Integrand size = 9, antiderivative size = 33 \[ \int \frac {1}{(\csc (x)-\sin (x))^7} \, dx=-\frac {1}{7} \sec ^7(x)+\frac {\sec ^9(x)}{3}-\frac {3 \sec ^{11}(x)}{11}+\frac {\sec ^{13}(x)}{13} \] Output:
-1/7*sec(x)^7+1/3*sec(x)^9-3/11*sec(x)^11+1/13*sec(x)^13
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(\csc (x)-\sin (x))^7} \, dx=-\frac {1}{7} \sec ^7(x)+\frac {\sec ^9(x)}{3}-\frac {3 \sec ^{11}(x)}{11}+\frac {\sec ^{13}(x)}{13} \] Input:
Integrate[(Csc[x] - Sin[x])^(-7),x]
Output:
-1/7*Sec[x]^7 + Sec[x]^9/3 - (3*Sec[x]^11)/11 + Sec[x]^13/13
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 4897, 3042, 3086, 25, 244, 2009}
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.
\(\displaystyle \int \frac {1}{(\csc (x)-\sin (x))^7} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(\csc (x)-\sin (x))^7}dx\) |
\(\Big \downarrow \) 4897 |
\(\displaystyle \int \tan ^7(x) \sec ^7(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (x)^7 \sec (x)^7dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \int -\sec ^6(x) \left (1-\sec ^2(x)\right )^3d\sec (x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \sec ^6(x) \left (1-\sec ^2(x)\right )^3d\sec (x)\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\int \left (-\sec ^{12}(x)+3 \sec ^{10}(x)-3 \sec ^8(x)+\sec ^6(x)\right )d\sec (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sec ^{13}(x)}{13}-\frac {3 \sec ^{11}(x)}{11}+\frac {\sec ^9(x)}{3}-\frac {\sec ^7(x)}{7}\) |
Input:
Int[(Csc[x] - Sin[x])^(-7),x]
Output:
-1/7*Sec[x]^7 + Sec[x]^9/3 - (3*Sec[x]^11)/11 + Sec[x]^13/13
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Time = 2.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {1}{13 \cos \left (x \right )^{13}}-\frac {1}{7 \cos \left (x \right )^{7}}-\frac {3}{11 \cos \left (x \right )^{11}}+\frac {1}{3 \cos \left (x \right )^{9}}\) | \(26\) |
risch | \(-\frac {128 \left (429 \,{\mathrm e}^{19 i x}-1430 \,{\mathrm e}^{17 i x}+3523 \,{\mathrm e}^{15 i x}-4020 \,{\mathrm e}^{13 i x}+3523 \,{\mathrm e}^{11 i x}-1430 \,{\mathrm e}^{9 i x}+429 \,{\mathrm e}^{7 i x}\right )}{3003 \left ({\mathrm e}^{2 i x}+1\right )^{13}}\) | \(62\) |
parallelrisch | \(\frac {\frac {85760}{1001}+\frac {960 \cos \left (x \right )}{7}-\frac {34688 \cos \left (2 x \right )}{231}+\frac {160 \cos \left (7 x \right )}{7}+\frac {80 \cos \left (11 x \right )}{77}+\frac {80 \cos \left (13 x \right )}{1001}+\frac {400 \cos \left (5 x \right )}{7}+\frac {720 \cos \left (3 x \right )}{7}+\frac {1280 \cos \left (4 x \right )}{21}-\frac {128 \cos \left (6 x \right )}{7}+\frac {480 \cos \left (9 x \right )}{77}}{\cos \left (13 x \right )+13 \cos \left (11 x \right )+78 \cos \left (9 x \right )+286 \cos \left (7 x \right )+715 \cos \left (5 x \right )+1287 \cos \left (3 x \right )+1716 \cos \left (x \right )}\) | \(104\) |
Input:
int(1/(csc(x)-sin(x))^7,x,method=_RETURNVERBOSE)
Output:
1/13/cos(x)^13-1/7/cos(x)^7-3/11/cos(x)^11+1/3/cos(x)^9
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(\csc (x)-\sin (x))^7} \, dx=-\frac {429 \, \cos \left (x\right )^{6} - 1001 \, \cos \left (x\right )^{4} + 819 \, \cos \left (x\right )^{2} - 231}{3003 \, \cos \left (x\right )^{13}} \] Input:
integrate(1/(csc(x)-sin(x))^7,x, algorithm="fricas")
Output:
-1/3003*(429*cos(x)^6 - 1001*cos(x)^4 + 819*cos(x)^2 - 231)/cos(x)^13
Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^7} \, dx=\text {Timed out} \] Input:
integrate(1/(csc(x)-sin(x))**7,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (25) = 50\).
