Integrand size = 11, antiderivative size = 99 \[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=-\frac {3 \arctan \left (\sqrt {-\sin (x)}\right ) \cos (x)}{32 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}+\frac {3 \text {arctanh}\left (\sqrt {-\sin (x)}\right ) \cos (x)}{32 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}-\frac {3 \tan (x)}{16 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^2(x) \tan (x)}{4 \sqrt {\cos (x) \cot (x)}} \]
-3/32*arctan((-sin(x))^(1/2))*cos(x)/(cos(x)*cot(x))^(1/2)/(-sin(x))^(1/2) +3/32*arctanh((-sin(x))^(1/2))*cos(x)/(cos(x)*cot(x))^(1/2)/(-sin(x))^(1/2 )-3/16*tan(x)/(cos(x)*cot(x))^(1/2)+1/4*sec(x)^2*tan(x)/(cos(x)*cot(x))^(1 /2)
Time = 0.43 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=-\frac {\sqrt {\cos (x) \cot (x)} \sin (x) \left (-3 \arctan \left (\sqrt [4]{\sin ^2(x)}\right )+3 \text {arctanh}\left (\sqrt [4]{\sin ^2(x)}\right )+(-5+3 \cos (2 x)) \sec ^4(x) \sin ^2(x)^{3/4}\right ) \tan (x)}{32 \sin ^2(x)^{3/4}} \]
-1/32*(Sqrt[Cos[x]*Cot[x]]*Sin[x]*(-3*ArcTan[(Sin[x]^2)^(1/4)] + 3*ArcTanh [(Sin[x]^2)^(1/4)] + (-5 + 3*Cos[2*x])*Sec[x]^4*(Sin[x]^2)^(3/4))*Tan[x])/ (Sin[x]^2)^(3/4)
Time = 0.58 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.455, Rules used = {3042, 4897, 3042, 4900, 3042, 3077, 3042, 3079, 3042, 3081, 3042, 3044, 266, 827, 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.
| \(\displaystyle \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\csc (x)-\sin (x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4897 |
| \(\displaystyle \int \frac {1}{(\cos (x) \cot (x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\cos (x) \cot (x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4900 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \int \frac {1}{\cos ^{\frac {5}{2}}(x) \cot ^{\frac {5}{2}}(x)}dx}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^{5/2} \left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3077 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \int \frac {1}{\cos ^{\frac {5}{2}}(x) \sqrt {\cot (x)}}dx\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^{5/2} \sqrt {-\tan \left (x+\frac {\pi }{2}\right )}}dx\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3079 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \left (\frac {1}{4} \int \frac {1}{\sqrt {\cos (x)} \sqrt {\cot (x)}}dx+\frac {1}{2 \sqrt {\cos (x)} \cot ^{\frac {3}{2}}(x)}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \left (\frac {1}{4} \int \frac {1}{\sqrt {\sin \left (x+\frac {\pi }{2}\right )} \sqrt {-\tan \left (x+\frac {\pi }{2}\right )}}dx+\frac {1}{2 \sqrt {\cos (x)} \cot ^{\frac {3}{2}}(x)}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3081 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \left (\frac {\sqrt {\cos (x)} \int \sec (x) \sqrt {-\sin (x)}dx}{4 \sqrt {-\sin (x)} \sqrt {\cot (x)}}+\frac {1}{2 \sqrt {\cos (x)} \cot ^{\frac {3}{2}}(x)}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \left (\frac {\sqrt {\cos (x)} \int \frac {\sqrt {-\sin (x)}}{\cos (x)}dx}{4 \sqrt {-\sin (x)} \sqrt {\cot (x)}}+\frac {1}{2 \sqrt {\cos (x)} \cot ^{\frac {3}{2}}(x)}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \left (\frac {1}{2 \sqrt {\cos (x)} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {\cos (x)} \int \frac {\sqrt {-\sin (x)}}{1-\sin ^2(x)}d(-\sin (x))}{4 \sqrt {-\sin (x)} \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \left (\frac {1}{2 \sqrt {\cos (x)} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {\cos (x)} \int \frac {\sin ^2(x)}{1-\sin ^4(x)}d\sqrt {-\sin (x)}}{2 \sqrt {-\sin (x)} \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \left (\frac {1}{2 \sqrt {\cos (x)} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {\cos (x)} \left (\frac {1}{2} \int \frac {1}{1-\sin ^2(x)}d\sqrt {-\sin (x)}-\frac {1}{2} \int \frac {1}{\sin ^2(x)+1}d\sqrt {-\sin (x)}\right )}{2 \sqrt {-\sin (x)} \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \left (\frac {1}{2 \sqrt {\cos (x)} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {\cos (x)} \left (\frac {1}{2} \int \frac {1}{1-\sin ^2(x)}d\sqrt {-\sin (x)}+\frac {1}{2} \arctan (\sin (x))\right )}{2 \sqrt {-\sin (x)} \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \left (\frac {1}{2 \sqrt {\cos (x)} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {\cos (x)} \left (\frac {1}{2} \arctan (\sin (x))-\frac {1}{2} \text {arctanh}(\sin (x))\right )}{2 \sqrt {-\sin (x)} \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
(Sqrt[Cos[x]]*Sqrt[Cot[x]]*(1/(4*Cos[x]^(5/2)*Cot[x]^(3/2)) - (3*(1/(2*Sqr t[Cos[x]]*Cot[x]^(3/2)) - ((ArcTan[Sin[x]]/2 - ArcTanh[Sin[x]]/2)*Sqrt[Cos [x]])/(2*Sqrt[Cot[x]]*Sqrt[-Sin[x]])))/8))/Sqrt[Cos[x]*Cot[x]]
3.4.20.3.1 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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) 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. \(157\) vs. \(2(75)=150\).
Time = 16.58 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.60
| method | result | size |
| default | \(\frac {3 \cos \left (x \right ) \sin \left (x \right )^{2} \operatorname {arctanh}\left (\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )+3 \cos \left (x \right ) \sin \left (x \right )^{2} \arctan \left (\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )+6 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )^{3}+6 \sin \left (x \right )^{2} \tan \left (x \right ) \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-8 \sin \left (x \right ) \tan \left (x \right )^{2} \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-8 \tan \left (x \right )^{3} \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}{32 \left (\cos \left (x \right )-1\right ) \left (\cos \left (x \right )+1\right )^{2} \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {\cot \left (x \right ) \cos \left (x \right )}}\) | \(158\) |
1/32/(cos(x)-1)/(cos(x)+1)^2/(sin(x)/(cos(x)+1)^2)^(1/2)/(cot(x)*cos(x))^( 1/2)*(3*cos(x)*sin(x)^2*arctanh((sin(x)/(cos(x)+1)^2)^(1/2)*(cot(x)+csc(x) ))+3*cos(x)*sin(x)^2*arctan((sin(x)/(cos(x)+1)^2)^(1/2)*(cot(x)+csc(x)))+6 *(sin(x)/(cos(x)+1)^2)^(1/2)*sin(x)^3+6*sin(x)^2*tan(x)*(sin(x)/(cos(x)+1) ^2)^(1/2)-8*sin(x)*tan(x)^2*(sin(x)/(cos(x)+1)^2)^(1/2)-8*tan(x)^3*(sin(x) /(cos(x)+1)^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (75) = 150\).
Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.67 \[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=-\frac {6 \, \arctan \left (\frac {2 \, \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{\cos \left (x\right ) \sin \left (x\right ) - \cos \left (x\right )}\right ) \cos \left (x\right )^{5} - 3 \, \cos \left (x\right )^{5} \log \left (\frac {\cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - 4 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} - 2 \, \cos \left (x\right ) + 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ) - 8 \, {\left (3 \, \cos \left (x\right )^{4} - 7 \, \cos \left (x\right )^{2} + 4\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}}}{128 \, \cos \left (x\right )^{5}} \]
-1/128*(6*arctan(2*sqrt(cos(x)^2/sin(x))*sin(x)/(cos(x)*sin(x) - cos(x)))* cos(x)^5 - 3*cos(x)^5*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*(3*cos(x)^4 - 7*cos(x)^2 + 4)*sqrt(cos(x)^2/sin(x)))/c os(x)^5
\[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=\int \frac {1}{\left (- \sin {\left (x \right )} + \csc {\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (x\right )}-\sin \left (x\right )\right )}^{5/2}} \,d x \]