Integrand size = 11, antiderivative size = 118 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}-\frac {5 \arctan \left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}-\frac {5 \text {arctanh}\left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}} \]
5/192*sec(x)/(cos(x)*cot(x))^(1/2)-5/48*sec(x)^3/(cos(x)*cot(x))^(1/2)-5/1 28*arctan((-sin(x))^(1/2))*cot(x)*(-sin(x))^(1/2)/(cos(x)*cot(x))^(1/2)-5/ 128*arctanh((-sin(x))^(1/2))*cot(x)*(-sin(x))^(1/2)/(cos(x)*cot(x))^(1/2)+ 1/6*sec(x)^3*tan(x)^2/(cos(x)*cot(x))^(1/2)
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\frac {15 \arctan \left (\sqrt [4]{\sin ^2(x)}\right ) \cos (x)+15 \text {arctanh}\left (\sqrt [4]{\sin ^2(x)}\right ) \cos (x)+2 \sec (x) \left (5-52 \sec ^2(x)+32 \sec ^4(x)\right ) \sqrt [4]{\sin ^2(x)}}{384 \sqrt {\cos (x) \cot (x)} \sqrt [4]{\sin ^2(x)}} \]
(15*ArcTan[(Sin[x]^2)^(1/4)]*Cos[x] + 15*ArcTanh[(Sin[x]^2)^(1/4)]*Cos[x] + 2*Sec[x]*(5 - 52*Sec[x]^2 + 32*Sec[x]^4)*(Sin[x]^2)^(1/4))/(384*Sqrt[Cos [x]*Cot[x]]*(Sin[x]^2)^(1/4))
Time = 0.68 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.02, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.636, Rules used = {3042, 4897, 3042, 4900, 3042, 3077, 3042, 3077, 3042, 3079, 3042, 3081, 3042, 3044, 266, 756, 216, 219}
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.
| \(\displaystyle \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\csc (x)-\sin (x))^{7/2}}dx\) |
| \(\Big \downarrow \) 4897 |
| \(\displaystyle \int \frac {1}{(\cos (x) \cot (x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\cos (x) \cot (x))^{7/2}}dx\) |
| \(\Big \downarrow \) 4900 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \int \frac {1}{\cos ^{\frac {7}{2}}(x) \cot ^{\frac {7}{2}}(x)}dx}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^{7/2} \left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3077 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \int \frac {1}{\cos ^{\frac {7}{2}}(x) \cot ^{\frac {3}{2}}(x)}dx\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^{7/2} \left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2}}dx\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3077 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}-\frac {1}{8} \int \frac {\sqrt {\cot (x)}}{\cos ^{\frac {7}{2}}(x)}dx\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}-\frac {1}{8} \int \frac {\sqrt {-\tan \left (x+\frac {\pi }{2}\right )}}{\sin \left (x+\frac {\pi }{2}\right )^{7/2}}dx\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3079 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{8} \left (-\frac {3}{4} \int \frac {\sqrt {\cot (x)}}{\cos ^{\frac {3}{2}}(x)}dx-\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )+\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{8} \left (-\frac {3}{4} \int \frac {\sqrt {-\tan \left (x+\frac {\pi }{2}\right )}}{\sin \left (x+\frac {\pi }{2}\right )^{3/2}}dx-\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )+\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3081 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{8} \left (-\frac {3 \sqrt {-\sin (x)} \sqrt {\cot (x)} \int \frac {\sec (x)}{\sqrt {-\sin (x)}}dx}{4 \sqrt {\cos (x)}}-\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )+\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{8} \left (-\frac {3 \sqrt {-\sin (x)} \sqrt {\cot (x)} \int \frac {1}{\cos (x) \sqrt {-\sin (x)}}dx}{4 \sqrt {\cos (x)}}-\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )+\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {-\sin (x)} \sqrt {\cot (x)} \int \frac {1}{\sqrt {-\sin (x)} \left (1-\sin ^2(x)\right )}d(-\sin (x))}{4 \sqrt {\cos (x)}}-\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )+\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {-\sin (x)} \sqrt {\cot (x)} \int \frac {1}{1-\sin ^4(x)}d\sqrt {-\sin (x)}}{2 \sqrt {\cos (x)}}-\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )+\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {-\sin (x)} \sqrt {\cot (x)} \left (\frac {1}{2} \int \frac {1}{1-\sin ^2(x)}d\sqrt {-\sin (x)}+\frac {1}{2} \int \frac {1}{\sin ^2(x)+1}d\sqrt {-\sin (x)}\right )}{2 \sqrt {\cos (x)}}-\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )+\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {-\sin (x)} \sqrt {\cot (x)} \left (\frac {1}{2} \int \frac {1}{1-\sin ^2(x)}d\sqrt {-\sin (x)}-\frac {1}{2} \arctan (\sin (x))\right )}{2 \sqrt {\cos (x)}}-\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )+\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {-\sin (x)} \sqrt {\cot (x)} \left (-\frac {1}{2} \arctan (\sin (x))-\frac {1}{2} \text {arctanh}(\sin (x))\right )}{2 \sqrt {\cos (x)}}-\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )+\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
