Integrand size = 10, antiderivative size = 132 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=-\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}+\frac {63 x \csc ^2(x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}} \]
-63/256*cot(x)/a^2/(a*csc(x)^4)^(1/2)+63/256*x*csc(x)^2/a^2/(a*csc(x)^4)^( 1/2)-21/128*cos(x)*sin(x)/a^2/(a*csc(x)^4)^(1/2)-21/160*cos(x)*sin(x)^3/a^ 2/(a*csc(x)^4)^(1/2)-9/80*cos(x)*sin(x)^5/a^2/(a*csc(x)^4)^(1/2)-1/10*cos( x)*sin(x)^7/a^2/(a*csc(x)^4)^(1/2)
Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\frac {\sqrt {a \csc ^4(x)} \sin ^2(x) (2520 x-2100 \sin (2 x)+600 \sin (4 x)-150 \sin (6 x)+25 \sin (8 x)-2 \sin (10 x))}{10240 a^3} \]
(Sqrt[a*Csc[x]^4]*Sin[x]^2*(2520*x - 2100*Sin[2*x] + 600*Sin[4*x] - 150*Si n[6*x] + 25*Sin[8*x] - 2*Sin[10*x]))/(10240*a^3)
Time = 0.46 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.70, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {3042, 4611, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sec \left (x+\frac {\pi }{2}\right )^4\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4611 |
| \(\displaystyle \frac {\csc ^2(x) \int \sin ^{10}(x)dx}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^2(x) \int \sin (x)^{10}dx}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\csc ^2(x) \left (\frac {9}{10} \int \sin ^8(x)dx-\frac {1}{10} \sin ^9(x) \cos (x)\right )}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^2(x) \left (\frac {9}{10} \int \sin (x)^8dx-\frac {1}{10} \sin ^9(x) \cos (x)\right )}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\csc ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \int \sin ^6(x)dx-\frac {1}{8} \sin ^7(x) \cos (x)\right )-\frac {1}{10} \sin ^9(x) \cos (x)\right )}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \int \sin (x)^6dx-\frac {1}{8} \sin ^7(x) \cos (x)\right )-\frac {1}{10} \sin ^9(x) \cos (x)\right )}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\csc ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \sin ^4(x)dx-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\right )-\frac {1}{10} \sin ^9(x) \cos (x)\right )}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \sin (x)^4dx-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\right )-\frac {1}{10} \sin ^9(x) \cos (x)\right )}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\csc ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin ^2(x)dx-\frac {1}{4} \sin ^3(x) \cos (x)\right )-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\right )-\frac {1}{10} \sin ^9(x) \cos (x)\right )}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\csc ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin (x)^2dx-\frac {1}{4} \sin ^3(x) \cos (x)\right )-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\right )-\frac {1}{10} \sin ^9(x) \cos (x)\right )}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\csc ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\right )-\frac {1}{10} \sin ^9(x) \cos (x)\right )}{a^2 \sqrt {a \csc ^4(x)}}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\csc ^2(x) \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sin (x) \cos (x)\right )-\frac {1}{4} \sin ^3(x) \cos (x)\right )-\frac {1}{6} \sin ^5(x) \cos (x)\right )-\frac {1}{8} \sin ^7(x) \cos (x)\right )-\frac {1}{10} \sin ^9(x) \cos (x)\right )}{a^2 \sqrt {a \csc ^4(x)}}\) |
(Csc[x]^2*(-1/10*(Cos[x]*Sin[x]^9) + (9*(-1/8*(Cos[x]*Sin[x]^7) + (7*(-1/6 *(Cos[x]*Sin[x]^5) + (5*(-1/4*(Cos[x]*Sin[x]^3) + (3*(x/2 - (Cos[x]*Sin[x] )/2))/4))/6))/8))/10))/(a^2*Sqrt[a*Csc[x]^4])
