Integrand size = 13, antiderivative size = 143 \[ \int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}+\frac {3 \log \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{8 \sqrt {2}}-\frac {3 \log \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{8 \sqrt {2}}-\frac {1}{2} \sqrt {\cos (x)} \sin ^{\frac {3}{2}}(x) \]
-3/8*arctan(1-2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2))*2^(1/2)+3/8*arctan(1+2^(1 /2)*sin(x)^(1/2)/cos(x)^(1/2))*2^(1/2)+3/16*ln(1-2^(1/2)*sin(x)^(1/2)/cos( x)^(1/2)+tan(x))*2^(1/2)-3/16*ln(1+2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2)+tan(x ))*2^(1/2)-1/2*sin(x)^(3/2)*cos(x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.27 \[ \int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx=\frac {2 \cos ^2(x)^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{4},\frac {11}{4},\sin ^2(x)\right ) \sin ^{\frac {7}{2}}(x)}{7 \cos ^{\frac {3}{2}}(x)} \]
(2*(Cos[x]^2)^(3/4)*Hypergeometric2F1[3/4, 7/4, 11/4, Sin[x]^2]*Sin[x]^(7/ 2))/(7*Cos[x]^(3/2))
Time = 0.39 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {3042, 3048, 3042, 3054, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (x)^{5/2}}{\sqrt {\cos (x)}}dx\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {3}{4} \int \frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}dx-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} \int \frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}dx-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {3}{2} \int \frac {\tan (x)}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \int \frac {\tan (x)+1}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\frac {1}{2} \int \frac {1}{\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (x)-1}d\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (x)-1}d\left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (x)}{\tan ^2(x)+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt {\sin (x)}}{\sqrt {\cos (x)}}}{\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}\right )\right )-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt {\sin (x)}}{\sqrt {\cos (x)}}}{\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}\right )\right )-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt {\sin (x)}}{\sqrt {\cos (x)}}}{\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}{\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1}d\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}\right )\right )-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{2 \sqrt {2}}\right )\right )-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}\) |
(3*((-(ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]/Sqrt[2])/2 + (Log[1 - (Sqrt[2]*Sqr t[Sin[x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2]) - Log[1 + (Sqrt[2]*Sqrt[Sin[ x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2]))/2))/2 - (Sqrt[Cos[x]]*Sin[x]^(3/2 ))/2
3.3.90.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> With[{k = Denominator[m]}, Simp[k*a*(b/f) Subst[Int[x^(k *(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
Leaf count of result is larger than twice the leaf count of optimal. \(387\) vs. \(2(97)=194\).
Time = 0.29 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.71
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (4 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\sin ^{3}\left (x \right )\right )+6 \cos \left (x \right ) \arctan \left (\frac {\sin \left (x \right ) \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {2}+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )-6 \cos \left (x \right ) \arctan \left (\frac {-\sin \left (x \right ) \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {2}+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )-3 \cos \left (x \right ) \ln \left (2 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cot \left (x \right )+2 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \csc \left (x \right )+2 \cot \left (x \right )+2\right )+3 \cos \left (x \right ) \ln \left (-2 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cot \left (x \right )-2 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \csc \left (x \right )+2 \cot \left (x \right )+2\right )-6 \arctan \left (\frac {\sin \left (x \right ) \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {2}+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )+6 \arctan \left (\frac {-\sin \left (x \right ) \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {2}+\cos \left (x \right )-1}{-1+\cos \left (x \right )}\right )+3 \ln \left (2 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cot \left (x \right )+2 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \csc \left (x \right )+2 \cot \left (x \right )+2\right )-3 \ln \left (-2 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cot \left (x \right )-2 \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \csc \left (x \right )+2 \cot \left (x \right )+2\right )\right ) \left (\sqrt {\cos }\left (x \right )\right )}{16 \sqrt {\frac {\sin \left (x \right ) \cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )^{\frac {3}{2}}}\) | \(388\) |
