Integrand size = 45, antiderivative size = 284 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (32 A+44 B+33 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a (5 A+11 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
2/99*a*(5*A+11*B)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(9/2)+2/1 1*A*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/d/cos(d*x+c)^(11/2)+2/3465*a^3*(1160 *A+1364*B+1485*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(1/2)+2/3 465*a^3*(2840*A+3212*B+3795*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)/(a+a*cos(d*x+ c))^(1/2)+4/3465*a^3*(2840*A+3212*B+3795*C)*sin(d*x+c)/d/cos(d*x+c)^(1/2)/ (a+a*cos(d*x+c))^(1/2)+2/231*a^2*(32*A+44*B+33*C)*sin(d*x+c)*(a+a*cos(d*x+ c))^(1/2)/d/cos(d*x+c)^(7/2)
Time = 1.00 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.67 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {a^2 \sqrt {a (1+\cos (c+d x))} (18140 A+15356 B+13365 C+(50140 A+49654 B+49830 C) \cos (c+d x)+4 (4615 A+4642 B+4290 C) \cos (2 (c+d x))+18460 A \cos (3 (c+d x))+20878 B \cos (3 (c+d x))+22935 C \cos (3 (c+d x))+2840 A \cos (4 (c+d x))+3212 B \cos (4 (c+d x))+3795 C \cos (4 (c+d x))+2840 A \cos (5 (c+d x))+3212 B \cos (5 (c+d x))+3795 C \cos (5 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{13860 d \cos ^{\frac {11}{2}}(c+d x)} \]
Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x] ^2))/Cos[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)])*Tan[(c + d*x)/2])/(13860*d*Cos[c + d*x]^(11/2))
Time = 1.72 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^{5/2} (a (5 A+11 B)+a (4 A+11 C) \cos (c+d x))}{2 \cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^{5/2} (a (5 A+11 B)+a (4 A+11 C) \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (5 A+11 B)+a (4 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {2}{9} \int \frac {(\cos (c+d x) a+a)^{3/2} \left (3 (32 A+44 B+33 C) a^2+(56 A+44 B+99 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {(\cos (c+d x) a+a)^{3/2} \left (3 (32 A+44 B+33 C) a^2+(56 A+44 B+99 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (3 (32 A+44 B+33 C) a^2+(56 A+44 B+99 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((1160 A+1364 B+1485 C) a^3+(776 A+836 B+1089 C) \cos (c+d x) a^3\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {1}{7} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((1160 A+1364 B+1485 C) a^3+(776 A+836 B+1089 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {1}{7} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left ((1160 A+1364 B+1485 C) a^3+(776 A+836 B+1089 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} a^3 (2840 A+3212 B+3795 C) \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^4 (1160 A+1364 B+1485 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} a^3 (2840 A+3212 B+3795 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^4 (1160 A+1364 B+1485 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} a^3 (2840 A+3212 B+3795 C) \left (\frac {2}{3} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (1160 A+1364 B+1485 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} a^3 (2840 A+3212 B+3795 C) \left (\frac {2}{3} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^4 (1160 A+1364 B+1485 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3250 |
| \(\displaystyle \frac {\frac {2 a^2 (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (\frac {6 a^3 (32 A+44 B+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2 a^4 (1160 A+1364 B+1485 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {3}{5} a^3 (2840 A+3212 B+3795 C) \left (\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4 a \sin (c+d x)}{3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )\right )\right )}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
Int[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/C os[c + d*x]^(13/2),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)*Si n[c + d*x])/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (3*a^3*(28 40*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)
3.6.1.3.1 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[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])
Time = 13.93 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(\frac {2 a^{2} \sin \left (d x +c \right ) \left (5680 A \left (\cos ^{5}\left (d x +c \right )\right )+6424 B \left (\cos ^{5}\left (d x +c \right )\right )+7590 C \left (\cos ^{5}\left (d x +c \right )\right )+2840 A \left (\cos ^{4}\left (d x +c \right )\right )+3212 B \left (\cos ^{4}\left (d x +c \right )\right )+3795 C \left (\cos ^{4}\left (d x +c \right )\right )+2130 A \left (\cos ^{3}\left (d x +c \right )\right )+2409 B \left (\cos ^{3}\left (d x +c \right )\right )+1980 C \left (\cos ^{3}\left (d x +c \right )\right )+1775 A \left (\cos ^{2}\left (d x +c \right )\right )+1430 B \left (\cos ^{2}\left (d x +c \right )\right )+495 C \left (\cos ^{2}\left (d x +c \right )\right )+1120 A \cos \left (d x +c \right )+385 B \cos \left (d x +c \right )+315 A \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{3465 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {11}{2}}}\) | \(199\) |
