Integrand size = 45, antiderivative size = 284 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {4 a^3 (2840 A+3212 B+3795 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (2840 A+3212 B+3795 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (1160 A+1364 B+1485 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (32 A+44 B+33 C) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {2 a (5 A+11 B) (a+a \cos (c+d x))^{3/2} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \]
2/99*a*(5*A+11*B)*(a+a*cos(d*x+c))^(3/2)*sec(d*x+c)^(9/2)*sin(d*x+c)/d+2/1 1*A*(a+a*cos(d*x+c))^(5/2)*sec(d*x+c)^(11/2)*sin(d*x+c)/d+2/3465*a^3*(2840 *A+3212*B+3795*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/3 465*a^3*(1160*A+1364*B+1485*C)*sec(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*cos(d*x+ c))^(1/2)+2/231*a^2*(32*A+44*B+33*C)*sec(d*x+c)^(7/2)*sin(d*x+c)*(a+a*cos( d*x+c))^(1/2)/d+4/3465*a^3*(2840*A+3212*B+3795*C)*sin(d*x+c)*sec(d*x+c)^(1 /2)/d/(a+a*cos(d*x+c))^(1/2)
Time = 0.96 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.67 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} (18140 A+15356 B+13365 C+(50140 A+49654 B+49830 C) \cos (c+d x)+4 (4615 A+4642 B+4290 C) \cos (2 (c+d x))+18460 A \cos (3 (c+d x))+20878 B \cos (3 (c+d x))+22935 C \cos (3 (c+d x))+2840 A \cos (4 (c+d x))+3212 B \cos (4 (c+d x))+3795 C \cos (4 (c+d x))+2840 A \cos (5 (c+d x))+3212 B \cos (5 (c+d x))+3795 C \cos (5 (c+d x))) \sec ^{\frac {11}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{13860 d} \]
Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2)*Sec[c + d*x]^(13/2),x]
(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(18140*A + 15356*B + 13365*C + (50140*A + 49654*B + 49830*C)*Cos[c + d*x] + 4*(4615*A + 4642*B + 4290*C)*Cos[2*(c + d*x)] + 18460*A*Cos[3*(c + d*x)] + 20878*B*Cos[3*(c + d*x)] + 22935*C*Cos[ 3*(c + d*x)] + 2840*A*Cos[4*(c + d*x)] + 3212*B*Cos[4*(c + d*x)] + 3795*C* Cos[4*(c + d*x)] + 2840*A*Cos[5*(c + d*x)] + 3212*B*Cos[5*(c + d*x)] + 379 5*C*Cos[5*(c + d*x)])*Sec[c + d*x]^(11/2)*Tan[(c + d*x)/2])/(13860*d)
Time = 1.97 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.12, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {3042, 4709, 3042, 3522, 27, 3042, 3454, 27, 3042, 3454, 27, 3042, 3459, 3042, 3251, 3042, 3250}
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.
\(\displaystyle \int \sec ^{\frac {13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec (c+d x)^{13/2} (a \cos (c+d x)+a)^{5/2} \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )dx\) |
\(\Big \downarrow \) 4709 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(\cos (c+d x) a+a)^{5/2} \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )}{\cos ^{\frac {13}{2}}(c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
\(\Big \downarrow \) 3522 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \int \frac {(\cos (c+d x) a+a)^{5/2} (a (5 A+11 B)+a (4 A+11 C) \cos (c+d x))}{2 \cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {(\cos (c+d x) a+a)^{5/2} (a (5 A+11 B)+a (4 A+11 C) \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (5 A+11 B)+a (4 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3454 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {2}{9} \int \frac {(\cos (c+d x) a+a)^{3/2} \left (3 (32 A+44 B+33 C) a^2+(56 A+44 B+99 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{9} \int \frac {(\cos (c+d x) a+a)^{3/2} \left (3 (32 A+44 B+33 C) a^2+(56 A+44 B+99 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{9} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (3 (32 A+44 B+33 C) a^2+(56 A+44 B+99 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3454 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((1160 A+1364 B+1485 C) a^3+(776 A+836 B+1089 C) \cos (c+d x) a^3\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{9} \left (\frac {1}{7} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((1160 A+1364 B+1485 C) a^3+(776 A+836 B+1089 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{9} \left (\frac {1}{7} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left ((1160 A+1364 B+1485 C) a^3+(776 A+836 B+1089 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3459 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} a^3 (2840 A+3212 B+3795 C) \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^4 (1160 A+1364 B+1485 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} a^3 (2840 A+3212 B+3795 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^4 (1160 A+1364 B+1485 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3251 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} a^3 (2840 A+3212 B+3795 C) \left (\frac {2}{3} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (1160 A+1364 B+1485 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} a^3 (2840 A+3212 B+3795 C) \left (\frac {2}{3} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (1160 A+1364 B+1485 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
