Integrand size = 43, antiderivative size = 273 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {7 (7 A-17 B+33 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(13 A-33 B+63 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{6 a^3 d}-\frac {(13 A-33 B+63 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{6 a^3 d}+\frac {7 (7 A-17 B+33 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 a^3 d}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(2 A-7 B+12 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(13 A-33 B+63 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )} \] Output:
7/10*(7*A-17*B+33*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^3/d-1/6*(13*A -33*B+63*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/a^3/d-1/6*(13*A-33*B+63 *C)*cos(d*x+c)^(1/2)*sin(d*x+c)/a^3/d+7/30*(7*A-17*B+33*C)*cos(d*x+c)^(3/2 )*sin(d*x+c)/a^3/d-1/5*(A-B+C)*cos(d*x+c)^(9/2)*sin(d*x+c)/d/(a+a*cos(d*x+ c))^3-1/15*(2*A-7*B+12*C)*cos(d*x+c)^(7/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c)) ^2-1/10*(13*A-33*B+63*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/d/(a^3+a^3*cos(d*x+c) )
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.25 (sec) , antiderivative size = 1534, normalized size of antiderivative = 5.62 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx =\text {Too large to display} \] Input:
Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]
Output:
(26*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, S in[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Si n[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan [Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x]) ^3*Sqrt[1 + Cot[c]^2]) - (22*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Hypergeometri cPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - Ar cTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2 ]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/ (d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (42*C*Cos[c/2 + (d*x)/2]^6 *Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2 ]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*S qrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d *x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Co s[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(29*A - 59*B + 99*C + 20*A*Cos[ c] - 60*B*Cos[c] + 132*C*Cos[c])*Csc[c])/(5*d) + (16*(B - 3*C)*Cos[d*x]*Si n[c])/(3*d) + (8*C*Cos[2*d*x]*Sin[2*c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x )/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (4*Sec[ c/2]*Sec[c/2 + (d*x)/2]^3*(14*A*Sin[(d*x)/2] - 19*B*Sin[(d*x)/2] + 24*C*Si n[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(29*A*Sin[(d*x)/2] - 59*B*Sin[(d*x)/2] + 99*C*Sin[(d*x)/2]))/(5*d) + (16*(B - 3*C)*Cos[c]*Si...
Time = 1.46 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.372, Rules used = {3042, 3520, 27, 3042, 3456, 25, 3042, 3456, 27, 3042, 3227, 3042, 3115, 3042, 3119, 3120}
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.
| \(\displaystyle \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 3520 |
| \(\displaystyle \frac {\int \frac {\cos ^{\frac {7}{2}}(c+d x) (a (A+9 B-9 C)+5 a (A-B+3 C) \cos (c+d x))}{2 (\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^{\frac {7}{2}}(c+d x) (a (A+9 B-9 C)+5 a (A-B+3 C) \cos (c+d x))}{(\cos (c+d x) a+a)^2}dx}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a (A+9 B-9 C)+5 a (A-B+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int -\frac {\cos ^{\frac {5}{2}}(c+d x) \left (7 a^2 (2 A-7 B+12 C)-5 a^2 (5 A-10 B+21 C) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (7 a^2 (2 A-7 B+12 C)-5 a^2 (5 A-10 B+21 C) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (7 a^2 (2 A-7 B+12 C)-5 a^2 (5 A-10 B+21 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {5}{2} \cos ^{\frac {3}{2}}(c+d x) \left (3 a^3 (13 A-33 B+63 C)-7 a^3 (7 A-17 B+33 C) \cos (c+d x)\right )dx}{a^2}+\frac {3 a^2 (13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {5 \int \cos ^{\frac {3}{2}}(c+d x) \left (3 a^3 (13 A-33 B+63 C)-7 a^3 (7 A-17 B+33 C) \cos (c+d x)\right )dx}{2 a^2}+\frac {3 a^2 (13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {5 \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (3 a^3 (13 A-33 B+63 C)-7 a^3 (7 A-17 B+33 