Integrand size = 21, antiderivative size = 226 \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}+\frac {\log \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\tan (a+b x)\right )}{2 \sqrt {2} b}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\tan (a+b x)\right )}{2 \sqrt {2} b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}} \]
-2/5*cos(b*x+a)^(5/2)/b/sin(b*x+a)^(5/2)-1/2*arctan(1-2^(1/2)*sin(b*x+a)^( 1/2)/cos(b*x+a)^(1/2))/b*2^(1/2)+1/2*arctan(1+2^(1/2)*sin(b*x+a)^(1/2)/cos (b*x+a)^(1/2))/b*2^(1/2)+1/4*ln(1-2^(1/2)*sin(b*x+a)^(1/2)/cos(b*x+a)^(1/2 )+tan(b*x+a))/b*2^(1/2)-1/4*ln(1+2^(1/2)*sin(b*x+a)^(1/2)/cos(b*x+a)^(1/2) +tan(b*x+a))/b*2^(1/2)+2*cos(b*x+a)^(1/2)/b/sin(b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.25 \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2 \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {5}{4},-\frac {1}{4},\sin ^2(a+b x)\right )}{5 b \cos ^{\frac {3}{2}}(a+b x) \sin ^{\frac {5}{2}}(a+b x)} \]
(-2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-5/4, -5/4, -1/4, Sin[a + b*x ]^2])/(5*b*Cos[a + b*x]^(3/2)*Sin[a + b*x]^(5/2))
Time = 0.55 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3047, 3042, 3047, 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.
| \(\displaystyle \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (a+b x)^{7/2}}{\sin (a+b x)^{7/2}}dx\) |
| \(\Big \downarrow \) 3047 |
| \(\displaystyle -\int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sin ^{\frac {3}{2}}(a+b x)}dx-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {\cos (a+b x)^{3/2}}{\sin (a+b x)^{3/2}}dx-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}\) |
| \(\Big \downarrow \) 3047 |
| \(\displaystyle \int \frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}dx-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}dx-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle \frac {2 \int \frac {\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {\tan (a+b x)+1}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\frac {1}{2} \int \frac {1}{\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )\right )}{b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )\right )}{b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}{\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )\right )}{b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{2 \sqrt {2}}\right )\right )}{b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
(2*((-(ArcTan[1 - (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]]]/Sqrt[2] ) + ArcTan[1 + (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]]]/Sqrt[2])/2 + (Log[1 - (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]] + Tan[a + b*x] ]/(2*Sqrt[2]) - Log[1 + (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2]))/2))/b - (2*Cos[a + b*x]^(5/2))/(5*b*Sin[a + b*x ]^(5/2)) + (2*Sqrt[Cos[a + b*x]])/(b*Sqrt[Sin[a + b*x]])
3.4.3.3.1 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_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[a*(a*Cos[e + f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/ (b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Cos[e + f*x] )^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ [m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
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. \(798\) vs. \(2(181)=362\).
Time = 0.37 (sec) , antiderivative size = 799, normalized size of antiderivative = 3.54
