Integrand size = 31, antiderivative size = 119 \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}+\frac {4 \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 a d} \]
arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/d/a^(1/2)+4/3*cos(d*x+c )/d/(a+a*sin(d*x+c))^(1/2)-cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)-2/3*cos(d*x +c)*(a+a*sin(d*x+c))^(1/2)/a/d
Time = 0.93 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.60 \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\csc \left (\frac {1}{4} (c+d x)\right ) \sec \left (\frac {1}{4} (c+d x)\right ) \left (-10 \cos \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {3}{2} (c+d x)\right )+\cos \left (\frac {5}{2} (c+d x)\right )+10 \sin \left (\frac {1}{2} (c+d x)\right )+3 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-3 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+3 \sin \left (\frac {3}{2} (c+d x)\right )-\sin \left (\frac {5}{2} (c+d x)\right )\right ) \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{24 d \sqrt {a (1+\sin (c+d x))}} \]
(Csc[(c + d*x)/4]*Sec[(c + d*x)/4]*(-10*Cos[(c + d*x)/2] + 3*Cos[(3*(c + d *x))/2] + Cos[(5*(c + d*x))/2] + 10*Sin[(c + d*x)/2] + 3*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 3*Log[1 - Cos[(c + d*x)/2] + Si n[(c + d*x)/2]]*Sin[c + d*x] + 3*Sin[(3*(c + d*x))/2] - Sin[(5*(c + d*x))/ 2])*(1 + Tan[(c + d*x)/2]))/(24*d*Sqrt[a*(1 + Sin[c + d*x])])
Time = 1.56 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.92, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.613, Rules used = {3042, 3360, 3042, 3238, 27, 3042, 3230, 3042, 3128, 219, 3523, 27, 3042, 3464, 3042, 3128, 219, 3252, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a \sin (c+d x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^2 \sqrt {a \sin (c+d x)+a}}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \frac {\sin ^2(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx+\int \frac {\csc ^2(c+d x) \left (1-2 \sin ^2(c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2}{\sqrt {\sin (c+d x) a+a}}dx+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx\) |
| \(\Big \downarrow \) 3238 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx+\frac {2 \int \frac {a-2 a \sin (c+d x)}{2 \sqrt {\sin (c+d x) a+a}}dx}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx+\frac {\int \frac {a-2 a \sin (c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a-2 a \sin (c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{3 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx+\frac {3 a \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx+\frac {3 a \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {6 a \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{3 a}+\int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {1-2 \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {\sin (c+d x) a+a}}dx+\frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int -\frac {\csc (c+d x) (3 \sin (c+d x) a+a)}{2 \sqrt {\sin (c+d x) a+a}}dx}{a}+\frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\csc (c+d x) (3 \sin (c+d x) a+a)}{\sqrt {\sin (c+d x) a+a}}dx}{2 a}+\frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {3 \sin (c+d x) a+a}{\sin (c+d x) \sqrt {\sin (c+d x) a+a}}dx}{2 a}+\frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3464 |
| \(\displaystyle -\frac {2 a \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx}{2 a}+\frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 a \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx}{2 a}+\frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle -\frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {4 a \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{2 a}+\frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}+\frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle -\frac {-\frac {2 a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}+\frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{2 a}+\frac {\frac {4 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {3 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 a d}-\frac {\cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
-1/2*((-2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]] )/d - (2*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/d)/a - Cot[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]]) - (2*C os[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a*d) + ((-3*Sqrt[2]*Sqrt[a]*ArcTa nh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/d + (4*a*Co s[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]))/(3*a)
3.5.67.3.1 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[(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[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2 ))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*(b*(m + 1) - a*Si n[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && ! LtQ[m, -2^(-1)]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/d^4 Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IGtQ[m, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A *b - a*B)/(b*c - a*d) Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Simp[(B*c - A*d)/(b*c - a*d) Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (2 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right ) \sqrt {a}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right ) a^{2} \sin \left (d x +c \right )-3 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {3}{2}}\right )}{3 a^{\frac {5}{2}} \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(130\) |
1/3*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)/a^(5/2)*(2*(a-a*sin(d*x+c))^( 3/2)*sin(d*x+c)*a^(1/2)+3*arctanh((a-a*sin(d*x+c))^(1/2)/a^(1/2))*a^2*sin( d*x+c)-3*(a-a*sin(d*x+c))^(1/2)*a^(3/2))/sin(d*x+c)/cos(d*x+c)/(a+a*sin(d* x+c))^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (103) = 206\).
Time = 0.28 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.57 \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {3 \, {\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} - {\left (2 \, \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 7\right )} \sin \left (d x + c\right ) - 5 \, \cos \left (d x + c\right ) - 7\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{12 \, {\left (a d \cos \left (d x + c\right )^{2} - a d - {\left (a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )\right )}} \]
1/12*(3*(cos(d*x + c)^2 - (cos(d*x + c) + 1)*sin(d*x + c) - 1)*sqrt(a)*log ((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 + (cos(d*x + c ) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) - 4*(2*cos(d*x + c)^3 + 4*cos(d*x + c)^2 - (2*cos( d*x + c)^2 - 2*cos(d*x + c) - 7)*sin(d*x + c) - 5*cos(d*x + c) - 7)*sqrt(a *sin(d*x + c) + a))/(a*d*cos(d*x + c)^2 - a*d - (a*d*cos(d*x + c) + a*d)*s in(d*x + c))
\[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]
\[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2}}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.50 \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {\frac {8 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {3 \, \log \left ({\left | \frac {1}{2} \, \sqrt {2} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3 \, \log \left ({\left | -\frac {1}{2} \, \sqrt {2} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {6 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} \sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{6 \, d} \]
-1/6*(8*sqrt(2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3/(sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) + 3*log(abs(1/2*sqrt(2) + sin(-1/4*pi + 1/2*d*x + 1/2 *c)))/(sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 3*log(abs(-1/2*sqrt( 2) + sin(-1/4*pi + 1/2*d*x + 1/2*c)))/(sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) + 6*sqrt(2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)/((2*sin(-1/4*pi + 1/2 *d*x + 1/2*c)^2 - 1)*sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))))/d
Timed out. \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{{\sin \left (c+d\,x\right )}^2\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]