Integrand size = 11, antiderivative size = 110 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{7/2}} \, dx=-\frac {5 \csc (x)}{192 \sqrt {\sin (x) \tan (x)}}+\frac {5 \csc ^3(x)}{48 \sqrt {\sin (x) \tan (x)}}-\frac {\cot ^2(x) \csc ^3(x)}{6 \sqrt {\sin (x) \tan (x)}}-\frac {5 \arctan \left (\sqrt {\cos (x)}\right ) \sin (x)}{128 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}-\frac {5 \text {arctanh}\left (\sqrt {\cos (x)}\right ) \sin (x)}{128 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \]
-5/192*csc(x)/(sin(x)*tan(x))^(1/2)+5/48*csc(x)^3/(sin(x)*tan(x))^(1/2)-1/ 6*cot(x)^2*csc(x)^3/(sin(x)*tan(x))^(1/2)-5/128*arctan(cos(x)^(1/2))*sin(x )/cos(x)^(1/2)/(sin(x)*tan(x))^(1/2)-5/128*arctanh(cos(x)^(1/2))*sin(x)/co s(x)^(1/2)/(sin(x)*tan(x))^(1/2)
Time = 0.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{7/2}} \, dx=-\frac {\cot (x) \left (15 \arctan \left (\sqrt [4]{\cos ^2(x)}\right )+15 \text {arctanh}\left (\sqrt [4]{\cos ^2(x)}\right )+2 \sqrt [4]{\cos ^2(x)} \csc ^2(x) \left (5-52 \csc ^2(x)+32 \csc ^4(x)\right )\right ) \sqrt {\sin (x) \tan (x)}}{384 \sqrt [4]{\cos ^2(x)}} \]
-1/384*(Cot[x]*(15*ArcTan[(Cos[x]^2)^(1/4)] + 15*ArcTanh[(Cos[x]^2)^(1/4)] + 2*(Cos[x]^2)^(1/4)*Csc[x]^2*(5 - 52*Csc[x]^2 + 32*Csc[x]^4))*Sqrt[Sin[x ]*Tan[x]])/(Cos[x]^2)^(1/4)
Time = 0.63 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.15, 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, 3045, 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.
| \(\displaystyle \int \frac {1}{(\sec (x)-\cos (x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sec (x)-\cos (x))^{7/2}}dx\) |
| \(\Big \downarrow \) 4897 |
| \(\displaystyle \int \frac {1}{(\sin (x) \tan (x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (x) \tan (x))^{7/2}}dx\) |
| \(\Big \downarrow \) 4900 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \int \frac {1}{\sin ^{\frac {7}{2}}(x) \tan ^{\frac {7}{2}}(x)}dx}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \int \frac {1}{\sin (x)^{7/2} \tan (x)^{7/2}}dx}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 3077 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \int \frac {1}{\sin ^{\frac {7}{2}}(x) \tan ^{\frac {3}{2}}(x)}dx-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \int \frac {1}{\sin (x)^{7/2} \tan (x)^{3/2}}dx-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 3077 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (-\frac {1}{8} \int \frac {\sqrt {\tan (x)}}{\sin ^{\frac {7}{2}}(x)}dx-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (-\frac {1}{8} \int \frac {\sqrt {\tan (x)}}{\sin (x)^{7/2}}dx-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 3079 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (\frac {1}{8} \left (\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}-\frac {3}{4} \int \frac {\sqrt {\tan (x)}}{\sin ^{\frac {3}{2}}(x)}dx\right )-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (\frac {1}{8} \left (\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}-\frac {3}{4} \int \frac {\sqrt {\tan (x)}}{\sin (x)^{3/2}}dx\right )-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 3081 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (\frac {1}{8} \left (\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}-\frac {3 \sqrt {\cos (x)} \sqrt {\tan (x)} \int \frac {\csc (x)}{\sqrt {\cos (x)}}dx}{4 \sqrt {\sin (x)}}\right )-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (\frac {1}{8} \left (\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}-\frac {3 \sqrt {\cos (x)} \sqrt {\tan (x)} \int \frac {1}{\sqrt {\cos (x)} \sin (x)}dx}{4 \sqrt {\sin (x)}}\right )-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {\cos (x)} \sqrt {\tan (x)} \int \frac {1}{\sqrt {\cos (x)} \left (1-\cos ^2(x)\right )}d\cos (x)}{4 \sqrt {\sin (x)}}+\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {\cos (x)} \sqrt {\tan (x)} \int \frac {1}{1-\cos ^2(x)}d\sqrt {\cos (x)}}{2 \sqrt {\sin (x)}}+\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {\cos (x)} \sqrt {\tan (x)} \left (\frac {1}{2} \int \frac {1}{1-\cos (x)}d\sqrt {\cos (x)}+\frac {1}{2} \int \frac {1}{\cos (x)+1}d\sqrt {\cos (x)}\right )}{2 \sqrt {\sin (x)}}+\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {\cos (x)} \sqrt {\tan (x)} \left (\frac {1}{2} \int \frac {1}{1-\cos (x)}d\sqrt {\cos (x)}+\frac {1}{2} \arctan \left (\sqrt {\cos (x)}\right )\right )}{2 \sqrt {\sin (x)}}+\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {\sin (x)} \sqrt {\tan (x)} \left (-\frac {5}{12} \left (\frac {1}{8} \left (\frac {3 \sqrt {\cos (x)} \sqrt {\tan (x)} \left (\frac {1}{2} \arctan \left (\sqrt {\cos (x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\cos (x)}\right )\right )}{2 \sqrt {\sin (x)}}+\frac {1}{2 \sin ^{\frac {3}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{4 \sin ^{\frac {7}{2}}(x) \sqrt {\tan (x)}}\right )-\frac {1}{6 \sin ^{\frac {7}{2}}(x) \tan ^{\frac {5}{2}}(x)}\right )}{\sqrt {\sin (x) \tan (x)}}\) |
(Sqrt[Sin[x]]*((-5*((1/(2*Sin[x]^(3/2)*Sqrt[Tan[x]]) + (3*(ArcTan[Sqrt[Cos [x]]]/2 + ArcTanh[Sqrt[Cos[x]]]/2)*Sqrt[Cos[x]]*Sqrt[Tan[x]])/(2*Sqrt[Sin[ x]]))/8 - 1/(4*Sin[x]^(7/2)*Sqrt[Tan[x]])))/12 - 1/(6*Sin[x]^(7/2)*Tan[x]^ (5/2)))*Sqrt[Tan[x]])/Sqrt[Sin[x]*Tan[x]]
3.4.40.3.1 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_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[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. \(216\) vs. \(2(82)=164\).
Time = 1.22 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.97
| method | result | size |
| default | \(-\frac {15 \cot \left (x \right ) \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )-15 \cot \left (x \right ) \ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right )+20 \cot \left (x \right )^{4} \csc \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-15 \csc \left (x \right ) \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )+15 \csc \left (x \right ) \ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right )+168 \cot \left (x \right )^{2} \csc \left (x \right )^{3} \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-60 \csc \left (x \right )^{5} \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}{768 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {\sin \left (x \right ) \tan \left (x \right )}}\) | \(217\) |
-1/768/(-cos(x)/(cos(x)+1)^2)^(1/2)/(sin(x)*tan(x))^(1/2)*(15*cot(x)*arcta n(1/2/(-cos(x)/(cos(x)+1)^2)^(1/2))-15*cot(x)*ln((2*cos(x)*(-cos(x)/(cos(x )+1)^2)^(1/2)+2*(-cos(x)/(cos(x)+1)^2)^(1/2)-cos(x)+1)/(cos(x)+1))+20*cot( x)^4*csc(x)*(-cos(x)/(cos(x)+1)^2)^(1/2)-15*csc(x)*arctan(1/2/(-cos(x)/(co s(x)+1)^2)^(1/2))+15*csc(x)*ln((2*cos(x)*(-cos(x)/(cos(x)+1)^2)^(1/2)+2*(- cos(x)/(cos(x)+1)^2)^(1/2)-cos(x)+1)/(cos(x)+1))+168*cot(x)^2*csc(x)^3*(-c os(x)/(cos(x)+1)^2)^(1/2)-60*csc(x)^5*(-cos(x)/(cos(x)+1)^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (82) = 164\).
Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.55 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{7/2}} \, dx=\frac {15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \arctan \left (\frac {2 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) + 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac {{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 2 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) + 4 \, {\left (5 \, \cos \left (x\right )^{5} + 42 \, \cos \left (x\right )^{3} - 15 \, \cos \left (x\right )\right )} \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}}}{768 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]
1/768*(15*(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*arctan(2*sqrt(-(cos(x)^ 2 - 1)/cos(x))*cos(x)/((cos(x) - 1)*sin(x)))*sin(x) + 15*(cos(x)^6 - 3*cos (x)^4 + 3*cos(x)^2 - 1)*log(((cos(x) + 1)*sin(x) - 2*sqrt(-(cos(x)^2 - 1)/ cos(x))*cos(x))/((cos(x) - 1)*sin(x)))*sin(x) + 4*(5*cos(x)^5 + 42*cos(x)^ 3 - 15*cos(x))*sqrt(-(cos(x)^2 - 1)/cos(x)))/((cos(x)^6 - 3*cos(x)^4 + 3*c os(x)^2 - 1)*sin(x))
Timed out. \[ \int \frac {1}{(-\cos (x)+\sec (x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(-\cos (x)+\sec (x))^{7/2}} \, dx=\int { \frac {1}{{\left (-\cos \left (x\right ) + \sec \left (x\right )\right )}^{\frac {7}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (82) = 164\).
Time = 0.46 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.91 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{7/2}} \, dx=-\frac {{\left (\frac {3 \, {\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2}} - \frac {27 \, {\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}^{2}}{\tan \left (\frac {1}{2} \, x\right )^{4}} + 1\right )} \tan \left (\frac {1}{2} \, x\right )^{6}}{3072 \, {\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}^{3}} + \frac {1}{768} \, \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} {\left ({\left (2 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 3\right )} \tan \left (\frac {1}{2} \, x\right )^{2} - 14\right )} - \frac {9 \, {\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}}{1024 \, \tan \left (\frac {1}{2} \, x\right )^{2}} + \frac {{\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}^{2}}{1024 \, \tan \left (\frac {1}{2} \, x\right )^{4}} + \frac {{\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}^{3}}{3072 \, \tan \left (\frac {1}{2} \, x\right )^{6}} - \frac {5}{256} \, \arcsin \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) - \frac {5}{256} \, \log \left (-\frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1}{\tan \left (\frac {1}{2} \, x\right )^{2}}\right ) \]
-1/3072*(3*(sqrt(-tan(1/2*x)^4 + 1) - 1)/tan(1/2*x)^2 - 27*(sqrt(-tan(1/2* x)^4 + 1) - 1)^2/tan(1/2*x)^4 + 1)*tan(1/2*x)^6/(sqrt(-tan(1/2*x)^4 + 1) - 1)^3 + 1/768*sqrt(-tan(1/2*x)^4 + 1)*((2*tan(1/2*x)^2 - 3)*tan(1/2*x)^2 - 14) - 9/1024*(sqrt(-tan(1/2*x)^4 + 1) - 1)/tan(1/2*x)^2 + 1/1024*(sqrt(-t an(1/2*x)^4 + 1) - 1)^2/tan(1/2*x)^4 + 1/3072*(sqrt(-tan(1/2*x)^4 + 1) - 1 )^3/tan(1/2*x)^6 - 5/256*arcsin(tan(1/2*x)^2) - 5/256*log(-(sqrt(-tan(1/2* x)^4 + 1) - 1)/tan(1/2*x)^2)
Timed out. \[ \int \frac {1}{(-\cos (x)+\sec (x))^{7/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\cos \left (x\right )}-\cos \left (x\right )\right )}^{7/2}} \,d x \]