| parts | \(\frac {2 A \sin \left (d x +c \right ) \left (1136 \left (\cos ^{5}\left (d x +c \right )\right )+568 \left (\cos ^{4}\left (d x +c \right )\right )+426 \left (\cos ^{3}\left (d x +c \right )\right )+355 \left (\cos ^{2}\left (d x +c \right )\right )+224 \cos \left (d x +c \right )+63\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{693 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {11}{2}}}+\frac {2 B \sin \left (d x +c \right ) \left (584 \left (\cos ^{4}\left (d x +c \right )\right )+292 \left (\cos ^{3}\left (d x +c \right )\right )+219 \left (\cos ^{2}\left (d x +c \right )\right )+130 \cos \left (d x +c \right )+35\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{315 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {9}{2}}}+\frac {2 C \sin \left (d x +c \right ) \left (46 \left (\cos ^{3}\left (d x +c \right )\right )+23 \left (\cos ^{2}\left (d x +c \right )\right )+12 \cos \left (d x +c \right )+3\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{21 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {7}{2}}}\) | \(257\) |
int((a+cos(d*x+c)*a)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(13/ 2),x,method=_RETURNVERBOSE)
2/3465*a^2/d*sin(d*x+c)*(5680*A*cos(d*x+c)^5+6424*B*cos(d*x+c)^5+7590*C*co s(d*x+c)^5+2840*A*cos(d*x+c)^4+3212*B*cos(d*x+c)^4+3795*C*cos(d*x+c)^4+213 0*A*cos(d*x+c)^3+2409*B*cos(d*x+c)^3+1980*C*cos(d*x+c)^3+1775*A*cos(d*x+c) ^2+1430*B*cos(d*x+c)^2+495*C*cos(d*x+c)^2+1120*A*cos(d*x+c)+385*B*cos(d*x+ c)+315*A)*(a*(1+cos(d*x+c)))^(1/2)/(1+cos(d*x+c))/cos(d*x+c)^(11/2)
Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.59 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 \, {\left (2 \, {\left (2840 \, A + 3212 \, B + 3795 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + {\left (2840 \, A + 3212 \, B + 3795 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 3 \, {\left (710 \, A + 803 \, B + 660 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 5 \, {\left (355 \, A + 286 \, B + 99 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 35 \, {\left (32 \, A + 11 \, B\right )} a^{2} \cos \left (d x + c\right ) + 315 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}} \]
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(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)*sqrt(cos(d*x + c))*sin(d* x + c)/(d*cos(d*x + c)^7 + d*cos(d*x + c)^6)
Timed out. \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 867 vs. \(2 (248) = 496\).
Time = 0.41 (sec) , antiderivative size = 867, normalized size of antiderivative = 3.05 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Too large to display} \]
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c )^(13/2),x, algorithm="maxima")
8/3465*(165*(21*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 56*sqrt( 2)*a^(5/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*sqrt(2)*a^(5/2)*sin(d* x + c)^5/(cos(d*x + c) + 1)^5 - 36*sqrt(2)*a^(5/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 8*sqrt(2)*a^(5/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)*C*(s in(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^2/((sin(d*x + c)/(cos(d*x + c) + 1 ) + 1)^(9/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2)*(2*sin(d*x + c)^ 2/(cos(d*x + c) + 1)^2 + sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1)) + 11*(3 15*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 945*sqrt(2)*a^(5/2)*s in(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1449*sqrt(2)*a^(5/2)*sin(d*x + c)^5/( cos(d*x + c) + 1)^5 - 1287*sqrt(2)*a^(5/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 572*sqrt(2)*a^(5/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 104*sqrt( 2)*a^(5/2)*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)*B*(sin(d*x + c)^2/(cos(d *x + c) + 1)^2 + 1)^3/((sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)*(-sin( d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)*(3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + sin(d*x + c)^6/(cos(d*x + c ) + 1)^6 + 1)) + 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/(cos(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)^1...
Timed out. \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]
integrate((a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c )^(13/2),x, algorithm="giac")
Time = 9.70 (sec) , antiderivative size = 809, normalized size of antiderivative = 2.85 \[ \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Too large to display} \]
int(((a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/c os(c + d*x)^(13/2),x)
((a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((a^2*(2840 *A + 3212*B + 3795*C)*4i)/(3465*d) - (a^2*exp(c*3i + d*x*3i)*(2*B + 5*C)*4 i)/(3*d) + (a^2*exp(c*8i + d*x*8i)*(2*B + 5*C)*4i)/(3*d) - (a^2*exp(c*5i + d*x*5i)*(30*A + 41*B + 50*C)*8i)/(15*d) + (a^2*exp(c*6i + d*x*6i)*(30*A + 41*B + 50*C)*8i)/(15*d) + (a^2*exp(c*4i + d*x*4i)*(160*A + 157*B + 165*C) *8i)/(35*d) - (a^2*exp(c*7i + d*x*7i)*(160*A + 157*B + 165*C)*8i)/(35*d) + (a^2*exp(c*2i + d*x*2i)*(710*A + 803*B + 870*C)*8i)/(315*d) - (a^2*exp(c* 9i + d*x*9i)*(710*A + 803*B + 870*C)*8i)/(315*d) - (a^2*exp(c*11i + d*x*11 i)*(2840*A + 3212*B + 3795*C)*4i)/(3465*d)))/((exp(- c*1i - d*x*1i)/2 + ex p(c*1i + d*x*1i)/2)^(1/2) + exp(c*1i + d*x*1i)*(exp(- c*1i - d*x*1i)/2 + e xp(c*1i + d*x*1i)/2)^(1/2) + 5*exp(c*2i + d*x*2i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 5*exp(c*3i + d*x*3i)*(exp(- c*1i - d*x*1i) /2 + exp(c*1i + d*x*1i)/2)^(1/2) + 10*exp(c*4i + d*x*4i)*(exp(- c*1i - d*x *1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 10*exp(c*5i + d*x*5i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 10*exp(c*6i + d*x*6i)*(exp(- c* 1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 10*exp(c*7i + d*x*7i)*(exp( - c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 5*exp(c*8i + d*x*8i)*(e xp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 5*exp(c*9i + d*x*9i) *(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + exp(c*10i + d*x*1 0i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + exp(c*11i +...