\(\Big \downarrow \) 3250 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2 a^4 (1160 A+1364 B+1485 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {3}{5} a^3 (2840 A+3212 B+3795 C) \left (\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4 a \sin (c+d x)}{3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )\right )\right )}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\right )\) |
Int[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec [c + d*x]^(13/2),x]
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*(a + a*Cos[c + d*x])^(5/2)*Sin [c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + ((2*a^2*(5*A + 11*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((6*a^3*(32*A + 44* B + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*a^4*(1160*A + 1364*B + 1485*C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2) *Sqrt[a + a*Cos[c + d*x]]) + (3*a^3*(2840*A + 3212*B + 3795*C)*((2*a*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])))/5)/7)/9)/(11*a))
3.14.28.3.1 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[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(3/2), x_Symbol] :> Simp[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sq rt[a + b*Sin[e + f*x]]*Sqrt[c + d*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[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), 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] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
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^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], 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] && 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 [(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) *(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d)) Int[Sqrt[a + b*Sin[e + f*x] ]*(c + d*Sin[e + f*x])^(n + 1), 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] && LtQ[n, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + 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 - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(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, B, 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])
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Time = 0.77 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.65
\[-\frac {2 a^{2} \left (\cos \left (d x +c \right )-1\right ) \left (\left (5680 \left (\cos ^{5}\left (d x +c \right )\right )+2840 \left (\cos ^{4}\left (d x +c \right )\right )+2130 \left (\cos ^{3}\left (d x +c \right )\right )+1775 \left (\cos ^{2}\left (d x +c \right )\right )+1120 \cos \left (d x +c \right )+315\right ) A +\cos \left (d x +c \right ) \left (6424 \left (\cos ^{4}\left (d x +c \right )\right )+3212 \left (\cos ^{3}\left (d x +c \right )\right )+2409 \left (\cos ^{2}\left (d x +c \right )\right )+1430 \cos \left (d x +c \right )+385\right ) B +\left (\cos ^{2}\left (d x +c \right )\right ) \left (7590 \left (\cos ^{3}\left (d x +c \right )\right )+3795 \left (\cos ^{2}\left (d x +c \right )\right )+1980 \cos \left (d x +c \right )+495\right ) C \right ) \sqrt {\left (1+\cos \left (d x +c \right )\right ) a}\, \left (\sec ^{\frac {13}{2}}\left (d x +c \right )\right ) \cot \left (d x +c \right )}{3465 d}\]
-2/3465*a^2/d*(cos(d*x+c)-1)*((5680*cos(d*x+c)^5+2840*cos(d*x+c)^4+2130*co s(d*x+c)^3+1775*cos(d*x+c)^2+1120*cos(d*x+c)+315)*A+cos(d*x+c)*(6424*cos(d *x+c)^4+3212*cos(d*x+c)^3+2409*cos(d*x+c)^2+1430*cos(d*x+c)+385)*B+cos(d*x +c)^2*(7590*cos(d*x+c)^3+3795*cos(d*x+c)^2+1980*cos(d*x+c)+495)*C)*((1+cos (d*x+c))*a)^(1/2)*sec(d*x+c)^(13/2)*cot(d*x+c)
Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.59 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {2 \, {\left (2 \, {\left (2840 \, A + 3212 \, B + 3795 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + {\left (2840 \, A + 3212 \, B + 3795 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 3 \, {\left (710 \, A + 803 \, B + 660 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 5 \, {\left (355 \, A + 286 \, B + 99 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 35 \, {\left (32 \, A + 11 \, B\right )} a^{2} \cos \left (d x + c\right ) + 315 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )} \sqrt {\cos \left (d x + c\right )}} \]
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(13/2),x, algorithm="fricas")
2/3465*(2*(2840*A + 3212*B + 3795*C)*a^2*cos(d*x + c)^5 + (2840*A + 3212*B + 3795*C)*a^2*cos(d*x + c)^4 + 3*(710*A + 803*B + 660*C)*a^2*cos(d*x + c) ^3 + 5*(355*A + 286*B + 99*C)*a^2*cos(d*x + c)^2 + 35*(32*A + 11*B)*a^2*co s(d*x + c) + 315*A*a^2)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/((d*cos(d*x + c)^6 + d*cos(d*x + c)^5)*sqrt(cos(d*x + c)))
Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (248) = 496\).
