Integrand size = 21, antiderivative size = 178 \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)} (1+\tan (a+b x))}\right )}{\sqrt {2} b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}} \] Output:
-1/2*arctan(1-2^(1/2)*sin(b*x+a)^(1/2)/cos(b*x+a)^(1/2))*2^(1/2)/b+1/2*arc tan(1+2^(1/2)*sin(b*x+a)^(1/2)/cos(b*x+a)^(1/2))*2^(1/2)/b-1/2*arctanh(2^( 1/2)*sin(b*x+a)^(1/2)/cos(b*x+a)^(1/2)/(1+tan(b*x+a)))*2^(1/2)/b-2/5*cos(b *x+a)^(5/2)/b/sin(b*x+a)^(5/2)+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 = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.32 \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2 \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {5}{4},-\frac {1}{4},\sin ^2(a+b x)\right )}{5 b \cos ^{\frac {3}{2}}(a+b x) \sin ^{\frac {5}{2}}(a+b x)} \] Input:
Integrate[Cos[a + b*x]^(7/2)/Sin[a + b*x]^(7/2),x]
Output:
(-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 = 0.54 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.29, 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 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{2 \sqrt {2}}\right )\right )}{b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
Input:
Int[Cos[a + b*x]^(7/2)/Sin[a + b*x]^(7/2),x]
Output:
(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]])
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. \(459\) vs. \(2(145)=290\).
Time = 1.03 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.58
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (\left (-5 \cos \left (b x +a \right )+5\right ) \sin \left (b x +a \right ) \ln \left (-\frac {2 \sqrt {2}\, \sqrt {\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right ) \cot \left (b x +a \right )+2 \cot \left (b x +a \right )-\csc \left (b x +a \right )-2 \cos \left (b x +a \right )+\sin \left (b x +a \right )+2}{\cos \left (b x +a \right )-1}\right )+\left (5 \cos \left (b x +a \right )-5\right ) \sin \left (b x +a \right ) \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right ) \cot \left (b x +a \right )-2 \cot \left (b x +a \right )+\csc \left (b x +a \right )+2 \cos \left (b x +a \right )-\sin \left (b x +a \right )-2}{\cos \left (b x +a \right )-1}\right )+\left (10 \cos \left (b x +a \right )-10\right ) \sin \left (b x +a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )-\cos \left (b x +a \right )+1}{\cos \left (b x +a \right )-1}\right )+\left (10 \cos \left (b x +a \right )-10\right ) \sin \left (b x +a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{\cos \left (b x +a \right )-1}\right )+\left (24 \cos \left (b x +a \right )^{2}-20\right ) \sqrt {2}\, \sqrt {\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\right ) \sqrt {\cos \left (b x +a \right )}}{20 b \sin \left (b x +a \right )^{\frac {5}{2}} \sqrt {\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}}\) | \(460\) |
Input:
int(cos(b*x+a)^(7/2)/sin(b*x+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
-1/20/b*2^(1/2)*((-5*cos(b*x+a)+5)*sin(b*x+a)*ln(-(2*2^(1/2)*(sin(b*x+a)*c os(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*sin(b*x+a)-cos(b*x+a)*cot(b*x+a)+2*cot(b *x+a)-csc(b*x+a)-2*cos(b*x+a)+sin(b*x+a)+2)/(cos(b*x+a)-1))+(5*cos(b*x+a)- 5)*sin(b*x+a)*ln((2*2^(1/2)*(sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2) *sin(b*x+a)+cos(b*x+a)*cot(b*x+a)-2*cot(b*x+a)+csc(b*x+a)+2*cos(b*x+a)-sin (b*x+a)-2)/(cos(b*x+a)-1))+(10*cos(b*x+a)-10)*sin(b*x+a)*arctan((2^(1/2)*( sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*sin(b*x+a)-cos(b*x+a)+1)/(co s(b*x+a)-1))+(10*cos(b*x+a)-10)*sin(b*x+a)*arctan((2^(1/2)*(sin(b*x+a)*cos (b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*sin(b*x+a)+cos(b*x+a)-1)/(cos(b*x+a)-1))+( 24*cos(b*x+a)^2-20)*2^(1/2)*(sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^2)^(1/2) )*cos(b*x+a)^(1/2)/sin(b*x+a)^(5/2)/(sin(b*x+a)*cos(b*x+a)/(cos(b*x+a)+1)^ 2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (145) = 290\).