(Sqrt[Cos[x]]*Sqrt[Cot[x]]*(1/(6*Cos[x]^(7/2)*Cot[x]^(5/2)) - (5*(1/(4*Cos [x]^(7/2)*Sqrt[Cot[x]]) + (-1/2*1/(Cos[x]^(3/2)*Sqrt[Cot[x]]) + (3*(-1/2*A rcTan[Sin[x]] - ArcTanh[Sin[x]]/2)*Sqrt[Cot[x]]*Sqrt[-Sin[x]])/(2*Sqrt[Cos [x]]))/8))/12))/Sqrt[Cos[x]*Cot[x]]
3.4.21.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n + 1)/(b*f*(m + n + 1))), x] - Simp[(n + 1)/(b^2*(m + n + 1)) Int[(a*Sin[e + f*x])^m*( b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n] && !(EqQ[n, -3/2] && EqQ[m, 1] )
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Simp[b*(a*Sin[e + f*x])^(m + 2)*((b*Tan[e + f*x])^(n - 1) /(a^2*f*(m + n + 1))), x] + Simp[(m + 2)/(a^2*(m + n + 1)) Int[(a*Sin[e + f*x])^(m + 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && L tQ[m, -1] && NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[Cos[e + f*x]^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^ n) Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(- 1)]) || IntegersQ[m - 1/2, n - 1/2])
Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTri g[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, Simp[(vv^m*ww^n)^FracPar t[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])) Int[uu*vv^(m*p)*ww^(n*p), x] , x]] /; FreeQ[{m, n, p}, x] && !IntegerQ[p] && ( !InertTrigFreeQ[v] || ! InertTrigFreeQ[w])
Leaf count of result is larger than twice the leaf count of optimal. \(185\) vs. \(2(90)=180\).
Time = 16.42 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.58
| method | result | size |
| default | \(\frac {\tan \left (x \right )^{2} \sec \left (x \right )^{3} \left (15 \cos \left (x \right )^{6} \arctan \left (\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )-15 \cos \left (x \right )^{6} \operatorname {arctanh}\left (\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )-10 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{5}-10 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{4}+104 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{3}+104 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{2}-64 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )-64 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\right )}{384 \left (\cos \left (x \right )-1\right ) \left (\cos \left (x \right )+1\right )^{2} \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {\cot \left (x \right ) \cos \left (x \right )}}\) | \(186\) |
1/384*tan(x)^2*sec(x)^3*(15*cos(x)^6*arctan((sin(x)/(cos(x)+1)^2)^(1/2)*(c ot(x)+csc(x)))-15*cos(x)^6*arctanh((sin(x)/(cos(x)+1)^2)^(1/2)*(cot(x)+csc (x)))-10*(sin(x)/(cos(x)+1)^2)^(1/2)*cos(x)^5-10*(sin(x)/(cos(x)+1)^2)^(1/ 2)*cos(x)^4+104*(sin(x)/(cos(x)+1)^2)^(1/2)*cos(x)^3+104*(sin(x)/(cos(x)+1 )^2)^(1/2)*cos(x)^2-64*(sin(x)/(cos(x)+1)^2)^(1/2)*cos(x)-64*(sin(x)/(cos( x)+1)^2)^(1/2))/(cos(x)-1)/(cos(x)+1)^2/(sin(x)/(cos(x)+1)^2)^(1/2)/(cot(x )*cos(x))^(1/2)
Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=-\frac {30 \, \arctan \left (\frac {2 \, \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{\cos \left (x\right ) \sin \left (x\right ) - \cos \left (x\right )}\right ) \cos \left (x\right )^{7} - 15 \, \cos \left (x\right )^{7} \log \left (\frac {\cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 4 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} - 2 \, \cos \left (x\right ) + 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ) - 8 \, {\left (5 \, \cos \left (x\right )^{4} - 52 \, \cos \left (x\right )^{2} + 32\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{1536 \, \cos \left (x\right )^{7}} \]
-1/1536*(30*arctan(2*sqrt(cos(x)^2/sin(x))*sin(x)/(cos(x)*sin(x) - cos(x)) )*cos(x)^7 - 15*cos(x)^7*log((cos(x)^3 - 5*cos(x)^2 - (cos(x)^2 + 6*cos(x) + 4)*sin(x) + 4*(cos(x)^2 - (cos(x) + 1)*sin(x) - 1)*sqrt(cos(x)^2/sin(x) ) - 2*cos(x) + 4)/(cos(x)^3 + 3*cos(x)^2 - (cos(x)^2 - 2*cos(x) - 4)*sin(x ) - 2*cos(x) - 4)) - 8*(5*cos(x)^4 - 52*cos(x)^2 + 32)*sqrt(cos(x)^2/sin(x ))*sin(x))/cos(x)^7
Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (x\right )}-\sin \left (x\right )\right )}^{7/2}} \,d x \]