3.1.67.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[b^ IntPart[p]*((b*(c*Sec[e + f*x])^n)^FracPart[p]/(c*Sec[e + f*x])^(n*FracPart [p])) Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p]
Time = 0.89 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.46
| method | result | size |
| default | \(-\frac {\left (128 \cos \left (x \right )^{8} \cot \left (x \right )-656 \cos \left (x \right )^{6} \cot \left (x \right )+1368 \cos \left (x \right )^{4} \cot \left (x \right )-1490 \cos \left (x \right )^{2} \cot \left (x \right )+965 \cot \left (x \right )-315 \csc \left (x \right )^{2} x \right ) \sqrt {16}}{5120 \sqrt {a \csc \left (x \right )^{4}}\, a^{2}}\) | \(61\) |
| risch | \(-\frac {63 \,{\mathrm e}^{2 i x} x}{256 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {i {\mathrm e}^{12 i x}}{10240 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {5 i {\mathrm e}^{10 i x}}{4096 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {105 i {\mathrm e}^{4 i x}}{1024 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {105 i}{1024 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {15 i {\mathrm e}^{-2 i x}}{512 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {15 i {\mathrm e}^{-4 i x}}{2048 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {37 i \cos \left (8 x \right )}{5120 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {19 \sin \left (8 x \right )}{2560 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {115 i \cos \left (6 x \right )}{4096 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {125 \sin \left (6 x \right )}{4096 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}\) | \(409\) |
-1/5120/(a*csc(x)^4)^(1/2)/a^2*(128*cos(x)^8*cot(x)-656*cos(x)^6*cot(x)+13 68*cos(x)^4*cot(x)-1490*cos(x)^2*cot(x)+965*cot(x)-315*csc(x)^2*x)*16^(1/2 )
Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=-\frac {{\left (315 \, x \cos \left (x\right )^{2} - {\left (128 \, \cos \left (x\right )^{11} - 784 \, \cos \left (x\right )^{9} + 2024 \, \cos \left (x\right )^{7} - 2858 \, \cos \left (x\right )^{5} + 2455 \, \cos \left (x\right )^{3} - 965 \, \cos \left (x\right )\right )} \sin \left (x\right ) - 315 \, x\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{1280 \, a^{3}} \]
-1/1280*(315*x*cos(x)^2 - (128*cos(x)^11 - 784*cos(x)^9 + 2024*cos(x)^7 - 2858*cos(x)^5 + 2455*cos(x)^3 - 965*cos(x))*sin(x) - 315*x)*sqrt(a/(cos(x) ^4 - 2*cos(x)^2 + 1))/a^3
\[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \csc ^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=-\frac {965 \, \tan \left (x\right )^{9} + 2370 \, \tan \left (x\right )^{7} + 2688 \, \tan \left (x\right )^{5} + 1470 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{1280 \, {\left (a^{\frac {5}{2}} \tan \left (x\right )^{10} + 5 \, a^{\frac {5}{2}} \tan \left (x\right )^{8} + 10 \, a^{\frac {5}{2}} \tan \left (x\right )^{6} + 10 \, a^{\frac {5}{2}} \tan \left (x\right )^{4} + 5 \, a^{\frac {5}{2}} \tan \left (x\right )^{2} + a^{\frac {5}{2}}\right )}} + \frac {63 \, x}{256 \, a^{\frac {5}{2}}} \]
-1/1280*(965*tan(x)^9 + 2370*tan(x)^7 + 2688*tan(x)^5 + 1470*tan(x)^3 + 31 5*tan(x))/(a^(5/2)*tan(x)^10 + 5*a^(5/2)*tan(x)^8 + 10*a^(5/2)*tan(x)^6 + 10*a^(5/2)*tan(x)^4 + 5*a^(5/2)*tan(x)^2 + a^(5/2)) + 63/256*x/a^(5/2)
Exception generated. \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^4}\right )}^{5/2}} \,d x \]