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{2 a^2}+\frac {3 a^2 (13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {-\frac {\frac {5 \left (3 a^3 (13 A-33 B+63 C) \int \cos ^{\frac {3}{2}}(c+d x)dx-7 a^3 (7 A-17 B+33 C) \int \cos ^{\frac {5}{2}}(c+d x)dx\right )}{2 a^2}+\frac {3 a^2 (13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {5 \left (3 a^3 (13 A-33 B+63 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx-7 a^3 (7 A-17 B+33 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )}{2 a^2}+\frac {3 a^2 (13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {-\frac {\frac {5 \left (3 a^3 (13 A-33 B+63 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )-7 a^3 (7 A-17 B+33 C) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )}{2 a^2}+\frac {3 a^2 (13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {5 \left (3 a^3 (13 A-33 B+63 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )-7 a^3 (7 A-17 B+33 C) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )}{2 a^2}+\frac {3 a^2 (13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {-\frac {\frac {5 \left (3 a^3 (13 A-33 B+63 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )-7 a^3 (7 A-17 B+33 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )}{2 a^2}+\frac {3 a^2 (13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {-\frac {\frac {3 a^2 (13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac {5 \left (3 a^3 (13 A-33 B+63 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )-7 a^3 (7 A-17 B+33 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )}{2 a^2}}{3 a^2}-\frac {2 a (2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
Input:
Int[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Co s[c + d*x])^3,x]
Output:
-1/5*((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]) ^3) + ((-2*a*(2*A - 7*B + 12*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2) - ((3*a^2*(13*A - 33*B + 63*C)*Cos[c + d*x]^(5/2)*Sin[ c + d*x])/(d*(a + a*Cos[c + d*x])) + (5*(3*a^3*(13*A - 33*B + 63*C)*((2*El lipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) ) - 7*a^3*(7*A - 17*B + 33*C)*((6*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*Co s[c + d*x]^(3/2)*Sin[c + d*x])/(5*d))))/(2*a^2))/(3*a^2))/(10*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
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[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{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[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
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[(a*A - b*B + a*C)*Cos[e + f*x]*(a + b* Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1))), x ] + Simp[1/(b*(b*c - a*d)*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(b*c*m + a *d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c *(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c ^2 - d^2, 0] && LtQ[m, -2^(-1)]
Leaf count of result is larger than twice the leaf count of optimal. \(665\) vs. \(2(252)=504\).
Time = 20.96 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.44
| method | result | size |
| default | \(\frac {\sqrt {\left (-1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-192 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-160 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+864 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+348 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+130 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+294 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-468 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-330 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-714 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+228 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+630 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+1386 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-578 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+1058 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-1590 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+264 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-474 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+744 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-37 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+47 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-57 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 A -3 B +3 C \right )}{60 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}\) | \(666\) |