1/20/b*2^(1/2)/(1/((1-cos(b*x+a))^2*csc(b*x+a)^2+1)*(csc(b*x+a)-cot(b*x+a) ))^(7/2)*(1-cos(b*x+a))*(-((1-cos(b*x+a))^2*csc(b*x+a)^2-1)/((1-cos(b*x+a) )^2*csc(b*x+a)^2+1))^(7/2)*((-(1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^ 2-1)*csc(b*x+a))^(1/2)*(1-cos(b*x+a))^4*csc(b*x+a)^4+5*ln(1/(1-cos(b*x+a)) *(-(1-cos(b*x+a))^2*csc(b*x+a)+2*(-(1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b* x+a)^2-1)*csc(b*x+a))^(1/2)*sin(b*x+a)+2-2*cos(b*x+a)+sin(b*x+a)))*(1-cos( b*x+a))^3*csc(b*x+a)^3+10*arctan(1/(1-cos(b*x+a))*((-(1-cos(b*x+a))*((1-co s(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)*sin(b*x+a)+1-cos(b*x+a)))*(1 -cos(b*x+a))^3*csc(b*x+a)^3-5*ln(-1/(1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b *x+a)+2*(-(1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/ 2)*sin(b*x+a)-2+2*cos(b*x+a)-sin(b*x+a)))*(1-cos(b*x+a))^3*csc(b*x+a)^3+10 *arctan(1/(1-cos(b*x+a))*((-(1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2- 1)*csc(b*x+a))^(1/2)*sin(b*x+a)+cos(b*x+a)-1))*(1-cos(b*x+a))^3*csc(b*x+a) ^3-22*(-(1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2) *(1-cos(b*x+a))^2*csc(b*x+a)^2+(-(1-cos(b*x+a))*((1-cos(b*x+a))^2*csc(b*x+ a)^2-1)*csc(b*x+a))^(1/2))/((1-cos(b*x+a))^2*csc(b*x+a)^2-1)^3/(-(1-cos(b* x+a))*((1-cos(b*x+a))^2*csc(b*x+a)^2-1)*csc(b*x+a))^(1/2)*csc(b*x+a)
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 901, normalized size of antiderivative = 3.99 \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=\text {Too large to display} \]
1/40*(5*(b*cos(b*x + a)^2 - b)*(-1/b^4)^(1/4)*log(1/2*(b^3*(-1/b^4)^(3/4)* cos(b*x + a) - b*(-1/b^4)^(1/4)*sin(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin( b*x + a)) + 1/2*cos(b*x + a)*sin(b*x + a) - 1/4*(2*b^2*cos(b*x + a)^2 - b^ 2)*sqrt(-1/b^4))*sin(b*x + a) - 5*(b*cos(b*x + a)^2 - b)*(-1/b^4)^(1/4)*lo g(-1/2*(b^3*(-1/b^4)^(3/4)*cos(b*x + a) - b*(-1/b^4)^(1/4)*sin(b*x + a))*s qrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 1/2*cos(b*x + a)*sin(b*x + a) - 1/4 *(2*b^2*cos(b*x + a)^2 - b^2)*sqrt(-1/b^4))*sin(b*x + a) - 5*(I*b*cos(b*x + a)^2 - I*b)*(-1/b^4)^(1/4)*log(1/2*(I*b^3*(-1/b^4)^(3/4)*cos(b*x + a) + I*b*(-1/b^4)^(1/4)*sin(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 1 /2*cos(b*x + a)*sin(b*x + a) + 1/4*(2*b^2*cos(b*x + a)^2 - b^2)*sqrt(-1/b^ 4))*sin(b*x + a) - 5*(-I*b*cos(b*x + a)^2 + I*b)*(-1/b^4)^(1/4)*log(1/2*(- I*b^3*(-1/b^4)^(3/4)*cos(b*x + a) - I*b*(-1/b^4)^(1/4)*sin(b*x + a))*sqrt( cos(b*x + a))*sqrt(sin(b*x + a)) + 1/2*cos(b*x + a)*sin(b*x + a) + 1/4*(2* b^2*cos(b*x + a)^2 - b^2)*sqrt(-1/b^4))*sin(b*x + a) + 5*(b*cos(b*x + a)^2 - b)*(-1/b^4)^(1/4)*log(2*(b^3*(-1/b^4)^(3/4)*sin(b*x + a) - b*(-1/b^4)^( 1/4)*cos(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 1)*sin(b*x + a) - 5*(b*cos(b*x + a)^2 - b)*(-1/b^4)^(1/4)*log(-2*(b^3*(-1/b^4)^(3/4)*sin( b*x + a) - b*(-1/b^4)^(1/4)*cos(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 1)*sin(b*x + a) - 5*(-I*b*cos(b*x + a)^2 + I*b)*(-1/b^4)^(1/4)*log (-2*(I*b^3*(-1/b^4)^(3/4)*sin(b*x + a) + I*b*(-1/b^4)^(1/4)*cos(b*x + a...
Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{\frac {7}{2}}}{\sin \left (b x + a\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{\frac {7}{2}}}{\sin \left (b x + a\right )^{\frac {7}{2}}} \,d x } \]
Time = 1.79 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.19 \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2\,{\cos \left (a+b\,x\right )}^{9/2}\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{5/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {9}{4},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (a+b\,x\right )}^2\right )}{9\,b\,{\sin \left (a+b\,x\right )}^{5/2}} \]