Integrand size = 31, antiderivative size = 178 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {3 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}+\frac {171 a^2 \cos (c+d x)}{35 d \sqrt {a+a \sin (c+d x)}}+\frac {69 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}+\frac {4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac {\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d} \]
-3*a^(3/2)*arctanh(cos(d*x+c)*a^(1/2)/(a+a*sin(d*x+c))^(1/2))/d+4/35*cos(d *x+c)*(a+a*sin(d*x+c))^(3/2)/d-cot(d*x+c)*(a+a*sin(d*x+c))^(3/2)/d-2/7*cos (d*x+c)*(a+a*sin(d*x+c))^(5/2)/a/d+171/35*a^2*cos(d*x+c)/d/(a+a*sin(d*x+c) )^(1/2)+69/35*a*cos(d*x+c)*(a+a*sin(d*x+c))^(1/2)/d
Time = 6.50 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.59 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (840 \cos \left (\frac {1}{2} (c+d x)\right )-574 \cos \left (\frac {3}{2} (c+d x)\right )+30 \cos \left (\frac {5}{2} (c+d x)\right )-21 \cos \left (\frac {7}{2} (c+d x)\right )+5 \cos \left (\frac {9}{2} (c+d x)\right )-840 \sin \left (\frac {1}{2} (c+d x)\right )+420 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-420 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-574 \sin \left (\frac {3}{2} (c+d x)\right )-30 \sin \left (\frac {5}{2} (c+d x)\right )-21 \sin \left (\frac {7}{2} (c+d x)\right )-5 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{140 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc \left (\frac {1}{4} (c+d x)\right )-\sec \left (\frac {1}{4} (c+d x)\right )\right ) \left (\csc \left (\frac {1}{4} (c+d x)\right )+\sec \left (\frac {1}{4} (c+d x)\right )\right )} \]
-1/140*(a*Csc[(c + d*x)/2]^4*Sqrt[a*(1 + Sin[c + d*x])]*(840*Cos[(c + d*x) /2] - 574*Cos[(3*(c + d*x))/2] + 30*Cos[(5*(c + d*x))/2] - 21*Cos[(7*(c + d*x))/2] + 5*Cos[(9*(c + d*x))/2] - 840*Sin[(c + d*x)/2] + 420*Log[1 + Cos [(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 420*Log[1 - Cos[(c + d*x) /2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 574*Sin[(3*(c + d*x))/2] - 30*Sin[( 5*(c + d*x))/2] - 21*Sin[(7*(c + d*x))/2] - 5*Sin[(9*(c + d*x))/2]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4] - Sec[(c + d*x)/4])*(Csc[(c + d*x)/ 4] + Sec[(c + d*x)/4]))
Time = 1.96 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.48, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.677, Rules used = {3042, 3360, 3042, 3238, 27, 3042, 3230, 3042, 3126, 3042, 3125, 3523, 27, 3042, 3455, 27, 3042, 3460, 3042, 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.
| \(\displaystyle \int \cos ^2(c+d x) \cot ^2(c+d x) (a \sin (c+d x)+a)^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^{3/2}}{\sin (c+d x)^2}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \sin ^2(c+d x) (\sin (c+d x) a+a)^{3/2}dx+\int \csc ^2(c+d x) (\sin (c+d x) a+a)^{3/2} \left (1-2 \sin ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^2 (\sin (c+d x) a+a)^{3/2}dx+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx\) |
| \(\Big \downarrow \) 3238 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx+\frac {2 \int \frac {1}{2} (5 a-2 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx+\frac {\int (5 a-2 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (5 a-2 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{7 a}+\int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx+\frac {\frac {19}{5} a \int (\sin (c+d x) a+a)^{3/2}dx+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx+\frac {\frac {19}{5} a \int (\sin (c+d x) a+a)^{3/2}dx+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx+\frac {\frac {19}{5} a \left (\frac {4}{3} a \int \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx+\frac {\frac {19}{5} a \left (\frac {4}{3} a \int \sqrt {\sin (c+d x) a+a}dx-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \int \frac {(\sin (c+d x) a+a)^{3/2} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx+\frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int \frac {1}{2} \csc (c+d x) (3 a-7 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{a}+\frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \csc (c+d x) (3 a-7 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}dx}{2 a}+\frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(3 a-7 a \sin (c+d x)) (\sin (c+d x) a+a)^{3/2}}{\sin (c+d x)}dx}{2 a}+\frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{3} \int \frac {1}{2} \csc (c+d x) \sqrt {\sin (c+d x) a+a} \left (9 a^2-19 a^2 \sin (c+d x)\right )dx+\frac {14 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{2 a}+\frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a} \left (9 a^2-19 a^2 \sin (c+d x)\right )dx+\frac {14 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{2 a}+\frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\sqrt {\sin (c+d x) a+a} \left (9 a^2-19 a^2 \sin (c+d x)\right )}{\sin (c+d x)}dx+\frac {14 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{2 a}+\frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {\frac {1}{3} \left (9 a^2 \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx+\frac {38 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )+\frac {14 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{2 a}+\frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (9 