Time = 0.05 (sec) , antiderivative size = 271, normalized size of antiderivative = 8.21 \[ \int \frac {1}{(\csc (x)-\sin (x))^7} \, dx=-\frac {32 \, {\left (\frac {13 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {78 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {286 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {2288 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {10296 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} + \frac {16302 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} + \frac {18018 \, \sin \left (x\right )^{14}}{{\left (\cos \left (x\right ) + 1\right )}^{14}} + \frac {9009 \, \sin \left (x\right )^{16}}{{\left (\cos \left (x\right ) + 1\right )}^{16}} + \frac {3003 \, \sin \left (x\right )^{18}}{{\left (\cos \left (x\right ) + 1\right )}^{18}} - 1\right )}}{3003 \, {\left (\frac {13 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {78 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {286 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {715 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {1287 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - \frac {1716 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} + \frac {1716 \, \sin \left (x\right )^{14}}{{\left (\cos \left (x\right ) + 1\right )}^{14}} - \frac {1287 \, \sin \left (x\right )^{16}}{{\left (\cos \left (x\right ) + 1\right )}^{16}} + \frac {715 \, \sin \left (x\right )^{18}}{{\left (\cos \left (x\right ) + 1\right )}^{18}} - \frac {286 \, \sin \left (x\right )^{20}}{{\left (\cos \left (x\right ) + 1\right )}^{20}} + \frac {78 \, \sin \left (x\right )^{22}}{{\left (\cos \left (x\right ) + 1\right )}^{22}} - \frac {13 \, \sin \left (x\right )^{24}}{{\left (\cos \left (x\right ) + 1\right )}^{24}} + \frac {\sin \left (x\right )^{26}}{{\left (\cos \left (x\right ) + 1\right )}^{26}} - 1\right )}} \] Input:
integrate(1/(csc(x)-sin(x))^7,x, algorithm="maxima")
Output:
-32/3003*(13*sin(x)^2/(cos(x) + 1)^2 - 78*sin(x)^4/(cos(x) + 1)^4 + 286*si n(x)^6/(cos(x) + 1)^6 + 2288*sin(x)^8/(cos(x) + 1)^8 + 10296*sin(x)^10/(co s(x) + 1)^10 + 16302*sin(x)^12/(cos(x) + 1)^12 + 18018*sin(x)^14/(cos(x) + 1)^14 + 9009*sin(x)^16/(cos(x) + 1)^16 + 3003*sin(x)^18/(cos(x) + 1)^18 - 1)/(13*sin(x)^2/(cos(x) + 1)^2 - 78*sin(x)^4/(cos(x) + 1)^4 + 286*sin(x)^ 6/(cos(x) + 1)^6 - 715*sin(x)^8/(cos(x) + 1)^8 + 1287*sin(x)^10/(cos(x) + 1)^10 - 1716*sin(x)^12/(cos(x) + 1)^12 + 1716*sin(x)^14/(cos(x) + 1)^14 - 1287*sin(x)^16/(cos(x) + 1)^16 + 715*sin(x)^18/(cos(x) + 1)^18 - 286*sin(x )^20/(cos(x) + 1)^20 + 78*sin(x)^22/(cos(x) + 1)^22 - 13*sin(x)^24/(cos(x) + 1)^24 + sin(x)^26/(cos(x) + 1)^26 - 1)
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (25) = 50\).