Input:
int(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x, method=_RETURNVERBOSE)
Output:
1/60*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-192*C*cos( 1/2*d*x+1/2*c)^12-160*B*cos(1/2*d*x+1/2*c)^10+864*C*cos(1/2*d*x+1/2*c)^10+ 348*A*cos(1/2*d*x+1/2*c)^8+130*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2* d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/ 2*c)^5+294*A*cos(1/2*d*x+1/2*c)^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2 *d*x+1/2*c)^2+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-468*B*cos(1/2 *d*x+1/2*c)^8-330*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+ 1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^5-714*B* cos(1/2*d*x+1/2*c)^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2 +1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+228*C*cos(1/2*d*x+1/2*c)^8 +630*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*Elli pticF(cos(1/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^5+1386*C*cos(1/2*d*x+ 1/2*c)^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*El lipticE(cos(1/2*d*x+1/2*c),2^(1/2))-578*A*cos(1/2*d*x+1/2*c)^6+1058*B*cos( 1/2*d*x+1/2*c)^6-1590*C*cos(1/2*d*x+1/2*c)^6+264*A*cos(1/2*d*x+1/2*c)^4-47 4*B*cos(1/2*d*x+1/2*c)^4+744*C*cos(1/2*d*x+1/2*c)^4-37*A*cos(1/2*d*x+1/2*c )^2+47*B*cos(1/2*d*x+1/2*c)^2-57*C*cos(1/2*d*x+1/2*c)^2+3*A-3*B+3*C)/a^3/c os(1/2*d*x+1/2*c)^5/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/s in(1/2*d*x+1/2*c)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.02 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c) )^3,x, algorithm="fricas")
Output:
1/60*(2*(12*C*cos(d*x + c)^4 + 4*(5*B - 6*C)*cos(d*x + c)^3 - 3*(29*A - 79 *B + 147*C)*cos(d*x + c)^2 - 2*(73*A - 188*B + 357*C)*cos(d*x + c) - 65*A + 165*B - 315*C)*sqrt(cos(d*x + c))*sin(d*x + c) - 5*(sqrt(2)*(-13*I*A + 3 3*I*B - 63*I*C)*cos(d*x + c)^3 + 3*sqrt(2)*(-13*I*A + 33*I*B - 63*I*C)*cos (d*x + c)^2 + 3*sqrt(2)*(-13*I*A + 33*I*B - 63*I*C)*cos(d*x + c) + sqrt(2) *(-13*I*A + 33*I*B - 63*I*C))*weierstrassPInverse(-4, 0, cos(d*x + c) + I* sin(d*x + c)) - 5*(sqrt(2)*(13*I*A - 33*I*B + 63*I*C)*cos(d*x + c)^3 + 3*s qrt(2)*(13*I*A - 33*I*B + 63*I*C)*cos(d*x + c)^2 + 3*sqrt(2)*(13*I*A - 33* I*B + 63*I*C)*cos(d*x + c) + sqrt(2)*(13*I*A - 33*I*B + 63*I*C))*weierstra ssPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*(sqrt(2)*(-7*I*A + 1 7*I*B - 33*I*C)*cos(d*x + c)^3 + 3*sqrt(2)*(-7*I*A + 17*I*B - 33*I*C)*cos( d*x + c)^2 + 3*sqrt(2)*(-7*I*A + 17*I*B - 33*I*C)*cos(d*x + c) + sqrt(2)*( -7*I*A + 17*I*B - 33*I*C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*(sqrt(2)*(7*I*A - 17*I*B + 33*I*C) *cos(d*x + c)^3 + 3*sqrt(2)*(7*I*A - 17*I*B + 33*I*C)*cos(d*x + c)^2 + 3*s qrt(2)*(7*I*A - 17*I*B + 33*I*C)*cos(d*x + c) + sqrt(2)*(7*I*A - 17*I*B + 33*I*C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d *cos(d*x + c) + a^3*d)
Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+a*cos(d*x+ c))**3,x)
Output:
Timed out
\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c) )^3,x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(7/2)/(a*co s(d*x + c) + a)^3, x)
\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c) )^3,x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(7/2)/(a*co s(d*x + c) + a)^3, x)
Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{7/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:
int((cos(c + d*x)^(7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + a*co s(c + d*x))^3,x)
Output:
int((cos(c + d*x)^(7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + a*co s(c + d*x))^3, x)
\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}}{\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1}d x \right ) a}{a^{3}} \] Input:
int(cos(d*x+c)^(7/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3,x)
Output:
(int((sqrt(cos(c + d*x))*cos(c + d*x)**5)/(cos(c + d*x)**3 + 3*cos(c + d*x )**2 + 3*cos(c + d*x) + 1),x)*c + int((sqrt(cos(c + d*x))*cos(c + d*x)**4) /(cos(c + d*x)**3 + 3*cos(c + d*x)**2 + 3*cos(c + d*x) + 1),x)*b + int((sq rt(cos(c + d*x))*cos(c + d*x)**3)/(cos(c + d*x)**3 + 3*cos(c + d*x)**2 + 3 *cos(c + d*x) + 1),x)*a)/a**3