a^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\frac {38 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )+\frac {14 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{2 a}+\frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {38 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {18 a^3 \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}\right )+\frac {14 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{2 a}+\frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {19}{5} a \left (-\frac {8 a^2 \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}\right )+\frac {4 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}}{7 a}+\frac {\frac {14 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}+\frac {1}{3} \left (\frac {38 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {18 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}\right )}{2 a}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}\) |
-((Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/d) - (2*Cos[c + d*x]*(a + a*Si n[c + d*x])^(5/2))/(7*a*d) + ((14*a^2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x] ])/(3*d) + ((-18*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d + (38*a^3*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]))/3)/(2*a) + ((4*a*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d) + (19*a*((-8*a^2*Co s[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]*Sqrt[a + a* Sin[c + d*x]])/(3*d)))/5)/(7*a)
3.5.56.3.1 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[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 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_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[n, -1]
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 = 0.12 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.01
| method | result | size |
| default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (\sin \left (d x +c \right ) \left (-10 \left (a -a \sin \left (d x +c \right )\right )^{\frac {7}{2}} \sqrt {a}+56 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}-70 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}-140 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {7}{2}}+105 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right ) a^{4}\right )+35 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {7}{2}}\right )}{35 a^{\frac {5}{2}} \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(180\) |
-1/35*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(sin(d*x+c)*(-10*(a-a*sin(d *x+c))^(7/2)*a^(1/2)+56*(a-a*sin(d*x+c))^(5/2)*a^(3/2)-70*(a-a*sin(d*x+c)) ^(3/2)*a^(5/2)-140*(a-a*sin(d*x+c))^(1/2)*a^(7/2)+105*arctanh((a-a*sin(d*x +c))^(1/2)/a^(1/2))*a^4)+35*(a-a*sin(d*x+c))^(1/2)*a^(7/2))/a^(5/2)/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. 360 vs. \(2 (154) = 308\).
Time = 0.29 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.02 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {105 \, {\left (a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) - a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (10 \, a \cos \left (d x + c\right )^{5} - 16 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{3} - 120 \, a \cos \left (d x + c\right )^{2} + 33 \, a \cos \left (d x + c\right ) - {\left (10 \, a \cos \left (d x + c\right )^{4} + 26 \, a \cos \left (d x + c\right )^{3} + 18 \, a \cos \left (d x + c\right )^{2} + 138 \, a \cos \left (d x + c\right ) + 171 \, a\right )} \sin \left (d x + c\right ) + 171 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{140 \, {\left (d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right ) - d\right )}} \]
1/140*(105*(a*cos(d*x + c)^2 - (a*cos(d*x + c) + a)*sin(d*x + c) - a)*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*(10*a*cos(d*x + c)^5 - 16*a*cos(d*x + c )^4 - 8*a*cos(d*x + c)^3 - 120*a*cos(d*x + c)^2 + 33*a*cos(d*x + c) - (10* a*cos(d*x + c)^4 + 26*a*cos(d*x + c)^3 + 18*a*cos(d*x + c)^2 + 138*a*cos(d *x + c) + 171*a)*sin(d*x + c) + 171*a)*sqrt(a*sin(d*x + c) + a))/(d*cos(d* x + c)^2 - (d*cos(d*x + c) + d)*sin(d*x + c) - d)
Timed out. \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2} \,d x } \]
Time = 0.43 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.37 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (320 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 896 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 560 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 560 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {140 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}\right )} \sqrt {a}}{140 \, d} \]
-1/140*sqrt(2)*(320*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/ 2*d*x + 1/2*c)^7 - 896*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 560*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*p i + 1/2*d*x + 1/2*c)^3 + 105*sqrt(2)*a*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c)))*sgn (cos(-1/4*pi + 1/2*d*x + 1/2*c)) + 560*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c ))*sin(-1/4*pi + 1/2*d*x + 1/2*c) + 140*a*sgn(cos(-1/4*pi + 1/2*d*x + 1/2* c))*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)/d
Timed out. \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^2} \,d x \]