\(\int \csc ^7(x) \, dx\) [338]

Optimal result

Integrand size = 4, antiderivative size = 36 \[ \int \csc ^7(x) \, dx=-\frac {5}{16} \text {arctanh}(\cos (x))-\frac {5}{16} \cot (x) \csc (x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {1}{6} \cot (x) \csc ^5(x) \]

[Out]

-5/16*arctanh(cos(x))-5/16*cot(x)*csc(x)-5/24*cot(x)*csc(x)^3-1/6*cot(x)*csc(x)^5

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3853, 3855} \[ \int \csc ^7(x) \, dx=-\frac {5}{16} \text {arctanh}(\cos (x))-\frac {1}{6} \cot (x) \csc ^5(x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {5}{16} \cot (x) \csc (x) \]

[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 3853

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 3855

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

Rubi steps \begin{align*} \text {integral}& = -\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} \text {arctanh}(\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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(36)=72\).

Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.64 \[ \int \csc ^7(x) \, dx=-\frac {5}{64} \csc ^2\left (\frac {x}{2}\right )-\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )-\frac {1}{384} \csc ^6\left (\frac {x}{2}\right )-\frac {5}{16} \log \left (\cos \left (\frac {x}{2}\right )\right )+\frac {5}{16} \log \left (\sin \left (\frac {x}{2}\right )\right )+\frac {5}{64} \sec ^2\left (\frac {x}{2}\right )+\frac {1}{64} \sec ^4\left (\frac {x}{2}\right )+\frac {1}{384} \sec ^6\left (\frac {x}{2}\right ) \]

[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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89

method	result	size
default	\(\left (-\frac {\left (\csc ^{5}\left (x \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (x \right )\right )}{24}-\frac {5 \csc \left (x \right )}{16}\right ) \cot \left (x \right )+\frac {5 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{16}\)	\(32\)
parallelrisch	\(-\frac {75 \left (\csc ^{6}\left (x \right )\right ) \left (\left (\cos \left (2 x \right )-\frac {2 \cos \left (4 x \right )}{5}+\frac {\cos \left (6 x \right )}{15}-\frac {2}{3}\right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+\frac {44 \cos \left (x \right )}{25}-\frac {34 \cos \left (3 x \right )}{45}+\frac {2 \cos \left (5 x \right )}{15}\right )}{512}\)	\(51\)
norman	\(\frac {-\frac {1}{384}-\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{128}-\frac {15 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{128}+\frac {15 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{128}+\frac {3 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{128}+\frac {\left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{384}}{\tan \left (\frac {x}{2}\right )^{6}}+\frac {5 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{16}\)	\(58\)
risch	\(\frac {15 \,{\mathrm e}^{11 i x}-85 \,{\mathrm e}^{9 i x}+198 \,{\mathrm e}^{7 i x}+198 \,{\mathrm e}^{5 i x}-85 \,{\mathrm e}^{3 i x}+15 \,{\mathrm e}^{i x}}{24 \left ({\mathrm e}^{2 i x}-1\right )^{6}}-\frac {5 \ln \left ({\mathrm e}^{i x}+1\right )}{16}+\frac {5 \ln \left ({\mathrm e}^{i x}-1\right )}{16}\)	\(76\)

[In]

int(csc(x)^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.58 \[ \int \csc ^7(x) \, dx=\frac {30 \, \cos \left (x\right )^{5} - 80 \, \cos \left (x\right )^{3} - 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 66 \, \cos \left (x\right )}{96 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.67 \[ \int \csc ^7(x) \, dx=- \frac {- 15 \cos ^{5}{\left (x \right )} + 40 \cos ^{3}{\left (x \right )} - 33 \cos {\left (x \right )}}{48 \cos ^{6}{\left (x \right )} - 144 \cos ^{4}{\left (x \right )} + 144 \cos ^{2}{\left (x \right )} - 48} + \frac {5 \log {\left (\cos {\left (x \right )} - 1 \right )}}{32} - \frac {5 \log {\left (\cos {\left (x \right )} + 1 \right )}}{32} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \csc ^7(x) \, dx=\frac {15 \, \cos \left (x\right )^{5} - 40 \, \cos \left (x\right )^{3} + 33 \, \cos \left (x\right )}{48 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} - \frac {5}{32} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac {5}{32} \, \log \left (\cos \left (x\right ) - 1\right ) \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (28) = 56\).

Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.11 \[ \int \csc ^7(x) \, dx=-\frac {{\left (\frac {9 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {45 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}^{3}}{384 \, {\left (\cos \left (x\right ) - 1\right )}^{3}} - \frac {15 \, {\left (\cos \left (x\right ) - 1\right )}}{128 \, {\left (\cos \left (x\right ) + 1\right )}} + \frac {3 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{128 \, {\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {{\left (\cos \left (x\right ) - 1\right )}^{3}}{384 \, {\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5}{32} \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \csc ^7(x) \, dx=\frac {\frac {5\,{\cos \left (x\right )}^5}{16}-\frac {5\,{\cos \left (x\right )}^3}{6}+\frac {11\,\cos \left (x\right )}{16}}{{\cos \left (x\right )}^6-3\,{\cos \left (x\right )}^4+3\,{\cos \left (x\right )}^2-1}-\frac {5\,\mathrm {atanh}\left (\cos \left (x\right )\right )}{16} \]

[In]

int(1/sin(x)^7,x)

[Out]

((11*cos(x))/16 - (5*cos(x)^3)/6 + (5*cos(x)^5)/16)/(3*cos(x)^2 - 3*cos(x)^4 + cos(x)^6 - 1) - (5*atanh(cos(x)
))/16