Time = 0.46 (sec) , antiderivative size = 1005, normalized size of antiderivative = 3.54 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Too large to display} \]
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(13/2),x, algorithm="maxima")
8/3465*(5*(693*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 2310*sqrt (2)*a^(5/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 4620*sqrt(2)*a^(5/2)*sin (d*x + c)^5/(cos(d*x + c) + 1)^5 - 5478*sqrt(2)*a^(5/2)*sin(d*x + c)^7/(co s(d*x + c) + 1)^7 + 3575*sqrt(2)*a^(5/2)*sin(d*x + c)^9/(cos(d*x + c) + 1) ^9 - 1300*sqrt(2)*a^(5/2)*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 200*sqrt (2)*a^(5/2)*sin(d*x + c)^13/(cos(d*x + c) + 1)^13)*A*(sin(d*x + c)^2/(cos( d*x + c) + 1)^2 + 1)^4/((sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(-sin (d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 1)) + 11*(315*sqrt(2)* a^(5/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 1260*sqrt(2)*a^(5/2)*sin(d*x + c )^3/(cos(d*x + c) + 1)^3 + 2394*sqrt(2)*a^(5/2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 2736*sqrt(2)*a^(5/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 185 9*sqrt(2)*a^(5/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 676*sqrt(2)*a^(5/2 )*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 104*sqrt(2)*a^(5/2)*sin(d*x + c) ^13/(cos(d*x + c) + 1)^13)*B*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^4/( (sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*sin(d*x + c)^4 /(cos(d*x + c) + 1)^4 + 4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 1)) + 165*(21*sqrt(2)*a^(5/2)*sin(d*x + c)/...
Timed out. \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c )^(13/2),x, algorithm="giac")
Time = 8.08 (sec) , antiderivative size = 787, normalized size of antiderivative = 2.77 \[ \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Too large to display} \]
int((1/cos(c + d*x))^(13/2)*(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
((1/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((a^2*(a + a*(e xp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(2840*A + 3212*B + 37 95*C)*4i)/(3465*d) - (a^2*exp(c*5i + d*x*5i)*(a + a*(exp(- c*1i - d*x*1i)/ 2 + exp(c*1i + d*x*1i)/2))^(1/2)*(30*A + 41*B + 50*C)*8i)/(15*d) + (a^2*ex p(c*6i + d*x*6i)*(a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^( 1/2)*(30*A + 41*B + 50*C)*8i)/(15*d) + (a^2*exp(c*4i + d*x*4i)*(a + a*(exp (- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(160*A + 157*B + 165*C) *8i)/(35*d) - (a^2*exp(c*7i + d*x*7i)*(a + a*(exp(- c*1i - d*x*1i)/2 + exp (c*1i + d*x*1i)/2))^(1/2)*(160*A + 157*B + 165*C)*8i)/(35*d) + (a^2*exp(c* 2i + d*x*2i)*(a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2) *(710*A + 803*B + 870*C)*8i)/(315*d) - (a^2*exp(c*9i + d*x*9i)*(a + a*(exp (- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(710*A + 803*B + 870*C) *8i)/(315*d) - (a^2*exp(c*11i + d*x*11i)*(a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(2840*A + 3212*B + 3795*C)*4i)/(3465*d) - (a^ 2*exp(c*3i + d*x*3i)*(a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2 ))^(1/2)*(2*B + 5*C)*4i)/(3*d) + (a^2*exp(c*8i + d*x*8i)*(a + a*(exp(- c*1 i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(2*B + 5*C)*4i)/(3*d)))/(exp( c*1i + d*x*1i) + 5*exp(c*2i + d*x*2i) + 5*exp(c*3i + d*x*3i) + 10*exp(c*4i + d*x*4i) + 10*exp(c*5i + d*x*5i) + 10*exp(c*6i + d*x*6i) + 10*exp(c*7i + d*x*7i) + 5*exp(c*8i + d*x*8i) + 5*exp(c*9i + d*x*9i) + exp(c*10i + d*...