Integrand size = 43, antiderivative size = 278 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {(176 A-57 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}+\frac {(339 A-108 B+17 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}+\frac {(339 A-108 B+17 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{42 a^4 d}-\frac {(43 A-15 B+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{42 a^4 d (1+\cos (c+d x))^2}-\frac {(176 A-57 B+8 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 a^4 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(13 A-6 B-C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]
-1/10*(176*A-57*B+8*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ell ipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^4/d+1/42*(339*A-108*B+17*C)*(cos(1/2* d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2 ))/a^4/d-1/42*(43*A-15*B+C)*cos(d*x+c)^(5/2)*sin(d*x+c)/a^4/d/(1+cos(d*x+c ))^2-1/30*(176*A-57*B+8*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/a^4/d/(1+cos(d*x+c) )-1/7*(A-B+C)*cos(d*x+c)^(9/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^4-1/35*(13*A- 6*B-C)*cos(d*x+c)^(7/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3+1/42*(339*A-108* B+17*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/a^4/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 18.86 (sec) , antiderivative size = 1965, normalized size of antiderivative = 7.07 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx =\text {Too large to display} \]
Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]
(-904*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Co t[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[ 1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2 *c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) + (288*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan [Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^ 2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sq rt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - Arc Tan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt [1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (136*C*Cos[c/2 + (d*x)/2]^8*Csc[c /2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[ c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcT an[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]]]])/(2 1*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*( a + a*Sec[c + d*x])^4) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec [c + d*x]^2)*((16*(96*A - 37*B + 8*C + 80*A*Cos[c] - 20*B*Cos[c])*Csc[c])/ (5*d) + (64*A*Cos[d*x]*Sin[c])/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^7...
Time = 1.86 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.05, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 4600, 3042, 3520, 27, 3042, 3456, 3042, 3456, 27, 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.
| \(\displaystyle \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a \sec (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^{3/2} \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )}{(a \sec (c+d x)+a)^4}dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{(a \cos (c+d x)+a)^4}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3520 |
| \(\displaystyle \frac {\int -\frac {\cos ^{\frac {7}{2}}(c+d x) (a (9 A-9 B-5 C)-a (17 A-3 B+3 C) \cos (c+d x))}{2 (\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\cos ^{\frac {7}{2}}(c+d x) (a (9 A-9 B-5 C)-a (17 A-3 B+3 C) \cos (c+d x))}{(\cos (c+d x) a+a)^3}dx}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a (9 A-9 B-5 C)-a (17 A-3 B+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (7 a^2 (13 A-6 B-C)-a^2 (124 A-33 B+12 C) \cos (c+d x)\right )}{(\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (7 a^2 (13 A-6 B-C)-a^2 (124 A-33 B+12 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (25 a^3 (43 A-15 B+C)-3 a^3 (463 A-141 B+29 C) \cos (c+d x)\right )}{2 (\cos (c+d x) a+a)}dx}{3 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (25 a^3 (43 A-15 B+C)-3 a^3 (463 A-141 B+29 C) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (25 a^3 (43 A-15 B+C)-3 a^3 (463 A-141 B+29 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int 3 \sqrt {\cos (c+d x)} \left (7 a^4 (176 A-57 B+8 C)-5 a^4 (339 A-108 B+17 C) \cos (c+d x)\right )dx}{a^2}+\frac {14 a^3 (176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {3 \int \sqrt {\cos (c+d x)} \left (7 a^4 (176 A-57 B+8 C)-5 a^4 (339 A-108 B+17 C) \cos (c+d x)\right )dx}{a^2}+\frac {14 a^3 (176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {3 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (7 a^4 (176 A-57 B+8 C)-5 a^4 (339 A-108 B+17 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {14 a^3 (176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle -\frac {\frac {\frac {\frac {3 \left (7 a^4 (176 A-57 B+8 C) \int \sqrt {\cos (c+d x)}dx-5 a^4 (339 A-108 B+17 C) \int \cos ^{\frac {3}{2}}(c+d x)dx\right )}{a^2}+\frac {14 a^3 (176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {3 \left (7 a^4 (176 A-57 B+8 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-5 a^4 (339 A-108 B+17 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )}{a^2}+\frac {14 a^3 (176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {\frac {\frac {3 \left (7 a^4 (176 A-57 B+8 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-5 a^4 (339 A-108 B+17 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )}{a^2}+\frac {14 a^3 (176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\frac {3 \left (7 a^4 (176 A-57 B+8 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-5 a^4 (339 A-108 B+17 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )}{a^2}+\frac {14 a^3 (176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {\frac {\frac {\frac {3 \left (\frac {14 a^4 (176 A-57 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-5 a^4 (339 A-108 B+17 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )}{a^2}+\frac {14 a^3 (176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -\frac {\frac {\frac {\frac {14 a^3 (176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac {3 \left (\frac {14 a^4 (176 A-57 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-5 a^4 (339 A-108 B+17 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )}{a^2}}{6 a^2}+\frac {5 (43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}+\frac {2 a (13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
-1/7*((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]) ^4) - ((2*a*(13*A - 6*B - C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a* Cos[c + d*x])^3) + ((5*(43*A - 15*B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/ (3*d*(1 + Cos[c + d*x])^2) + ((14*a^3*(176*A - 57*B + 8*C)*Cos[c + d*x]^(3 /2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + (3*((14*a^4*(176*A - 57*B + 8 *C)*EllipticE[(c + d*x)/2, 2])/d - 5*a^4*(339*A - 108*B + 17*C)*((2*Ellipt icF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d))))/ a^2)/(6*a^2))/(5*a^2))/(14*a^2)
3.13.39.3.1 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[((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)]
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x _)])^(m_.)*((A_.) + (B_.)*sec[(e_.) + (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.) *(x_)]^2), x_Symbol] :> Simp[d^(m + 2) Int[(b + a*Cos[e + f*x])^m*(d*Cos[ e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; Fr eeQ[{a, b, d, e, f, A, B, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(679\) vs. \(2(306)=612\).
