\(\int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx\) [214]

Optimal result

Integrand size = 11, antiderivative size = 118 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}-\frac {5 \arctan \left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}-\frac {5 \text {arctanh}\left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}} \] Output:

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)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\frac {15 \arctan \left (\sqrt [4]{\sin ^2(x)}\right ) \cos (x)+15 \text {arctanh}\left (\sqrt [4]{\sin ^2(x)}\right ) \cos (x)+2 \sec (x) \left (5-52 \sec ^2(x)+32 \sec ^4(x)\right ) \sqrt [4]{\sin ^2(x)}}{384 \sqrt {\cos (x) \cot (x)} \sqrt [4]{\sin ^2(x)}} \] Input:

Integrate[(Csc[x] - Sin[x])^(-7/2),x]
 

Output:

(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))
 

Rubi [A] (warning: unable to verify)

Time = 0.73 (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

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{6 \cos ^{\frac {7}{2}}(x) \cot ^{\frac {5}{2}}(x)}-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {-\sin (x)} \sqrt {\cot (x)} \left (-\frac {1}{2} \arctan (\sin (x))-\frac {1}{2} \text {arctanh}(\sin (x))\right )}{2 \sqrt {\cos (x)}}-\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )+\frac {1}{4 \cos ^{\frac {7}{2}}(x) \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\)

Input:

Int[(Csc[x] - Sin[x])^(-7/2),x]
 

Output:

(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]]
 

Defintions of rubi rules used

rule 216

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])
 

rule 219

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])
 

rule 266

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]
 

rule 756

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3044

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])
 

rule 3077

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] 
)
 

rule 3079

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]
 

rule 3081

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])
 

rule 4897

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]
 

rule 4900

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])
 

Maple [A] (verified)

Time = 23.75 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {15 \arctan \left (\frac {\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\cos \left (x \right )-1}\right ) \cos \left (x \right )-15 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\cos \left (x \right )-1}\right ) \cos \left (x \right )+2 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (32 \sec \left (x \right )^{5}+32 \sec \left (x \right )^{4}-52 \sec \left (x \right )^{3}-52 \sec \left (x \right )^{2}+5 \sec \left (x \right )+5\right )}{384 \left (\cos \left (x \right )+1\right ) \sqrt {\cos \left (x \right ) \cot \left (x \right )}\, \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\)	\(121\)

Input:

int(1/(csc(x)-sin(x))^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/384/(cos(x)+1)/(cos(x)*cot(x))^(1/2)/(sin(x)/(cos(x)+1)^2)^(1/2)*(15*arc 
tan((sin(x)/(cos(x)+1)^2)^(1/2)*sin(x)/(cos(x)-1))*cos(x)-15*arctanh((sin( 
x)/(cos(x)+1)^2)^(1/2)*sin(x)/(cos(x)-1))*cos(x)+2*(sin(x)/(cos(x)+1)^2)^( 
1/2)*(32*sec(x)^5+32*sec(x)^4-52*sec(x)^3-52*sec(x)^2+5*sec(x)+5))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\frac {30 \, \arctan \left (-\frac {\sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{2 \, {\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )}}\right ) \cos \left (x\right )^{7} + 15 \, \cos \left (x\right )^{7} \log \left (\frac {\cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 4 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} - 2 \, \cos \left (x\right ) + 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ) + 8 \, {\left (5 \, \cos \left (x\right )^{4} - 52 \, \cos \left (x\right )^{2} + 32\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{1536 \, \cos \left (x\right )^{7}} \] Input:

integrate(1/(csc(x)-sin(x))^(7/2),x, algorithm="fricas")
 

Output:

1/1536*(30*arctan(-1/2*sqrt(cos(x)^2/sin(x))*(cos(x) - sin(x) + 1)/(cos(x) 
 + sin(x) + 1))*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(c 
os(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
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(csc(x)-sin(x))**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(csc(x)-sin(x))^(7/2),x, algorithm="maxima")
 

Output:

integrate((csc(x) - sin(x))^(-7/2), x)
 

Giac [F]

\[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(csc(x)-sin(x))^(7/2),x, algorithm="giac")
 

Output:

integrate((csc(x) - sin(x))^(-7/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (x\right )}-\sin \left (x\right )\right )}^{7/2}} \,d x \] Input:

int(1/(1/sin(x) - sin(x))^(7/2),x)
 

Output:

int(1/(1/sin(x) - sin(x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int \frac {\sqrt {\csc \left (x \right )-\sin \left (x \right )}}{\csc \left (x \right )^{4}-4 \csc \left (x \right )^{3} \sin \left (x \right )+6 \csc \left (x \right )^{2} \sin \left (x \right )^{2}-4 \csc \left (x \right ) \sin \left (x \right )^{3}+\sin \left (x \right )^{4}}d x \] Input:

int(1/(csc(x)-sin(x))^(7/2),x)
 

Output:

int(sqrt(csc(x) - sin(x))/(csc(x)**4 - 4*csc(x)**3*sin(x) + 6*csc(x)**2*si 
n(x)**2 - 4*csc(x)*sin(x)**3 + sin(x)**4),x)