Time = 0.19 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.95 \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx =\text {Too large to display} \] Input:
integrate(cos(b*x+a)^(7/2)/sin(b*x+a)^(7/2),x, algorithm="fricas")
Output:
-1/40*(5*(sqrt(2)*cos(b*x + a)^2 - sqrt(2))*arctan(1/2*(2*cos(b*x + a)^3 - 2*cos(b*x + a)^2*sin(b*x + a) + sqrt(2)*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) - 2*cos(b*x + a))/(cos(b*x + a)^3 + cos(b*x + a)^2*sin(b*x + a) - cos (b*x + a)))*sin(b*x + a) + 5*(sqrt(2)*cos(b*x + a)^2 - sqrt(2))*arctan(-1/ 2*(2*cos(b*x + a)^3 - 2*cos(b*x + a)^2*sin(b*x + a) - sqrt(2)*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) - 2*cos(b*x + a))/(cos(b*x + a)^3 + cos(b*x + a) ^2*sin(b*x + a) - cos(b*x + a)))*sin(b*x + a) - 10*(sqrt(2)*cos(b*x + a)^2 - sqrt(2))*arctan(-1/2*(sqrt(2)*cos(b*x + a) - sqrt(2)*sin(b*x + a))/(sqr t(cos(b*x + a))*sqrt(sin(b*x + a))))*sin(b*x + a) + 5*(sqrt(2)*cos(b*x + a )^2 - sqrt(2))*log(2*(sqrt(2)*cos(b*x + a) + sqrt(2)*sin(b*x + a))*sqrt(co s(b*x + a))*sqrt(sin(b*x + a)) + 4*cos(b*x + a)*sin(b*x + a) + 1)*sin(b*x + a) - 5*(sqrt(2)*cos(b*x + a)^2 - sqrt(2))*log(-2*(sqrt(2)*cos(b*x + a) + sqrt(2)*sin(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 4*cos(b*x + a)*sin(b*x + a) + 1)*sin(b*x + a) - 16*(6*cos(b*x + a)^2 - 5)*sqrt(cos(b* x + a))*sqrt(sin(b*x + a)))/((b*cos(b*x + a)^2 - b)*sin(b*x + a))
Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)**(7/2)/sin(b*x+a)**(7/2),x)
Output:
Timed out
\[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{\frac {7}{2}}}{\sin \left (b x + a\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(cos(b*x+a)^(7/2)/sin(b*x+a)^(7/2),x, algorithm="maxima")
Output:
integrate(cos(b*x + a)^(7/2)/sin(b*x + a)^(7/2), x)
Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=\text {Timed out} \] Input:
integrate(cos(b*x+a)^(7/2)/sin(b*x+a)^(7/2),x, algorithm="giac")
Output:
Timed out
Time = 26.79 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.25 \[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2\,{\cos \left (a+b\,x\right )}^{9/2}\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{5/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {9}{4},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (a+b\,x\right )}^2\right )}{9\,b\,{\sin \left (a+b\,x\right )}^{5/2}} \] Input:
int(cos(a + b*x)^(7/2)/sin(a + b*x)^(7/2),x)
Output:
-(2*cos(a + b*x)^(9/2)*(sin(a + b*x)^2)^(5/4)*hypergeom([9/4, 9/4], 13/4, cos(a + b*x)^2))/(9*b*sin(a + b*x)^(5/2))
\[ \int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx=\frac {-2 \sqrt {\sin \left (b x +a \right )}\, \sqrt {\cos \left (b x +a \right )}\, \cos \left (b x +a \right )^{2}+10 \sqrt {\sin \left (b x +a \right )}\, \sqrt {\cos \left (b x +a \right )}\, \sin \left (b x +a \right )^{2}+5 \left (\int \frac {\sqrt {\sin \left (b x +a \right )}\, \sqrt {\cos \left (b x +a \right )}}{\cos \left (b x +a \right )}d x \right ) \sin \left (b x +a \right )^{3} b}{5 \sin \left (b x +a \right )^{3} b} \] Input:
int(cos(b*x+a)^(7/2)/sin(b*x+a)^(7/2),x)
Output:
( - 2*sqrt(sin(a + b*x))*sqrt(cos(a + b*x))*cos(a + b*x)**2 + 10*sqrt(sin( a + b*x))*sqrt(cos(a + b*x))*sin(a + b*x)**2 + 5*int((sqrt(sin(a + b*x))*s qrt(cos(a + b*x)))/cos(a + b*x),x)*sin(a + b*x)**3*b)/(5*sin(a + b*x)**3*b )