Integrand size = 4, antiderivative size = 36 \[ \int \csc ^7(x) \, dx=-\frac {5}{16} \text {arctanh}(\cos (x))-\frac {5}{16} \cot (x) \csc (x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {1}{6} \cot (x) \csc ^5(x) \] Output:
-5/16*arctanh(cos(x))-5/16*cot(x)*csc(x)-5/24*cot(x)*csc(x)^3-1/6*cot(x)*c sc(x)^5
Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(36)=72\).
Time = 0.01 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.64 \[ \int \csc ^7(x) \, dx=-\frac {5}{64} \csc ^2\left (\frac {x}{2}\right )-\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )-\frac {1}{384} \csc ^6\left (\frac {x}{2}\right )-\frac {5}{16} \log \left (\cos \left (\frac {x}{2}\right )\right )+\frac {5}{16} \log \left (\sin \left (\frac {x}{2}\right )\right )+\frac {5}{64} \sec ^2\left (\frac {x}{2}\right )+\frac {1}{64} \sec ^4\left (\frac {x}{2}\right )+\frac {1}{384} \sec ^6\left (\frac {x}{2}\right ) \] Input:
Integrate[Csc[x]^7,x]
Output:
(-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]^6/3 84
Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.28, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.000, Rules used = {3042, 4255, 3042, 4255, 3042, 4255, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \csc ^7(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (x)^7dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {5}{6} \int \csc ^5(x)dx-\frac {1}{6} \cot (x) \csc ^5(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} \int \csc (x)^5dx-\frac {1}{6} \cot (x) \csc ^5(x)\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int \csc ^3(x)dx-\frac {1}{4} \cot (x) \csc ^3(x)\right )-\frac {1}{6} \cot (x) \csc ^5(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} \left (\frac {3}{4} \int \csc (x)^3dx-\frac {1}{4} \cot (x) \csc ^3(x)\right )-\frac {1}{6} \cot (x) \csc ^5(x)\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )-\frac {1}{4} \cot (x) \csc ^3(x)\right )-\frac {1}{6} \cot (x) \csc ^5(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )-\frac {1}{4} \cot (x) \csc ^3(x)\right )-\frac {1}{6} \cot (x) \csc ^5(x)\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {5}{6} \left (\frac {3}{4} \left (-\frac {1}{2} \text {arctanh}(\cos (x))-\frac {1}{2} \cot (x) \csc (x)\right )-\frac {1}{4} \cot (x) \csc ^3(x)\right )-\frac {1}{6} \cot (x) \csc ^5(x)\) |
Input:
Int[Csc[x]^7,x]
Output:
-1/6*(Cot[x]*Csc[x]^5) + (5*(-1/4*(Cot[x]*Csc[x]^3) + (3*(-1/2*ArcTanh[Cos [x]] - (Cot[x]*Csc[x])/2))/4))/6
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] + Simp[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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89
method | result | size |
default | \(\left (-\frac {\csc \left (x \right )^{5}}{6}-\frac {5 \csc \left (x \right )^{3}}{24}-\frac {5 \csc \left (x \right )}{16}\right ) \cot \left (x \right )+\frac {5 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{16}\) | \(32\) |
parallelrisch | \(-\frac {\cot \left (\frac {x}{2}\right )^{6}}{384}+\frac {\tan \left (\frac {x}{2}\right )^{6}}{384}+\frac {3 \tan \left (\frac {x}{2}\right )^{4}}{128}+\frac {15 \tan \left (\frac {x}{2}\right )^{2}}{128}+\ln \left (\tan \left (\frac {x}{2}\right )^{\frac {5}{16}}\right )-\frac {15 \cot \left (\frac {x}{2}\right )^{2}}{128}-\frac {3 \cot \left (\frac {x}{2}\right )^{4}}{128}\) | \(57\) |
norman | \(\frac {-\frac {1}{384}-\frac {3 \tan \left (\frac {x}{2}\right )^{2}}{128}-\frac {15 \tan \left (\frac {x}{2}\right )^{4}}{128}+\frac {15 \tan \left (\frac {x}{2}\right )^{8}}{128}+\frac {3 \tan \left (\frac {x}{2}\right )^{10}}{128}+\frac {\tan \left (\frac {x}{2}\right )^{12}}{384}}{\tan \left (\frac {x}{2}\right )^{6}}+\frac {5 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{16}\) | \(58\) |
risch | \(\frac {15 \,{\mathrm e}^{11 i x}-85 \,{\mathrm e}^{9 i x}+198 \,{\mathrm e}^{7 i x}+198 \,{\mathrm e}^{5 i x}-85 \,{\mathrm e}^{3 i x}+15 \,{\mathrm e}^{i x}}{24 \left ({\mathrm e}^{2 i x}-1\right )^{6}}-\frac {5 \ln \left (1+{\mathrm e}^{i x}\right )}{16}+\frac {5 \ln \left ({\mathrm e}^{i x}-1\right )}{16}\) | \(76\) |
Input:
int(csc(x)^7,x,method=_RETURNVERBOSE)
Output:
(-1/6*csc(x)^5-5/24*csc(x)^3-5/16*csc(x))*cot(x)+5/16*ln(csc(x)-cot(x))
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (28) = 56\).
