Integrand size = 21, antiderivative size = 199 \[ \int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}-\frac {\log \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\tan (a+b x)\right )}{2 \sqrt {2} b}+\frac {\log \left (1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\tan (a+b x)\right )}{2 \sqrt {2} b}-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}} \]
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*arct an(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 = 55, normalized size of antiderivative = 0.28 \[ \int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=-\frac {2 \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-\frac {1}{4},\frac {3}{4},\sin ^2(a+b x)\right )}{b \cos ^{\frac {3}{2}}(a+b x) \sqrt {\sin (a+b x)}} \]
(-2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, -1/4, 3/4, Sin[a + b*x] ^2])/(b*Cos[a + b*x]^(3/2)*Sqrt[Sin[a + b*x]])
Time = 0.44 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {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.
| \(\displaystyle \int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (a+b x)^{3/2}}{\sin (a+b x)^{3/2}}dx\) |
| \(\Big \downarrow \) 3047 |
| \(\displaystyle -\int \frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}dx-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}dx-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 3054 |
| \(\displaystyle -\frac {2 \int \frac {\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{b}-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \int \frac {\tan (a+b x)+1}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\frac {1}{2} \int \frac {1}{\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (a+b x)-1}d\left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (a+b x)}{\tan ^2(a+b x)+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )\right )}{b}-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )\right )}{b}-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}{\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1}d\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )\right )}{b}-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{2 \sqrt {2}}\right )\right )}{b}-\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}\) |
(-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*Sqrt[Cos[a + b*x]])/(b*Sqrt[Sin[a + b*x]])
3.4.1.3.1 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. \(649\) vs. \(2(160)=320\).
Time = 0.34 (sec) , antiderivative size = 650, normalized size of antiderivative = 3.27
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (-\frac {\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1}{\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )+1}\right )^{\frac {3}{2}} \left (\ln \left (-\frac {\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )+2 \sqrt {-\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )-2+2 \cos \left (b x +a \right )-\sin \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right ) \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )-2 \arctan \left (\frac {\sqrt {-\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )+\cos \left (b x +a \right )-1}{1-\cos \left (b x +a \right )}\right ) \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )-\ln \left (\frac {-\left (1-\cos \left (b x +a \right )\right )^{2} \csc \left (b x +a \right )+2 \sqrt {-\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )+2-2 \cos \left (b x +a \right )+\sin \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right ) \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )-2 \arctan \left (\frac {\sqrt {-\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\, \sin \left (b x +a \right )+1-\cos \left (b x +a \right )}{1-\cos \left (b x +a \right )}\right ) \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )+4 \sqrt {-\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}\right ) \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{4 b \left (\frac {\csc \left (b x +a \right )-\cot \left (b x +a \right )}{\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )+1}\right )^{\frac {3}{2}} \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \sqrt {-\left (1-\cos \left (b x +a \right )\right ) \left (\left (1-\cos \left (b x +a \right )\right )^{2} \left (\csc ^{2}\left (b x +a \right )\right )-1\right ) \csc \left (b x +a \right )}}\) | \(650\) |
1/4/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)) )^(3/2)*(-((1-cos(b*x+a))^2*csc(b*x+a)^2-1)/((1-cos(b*x+a))^2*csc(b*x+a)^2 +1))^(3/2)*(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)))*(csc(b*x+a)-cot(b*x+a))-2*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))*(csc(b*x+a)-cot(b*x+a))-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)))*(csc(b*x+a)-cot( b*x+a))-2*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)+1-cos(b*x+a)))*(csc(b*x+a)-cot(b* x+a))+4*(-(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)/(-(1-cos(b*x+a))*((1-cos(b*x+a))^2*c sc(b*x+a)^2-1)*csc(b*x+a))^(1/2)*(csc(b*x+a)-cot(b*x+a))
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 766, normalized size of antiderivative = 3.85 \[ \int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\text {Too large to display} \]
1/8*(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) - 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) - 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) + 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) - 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) + 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) - 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))*sqrt(cos(b*x + a))*sqr t(sin(b*x + a)) + 1)*sin(b*x + a) + 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))*sqrt(cos(b*x...
\[ \int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {\cos ^{\frac {3}{2}}{\left (a + b x \right )}}{\sin ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{\frac {3}{2}}}{\sin \left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {\cos \left (b x + a\right )^{\frac {3}{2}}}{\sin \left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
Time = 1.50 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.22 \[ \int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=-\frac {2\,{\cos \left (a+b\,x\right )}^{5/2}\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{4},\frac {5}{4};\ \frac {9}{4};\ {\cos \left (a+b\,x\right )}^2\right )}{5\,b\,\sqrt {\sin \left (a+b\,x\right )}} \]