-1/16*2^(1/2)*(4*2^(1/2)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*sin(x)^3+6*cos (x)*arctan((sin(x)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*2^(1/2)+cos(x)-1)/(- 1+cos(x)))-6*cos(x)*arctan((-sin(x)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*2^( 1/2)+cos(x)-1)/(-1+cos(x)))-3*cos(x)*ln(2*2^(1/2)*(sin(x)*cos(x)/(cos(x)+1 )^2)^(1/2)*cot(x)+2*2^(1/2)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*csc(x)+2*co t(x)+2)+3*cos(x)*ln(-2*2^(1/2)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*cot(x)-2 *2^(1/2)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*csc(x)+2*cot(x)+2)-6*arctan((s in(x)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*2^(1/2)+cos(x)-1)/(-1+cos(x)))+6* arctan((-sin(x)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*2^(1/2)+cos(x)-1)/(-1+c os(x)))+3*ln(2*2^(1/2)*(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*cot(x)+2*2^(1/2) *(sin(x)*cos(x)/(cos(x)+1)^2)^(1/2)*csc(x)+2*cot(x)+2)-3*ln(-2*2^(1/2)*(si n(x)*cos(x)/(cos(x)+1)^2)^(1/2)*cot(x)-2*2^(1/2)*(sin(x)*cos(x)/(cos(x)+1) ^2)^(1/2)*csc(x)+2*cot(x)+2))*cos(x)^(1/2)/(sin(x)*cos(x)/(cos(x)+1)^2)^(1 /2)/sin(x)^(3/2)
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.20 \[ \int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx=-\frac {1}{2} \, \sqrt {\cos \left (x\right )} \sin \left (x\right )^{\frac {3}{2}} + \left (\frac {3}{64} i - \frac {3}{64}\right ) \, \sqrt {2} \log \left (2 i \, \cos \left (x\right )^{2} + {\left (\left (i + 1\right ) \, \sqrt {2} \cos \left (x\right ) - \left (i - 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - i\right ) - \left (\frac {3}{64} i - \frac {3}{64}\right ) \, \sqrt {2} \log \left (2 i \, \cos \left (x\right )^{2} + {\left (-\left (i + 1\right ) \, \sqrt {2} \cos \left (x\right ) + \left (i - 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - i\right ) - \left (\frac {3}{64} i + \frac {3}{64}\right ) \, \sqrt {2} \log \left (-2 i \, \cos \left (x\right )^{2} + {\left (-\left (i - 1\right ) \, \sqrt {2} \cos \left (x\right ) + \left (i + 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + i\right ) + \left (\frac {3}{64} i + \frac {3}{64}\right ) \, \sqrt {2} \log \left (-2 i \, \cos \left (x\right )^{2} + {\left (\left (i - 1\right ) \, \sqrt {2} \cos \left (x\right ) - \left (i + 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 2 \, \cos \left (x\right ) \sin \left (x\right ) + i\right ) - \left (\frac {3}{64} i + \frac {3}{64}\right ) \, \sqrt {2} \log \left ({\left (\left (i + 1\right ) \, \sqrt {2} \cos \left (x\right ) - \left (i - 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 1\right ) + \left (\frac {3}{64} i - \frac {3}{64}\right ) \, \sqrt {2} \log \left ({\left (-\left (i - 1\right ) \, \sqrt {2} \cos \left (x\right ) + \left (i + 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 1\right ) - \left (\frac {3}{64} i - \frac {3}{64}\right ) \, \sqrt {2} \log \left ({\left (\left (i - 1\right ) \, \sqrt {2} \cos \left (x\right ) - \left (i + 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 1\right ) + \left (\frac {3}{64} i + \frac {3}{64}\right ) \, \sqrt {2} \log \left ({\left (-\left (i + 1\right ) \, \sqrt {2} \cos \left (x\right ) + \left (i - 1\right ) \, \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 1\right ) \]
-1/2*sqrt(cos(x))*sin(x)^(3/2) + (3/64*I - 3/64)*sqrt(2)*log(2*I*cos(x)^2 + ((I + 1)*sqrt(2)*cos(x) - (I - 1)*sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin( x)) + 2*cos(x)*sin(x) - I) - (3/64*I - 3/64)*sqrt(2)*log(2*I*cos(x)^2 + (- (I + 1)*sqrt(2)*cos(x) + (I - 1)*sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 2*cos(x)*sin(x) - I) - (3/64*I + 3/64)*sqrt(2)*log(-2*I*cos(x)^2 + (-(I - 1)*sqrt(2)*cos(x) + (I + 1)*sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 2*cos(x)*sin(x) + I) + (3/64*I + 3/64)*sqrt(2)*log(-2*I*cos(x)^2 + ((I - 1)*sqrt(2)*cos(x) - (I + 1)*sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 2* cos(x)*sin(x) + I) - (3/64*I + 3/64)*sqrt(2)*log(((I + 1)*sqrt(2)*cos(x) - (I - 1)*sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 1) + (3/64*I - 3/64)* sqrt(2)*log((-(I - 1)*sqrt(2)*cos(x) + (I + 1)*sqrt(2)*sin(x))*sqrt(cos(x) )*sqrt(sin(x)) + 1) - (3/64*I - 3/64)*sqrt(2)*log(((I - 1)*sqrt(2)*cos(x) - (I + 1)*sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 1) + (3/64*I + 3/64) *sqrt(2)*log((-(I + 1)*sqrt(2)*cos(x) + (I - 1)*sqrt(2)*sin(x))*sqrt(cos(x ))*sqrt(sin(x)) + 1)
Timed out. \[ \int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx=\int { \frac {\sin \left (x\right )^{\frac {5}{2}}}{\sqrt {\cos \left (x\right )}} \,d x } \]
\[ \int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx=\int { \frac {\sin \left (x\right )^{\frac {5}{2}}}{\sqrt {\cos \left (x\right )}} \,d x } \]
Time = 0.55 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.17 \[ \int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx=-\frac {2\,\sqrt {\cos \left (x\right )}\,{\sin \left (x\right )}^{7/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{4};\ \frac {5}{4};\ {\cos \left (x\right )}^2\right )}{{\left ({\sin \left (x\right )}^2\right )}^{7/4}} \]