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.58 \[ \int \csc ^7(x) \, dx=\frac {30 \, \cos \left (x\right )^{5} - 80 \, \cos \left (x\right )^{3} - 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 66 \, \cos \left (x\right )}{96 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \] Input:
integrate(csc(x)^7,x, algorithm="fricas")
Output:
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)*lo g(-1/2*cos(x) + 1/2) + 66*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.67 \[ \int \csc ^7(x) \, dx=- \frac {- 15 \cos ^{5}{\left (x \right )} + 40 \cos ^{3}{\left (x \right )} - 33 \cos {\left (x \right )}}{48 \cos ^{6}{\left (x \right )} - 144 \cos ^{4}{\left (x \right )} + 144 \cos ^{2}{\left (x \right )} - 48} + \frac {5 \log {\left (\cos {\left (x \right )} - 1 \right )}}{32} - \frac {5 \log {\left (\cos {\left (x \right )} + 1 \right )}}{32} \] Input:
integrate(csc(x)**7,x)
Output:
-(-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
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \csc ^7(x) \, dx=\frac {15 \, \cos \left (x\right )^{5} - 40 \, \cos \left (x\right )^{3} + 33 \, \cos \left (x\right )}{48 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} - \frac {5}{32} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac {5}{32} \, \log \left (\cos \left (x\right ) - 1\right ) \] Input:
integrate(csc(x)^7,x, algorithm="maxima")
Output:
1/48*(15*cos(x)^5 - 40*cos(x)^3 + 33*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*co s(x)^2 - 1) - 5/32*log(cos(x) + 1) + 5/32*log(cos(x) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (28) = 56\).
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.11 \[ \int \csc ^7(x) \, dx=-\frac {{\left (\frac {9 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {45 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}^{3}}{384 \, {\left (\cos \left (x\right ) - 1\right )}^{3}} - \frac {15 \, {\left (\cos \left (x\right ) - 1\right )}}{128 \, {\left (\cos \left (x\right ) + 1\right )}} + \frac {3 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{128 \, {\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {{\left (\cos \left (x\right ) - 1\right )}^{3}}{384 \, {\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5}{32} \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \] Input:
integrate(csc(x)^7,x, algorithm="giac")
Output:
-1/384*(9*(cos(x) - 1)/(cos(x) + 1) - 45*(cos(x) - 1)^2/(cos(x) + 1)^2 + 1 10*(cos(x) - 1)^3/(cos(x) + 1)^3 - 1)*(cos(x) + 1)^3/(cos(x) - 1)^3 - 15/1 28*(cos(x) - 1)/(cos(x) + 1) + 3/128*(cos(x) - 1)^2/(cos(x) + 1)^2 - 1/384 *(cos(x) - 1)^3/(cos(x) + 1)^3 + 5/32*log(-(cos(x) - 1)/(cos(x) + 1))
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \csc ^7(x) \, dx=\frac {\frac {5\,{\cos \left (x\right )}^5}{16}-\frac {5\,{\cos \left (x\right )}^3}{6}+\frac {11\,\cos \left (x\right )}{16}}{{\cos \left (x\right )}^6-3\,{\cos \left (x\right )}^4+3\,{\cos \left (x\right )}^2-1}-\frac {5\,\mathrm {atanh}\left (\cos \left (x\right )\right )}{16} \] Input:
int(1/sin(x)^7,x)
Output:
((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
Time = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \csc ^7(x) \, dx=\frac {-15 \cos \left (x \right ) \sin \left (x \right )^{4}-10 \cos \left (x \right ) \sin \left (x \right )^{2}-8 \cos \left (x \right )+15 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{6}}{48 \sin \left (x \right )^{6}} \] Input:
int(csc(x)^7,x)
Output:
( - 15*cos(x)*sin(x)**4 - 10*cos(x)*sin(x)**2 - 8*cos(x) + 15*log(tan(x/2) )*sin(x)**6)/(48*sin(x)**6)