = 0.9 \begin{align*} \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}} \end{align*}

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 = 0.923332, size = 73, normalized size = 2.03 \begin{align*} \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 ) \end{align*}

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 [B]  time = 1.77784, size = 302, normalized size = 8.39 \begin{align*} \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 )}} \end{align*}

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)*log(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 = 0.158818, size = 60, normalized size = 1.67 \begin{align*} \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} \end{align*}

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*cos(x)**2 - 48) + 5*log(cos(x) -
1)/32 - 5*log(cos(x) + 1)/32

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Giac [B]  time = 1.05699, size = 151, normalized size = 4.19 \begin{align*} -\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} \, \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(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*(cos(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 - 1/3
84*(cos(x) - 1)^3/(cos(x) + 1)^3 + 5/32*log(-(cos(x) - 1)/(cos(x) + 1))