Time = 4.29 (sec) , antiderivative size = 680, normalized size of antiderivative = 2.45
| method | result | size |
| default | \(-\frac {\sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-15 C -15 A +15 B -5598 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+12768 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-25588 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+1224 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+12234 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+1902 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-706 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-6216 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+10776 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+6780 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 )^{7}+14784 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \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 )-2160 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 )^{7}-4788 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \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 )+340 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 )^{7}+672 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \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 )-1882 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+243 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-201 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1344 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+159 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} C +2240 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-2684 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}\right )}{840 a^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \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 {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(680\) |
int(cos(d*x+c)^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x, method=_RETURNVERBOSE)
-1/840*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-15*C-15*A +15*B-5598*B*cos(1/2*d*x+1/2*c)^6+12768*A*cos(1/2*d*x+1/2*c)^10-25588*A*co s(1/2*d*x+1/2*c)^8+1224*B*cos(1/2*d*x+1/2*c)^4+12234*A*cos(1/2*d*x+1/2*c)^ 6+1902*C*cos(1/2*d*x+1/2*c)^6-706*C*cos(1/2*d*x+1/2*c)^4-6216*B*cos(1/2*d* x+1/2*c)^10+10776*B*cos(1/2*d*x+1/2*c)^8-1882*A*cos(1/2*d*x+1/2*c)^4+243*A *cos(1/2*d*x+1/2*c)^2-201*B*cos(1/2*d*x+1/2*c)^2+1344*C*cos(1/2*d*x+1/2*c) ^10+159*cos(1/2*d*x+1/2*c)^2*C+2240*A*cos(1/2*d*x+1/2*c)^12-2684*C*cos(1/2 *d*x+1/2*c)^8+6780*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)^7+14784 *A*cos(1/2*d*x+1/2*c)^7*(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))-2160*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)^7-4788*B*cos(1/2*d*x+1/2*c)^7*(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))+340*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^7+672*C*c os(1/2*d*x+1/2*c)^7*(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)))/a^4/cos(1/2*d*x+1/2*c)^7/( -2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2* cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.37 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {2 \, {\left (140 \, A \cos \left (d x + c\right )^{4} + 7 \, {\left (368 \, A - 111 \, B + 24 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (6259 \, A - 1968 \, B + 337 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (5548 \, A - 1761 \, B + 284 \, C\right )} \cos \left (d x + c\right ) + 1695 \, A - 540 \, B + 85 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 5 \, {\left (\sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 \, {\left (\sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, {\left (\sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, {\left (\sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
integrate(cos(d*x+c)^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c) )^4,x, algorithm="fricas")
1/420*(2*(140*A*cos(d*x + c)^4 + 7*(368*A - 111*B + 24*C)*cos(d*x + c)^3 + (6259*A - 1968*B + 337*C)*cos(d*x + c)^2 + (5548*A - 1761*B + 284*C)*cos( d*x + c) + 1695*A - 540*B + 85*C)*sqrt(cos(d*x + c))*sin(d*x + c) - 5*(sqr t(2)*(339*I*A - 108*I*B + 17*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(339*I*A - 10 8*I*B + 17*I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(339*I*A - 108*I*B + 17*I*C)*co s(d*x + c)^2 + 4*sqrt(2)*(339*I*A - 108*I*B + 17*I*C)*cos(d*x + c) + sqrt( 2)*(339*I*A - 108*I*B + 17*I*C))*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 5*(sqrt(2)*(-339*I*A + 108*I*B - 17*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(-339*I*A + 108*I*B - 17*I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(-33 9*I*A + 108*I*B - 17*I*C)*cos(d*x + c)^2 + 4*sqrt(2)*(-339*I*A + 108*I*B - 17*I*C)*cos(d*x + c) + sqrt(2)*(-339*I*A + 108*I*B - 17*I*C))*weierstrass PInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*(sqrt(2)*(176*I*A - 57 *I*B + 8*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(176*I*A - 57*I*B + 8*I*C)*cos(d* x + c)^3 + 6*sqrt(2)*(176*I*A - 57*I*B + 8*I*C)*cos(d*x + c)^2 + 4*sqrt(2) *(176*I*A - 57*I*B + 8*I*C)*cos(d*x + c) + sqrt(2)*(176*I*A - 57*I*B + 8*I *C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*si n(d*x + c))) - 21*(sqrt(2)*(-176*I*A + 57*I*B - 8*I*C)*cos(d*x + c)^4 + 4* sqrt(2)*(-176*I*A + 57*I*B - 8*I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(-176*I*A + 57*I*B - 8*I*C)*cos(d*x + c)^2 + 4*sqrt(2)*(-176*I*A + 57*I*B - 8*I*C)*co s(d*x + c) + sqrt(2)*(-176*I*A + 57*I*B - 8*I*C))*weierstrassZeta(-4, 0...
Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c) )^4,x, algorithm="maxima")
\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{4}} \,d x } \]
integrate(cos(d*x+c)^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c) )^4,x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*cos(d*x + c)^(3/2)/(a*se c(d*x + c) + a)^4, x)
Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^4} \,d x \]