Integrand size = 11, antiderivative size = 99 \[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=-\frac {3 \arctan \left (\sqrt {-\sin (x)}\right ) \cos (x)}{32 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}+\frac {3 \text {arctanh}\left (\sqrt {-\sin (x)}\right ) \cos (x)}{32 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}-\frac {3 \tan (x)}{16 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^2(x) \tan (x)}{4 \sqrt {\cos (x) \cot (x)}} \] Output:
-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 \[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=-\frac {\sqrt {\cos (x) \cot (x)} \sin (x) \left (-3 \arctan \left (\sqrt [4]{\sin ^2(x)}\right )+3 \text {arctanh}\left (\sqrt [4]{\sin ^2(x)}\right )+(-5+3 \cos (2 x)) \sec ^4(x) \sin ^2(x)^{3/4}\right ) \tan (x)}{32 \sin ^2(x)^{3/4}} \] Input:
Integrate[(Csc[x] - Sin[x])^(-5/2),x]
Output:
-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 = 0.59 (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 |
\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{4 \cos ^{\frac {5}{2}}(x) \cot ^{\frac {3}{2}}(x)}-\frac {3}{8} \left (\frac {1}{2 \sqrt {\cos (x)} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {\cos (x)} \left (\frac {1}{2} \arctan (\sin (x))-\frac {1}{2} \text {arctanh}(\sin (x))\right )}{2 \sqrt {-\sin (x)} \sqrt {\cot (x)}}\right )\right )}{\sqrt {\cos (x) \cot (x)}}\) |
Input:
Int[(Csc[x] - Sin[x])^(-5/2),x]
Output:
(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]]
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])
Time = 11.66 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.14
method | result | size |
default | \(-\frac {-3 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\cos \left (x \right )-1}\right ) \cos \left (x \right )-3 \arctan \left (\frac {\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\cos \left (x \right )-1}\right ) \cos \left (x \right )+\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (6 \sin \left (x \right )+6 \tan \left (x \right )-8 \tan \left (x \right ) \sec \left (x \right )-8 \tan \left (x \right ) \sec \left (x \right )^{2}\right )}{32 \left (\cos \left (x \right )+1\right ) \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {\cos \left (x \right ) \cot \left (x \right )}}\) | \(113\) |
Input:
int(1/(csc(x)-sin(x))^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/32/(cos(x)+1)/(sin(x)/(cos(x)+1)^2)^(1/2)/(cos(x)*cot(x))^(1/2)*(-3*arc tanh((sin(x)/(cos(x)+1)^2)^(1/2)*sin(x)/(cos(x)-1))*cos(x)-3*arctan((sin(x )/(cos(x)+1)^2)^(1/2)*sin(x)/(cos(x)-1))*cos(x)+(sin(x)/(cos(x)+1)^2)^(1/2 )*(6*sin(x)+6*tan(x)-8*tan(x)*sec(x)-8*tan(x)*sec(x)^2))
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (75) = 150\).
Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.69 \[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=\frac {6 \, \arctan \left (-\frac {\sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{2 \, {\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )}}\right ) \cos \left (x\right )^{5} + 3 \, \cos \left (x\right )^{5} \log \left (\frac {\cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - 4 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} - 2 \, \cos \left (x\right ) + 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ) + 8 \, {\left (3 \, \cos \left (x\right )^{4} - 7 \, \cos \left (x\right )^{2} + 4\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}}}{128 \, \cos \left (x\right )^{5}} \] Input:
integrate(1/(csc(x)-sin(x))^(5/2),x, algorithm="fricas")
Output:
1/128*(6*arctan(-1/2*sqrt(cos(x)^2/sin(x))*(cos(x) - sin(x) + 1)/(cos(x) + sin(x) + 1))*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)))/cos(x)^5
\[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=\int \frac {1}{\left (- \sin {\left (x \right )} + \csc {\left (x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(csc(x)-sin(x))**(5/2),x)
Output:
Integral((-sin(x) + csc(x))**(-5/2), x)
\[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(csc(x)-sin(x))^(5/2),x, algorithm="maxima")
Output:
integrate((csc(x) - sin(x))^(-5/2), x)
\[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(csc(x)-sin(x))^(5/2),x, algorithm="giac")
Output:
integrate((csc(x) - sin(x))^(-5/2), x)
Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (x\right )}-\sin \left (x\right )\right )}^{5/2}} \,d x \] Input:
int(1/(1/sin(x) - sin(x))^(5/2),x)
Output:
int(1/(1/sin(x) - sin(x))^(5/2), x)
\[ \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx=\int \frac {\sqrt {\csc \left (x \right )-\sin \left (x \right )}}{\csc \left (x \right )^{3}-3 \csc \left (x \right )^{2} \sin \left (x \right )+3 \csc \left (x \right ) \sin \left (x \right )^{2}-\sin \left (x \right )^{3}}d x \] Input:
int(1/(csc(x)-sin(x))^(5/2),x)
Output:
int(sqrt(csc(x) - sin(x))/(csc(x)**3 - 3*csc(x)**2*sin(x) + 3*csc(x)*sin(x )**2 - sin(x)**3),x)