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.33 \[ \int \frac {1}{(\csc (x)-\sin (x))^7} \, dx=-\frac {32 \, {\left (\frac {13 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac {78 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {286 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {2288 \, {\left (\cos \left (x\right ) - 1\right )}^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {10296 \, {\left (\cos \left (x\right ) - 1\right )}^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {16302 \, {\left (\cos \left (x\right ) - 1\right )}^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {18018 \, {\left (\cos \left (x\right ) - 1\right )}^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}} - \frac {9009 \, {\left (\cos \left (x\right ) - 1\right )}^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {3003 \, {\left (\cos \left (x\right ) - 1\right )}^{9}}{{\left (\cos \left (x\right ) + 1\right )}^{9}} + 1\right )}}{3003 \, {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}^{13}} \] Input:
integrate(1/(csc(x)-sin(x))^7,x, algorithm="giac")
Output:
-32/3003*(13*(cos(x) - 1)/(cos(x) + 1) + 78*(cos(x) - 1)^2/(cos(x) + 1)^2 + 286*(cos(x) - 1)^3/(cos(x) + 1)^3 - 2288*(cos(x) - 1)^4/(cos(x) + 1)^4 + 10296*(cos(x) - 1)^5/(cos(x) + 1)^5 - 16302*(cos(x) - 1)^6/(cos(x) + 1)^6 + 18018*(cos(x) - 1)^7/(cos(x) + 1)^7 - 9009*(cos(x) - 1)^8/(cos(x) + 1)^ 8 + 3003*(cos(x) - 1)^9/(cos(x) + 1)^9 + 1)/((cos(x) - 1)/(cos(x) + 1) + 1 )^13
Time = 15.86 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(\csc (x)-\sin (x))^7} \, dx=\frac {1}{3\,{\cos \left (x\right )}^9}-\frac {1}{7\,{\cos \left (x\right )}^7}-\frac {3}{11\,{\cos \left (x\right )}^{11}}+\frac {1}{13\,{\cos \left (x\right )}^{13}} \] Input:
int(-1/(sin(x) - 1/sin(x))^7,x)
Output:
1/(3*cos(x)^9) - 1/(7*cos(x)^7) - 3/(11*cos(x)^11) + 1/(13*cos(x)^13)
Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.52 \[ \int \frac {1}{(\csc (x)-\sin (x))^7} \, dx=\frac {16 \cos \left (x \right ) \sin \left (x \right )^{12}-96 \cos \left (x \right ) \sin \left (x \right )^{10}+240 \cos \left (x \right ) \sin \left (x \right )^{8}-320 \cos \left (x \right ) \sin \left (x \right )^{6}+240 \cos \left (x \right ) \sin \left (x \right )^{4}-96 \cos \left (x \right ) \sin \left (x \right )^{2}+16 \cos \left (x \right )+429 \sin \left (x \right )^{6}-286 \sin \left (x \right )^{4}+104 \sin \left (x \right )^{2}-16}{3003 \cos \left (x \right ) \left (\sin \left (x \right )^{12}-6 \sin \left (x \right )^{10}+15 \sin \left (x \right )^{8}-20 \sin \left (x \right )^{6}+15 \sin \left (x \right )^{4}-6 \sin \left (x \right )^{2}+1\right )} \] Input:
int(1/(csc(x)-sin(x))^7,x)
Output:
(16*cos(x)*sin(x)**12 - 96*cos(x)*sin(x)**10 + 240*cos(x)*sin(x)**8 - 320* cos(x)*sin(x)**6 + 240*cos(x)*sin(x)**4 - 96*cos(x)*sin(x)**2 + 16*cos(x) + 429*sin(x)**6 - 286*sin(x)**4 + 104*sin(x)**2 - 16)/(3003*cos(x)*(sin(x) **12 - 6*sin(x)**10 + 15*sin(x)**8 - 20*sin(x)**6 + 15*sin(x)**4 - 6*sin(x )**2 + 1))