Integrand size = 43, antiderivative size = 277 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (3 a^3 B+15 a b^2 B+5 b^3 (A-C)+3 a^2 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (21 a^2 b B+21 b^3 B+21 a b^2 (A+3 C)+a^3 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a \left (21 a b B+6 b^2 (3 A-7 C)+a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a^2 (11 A b+7 a B-35 b C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 a (A-7 C) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 C (b+a \cos (c+d x))^3 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
2/5*(3*B*a^3+15*B*a*b^2+5*b^3*(A-C)+3*a^2*b*(3*A+5*C))*(cos(1/2*d*x+1/2*c) ^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/21* (21*B*a^2*b+21*B*b^3+21*a*b^2*(A+3*C)+a^3*(5*A+7*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))/d+2/35*a^ 2*(11*A*b+7*B*a-35*C*b)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2*C*(b+a*cos(d*x+c)) ^3*sin(d*x+c)/d/cos(d*x+c)^(1/2)+2/21*a*(21*B*a*b+6*b^2*(3*A-7*C)+a^2*(5*A +7*C))*sin(d*x+c)*cos(d*x+c)^(1/2)/d+2/7*a*(A-7*C)*(b+a*cos(d*x+c))^2*sin( d*x+c)*cos(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 27.86 (sec) , antiderivative size = 3207, normalized size of antiderivative = 11.58 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]
Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
(Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B + 15*a^2*b*C - 10*b^3*C + 9*a^2*A*b*Cos[2*c] + 5*A*b^3*Cos[2*c] + 3*a^3*B*Cos[2*c] + 15*a *b^2*B*Cos[2*c] + 15*a^2*b*C*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (a*(23*a^2*A + 84*A*b^2 + 84*a*b*B + 28*a^2*C)*Cos[d*x]*Sin[c])/(21*d) + (2*a^2*(3*A*b + a*B)*Cos[2*d*x]*Sin[2*c])/(5*d) + (a^3*A*Cos[3*d*x]*Sin[3*c])/(7*d) + ( a*(23*a^2*A + 84*A*b^2 + 84*a*b*B + 28*a^2*C)*Cos[c]*Sin[d*x])/(21*d) + (4 *b^3*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (2*a^2*(3*A*b + a*B)*Cos[2*c]*Sin [2*d*x])/(5*d) + (a^3*A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((b + a*Cos[c + d*x]) ^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (20*a^3*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c] ]]^2]*(a + b*Sec[c + d*x])^3*(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[-(Sqrt[1 + C ot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[ c]]]])/(21*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2* c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a*A*b^2*Cos[c + d*x]^5*Csc[c]*Hyperge ometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(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[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d *x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*C...
Time = 1.81 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.04, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {3042, 4600, 3042, 3526, 27, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 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.
| \(\displaystyle \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{7/2} (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+b)^3 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^3 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle 2 \int \frac {(b+a \cos (c+d x))^2 \left (a (A-7 C) \cos ^2(c+d x)+(A b-C b+a B) \cos (c+d x)+b B+6 a C\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(b+a \cos (c+d x))^2 \left (a (A-7 C) \cos ^2(c+d x)+(A b-C b+a B) \cos (c+d x)+b B+6 a C\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (a (A-7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(A b-C b+a B) \sin \left (c+d x+\frac {\pi }{2}\right )+b B+6 a C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2}{7} \int \frac {(b+a \cos (c+d x)) \left (a (11 A b-35 C b+7 a B) \cos ^2(c+d x)+\left ((5 A+7 C) a^2+14 b B a+7 b^2 (A-C)\right ) \cos (c+d x)+b (7 b B+a (A+35 C))\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {(b+a \cos (c+d x)) \left (a (11 A b-35 C b+7 a B) \cos ^2(c+d x)+\left ((5 A+7 C) a^2+14 b B a+7 b^2 (A-C)\right ) \cos (c+d x)+b (7 b B+a (A+35 C))\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a (11 A b-35 C b+7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left ((5 A+7 C) a^2+14 b B a+7 b^2 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+b (7 b B+a (A+35 C))\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {5 (7 b B+a (A+35 C)) b^2+5 a \left ((5 A+7 C) a^2+21 b B a+6 b^2 (3 A-7 C)\right ) \cos ^2(c+d x)+7 \left (3 B a^3+3 b (3 A+5 C) a^2+15 b^2 B a+5 b^3 (A-C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{5 d}\right )+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {5 (7 b B+a (A+35 C)) b^2+5 a \left ((5 A+7 C) a^2+21 b B a+6 b^2 (3 A-7 C)\right ) \cos ^2(c+d x)+7 \left (3 B a^3+3 b (3 A+5 C) a^2+15 b^2 B a+5 b^3 (A-C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{5 d}\right )+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {5 (7 b B+a (A+35 C)) b^2+5 a \left ((5 A+7 C) a^2+21 b B a+6 b^2 (3 A-7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (3 B a^3+3 b (3 A+5 C) a^2+15 b^2 B a+5 b^3 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{5 d}\right )+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right )+21 \left (3 B a^3+3 b (3 A+5 C) a^2+15 b^2 B a+5 b^3 (A-C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+21 a b B+6 b^2 (3 A-7 C)\right )}{3 d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{5 d}\right )+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right )+21 \left (3 B a^3+3 b (3 A+5 C) a^2+15 b^2 B a+5 b^3 (A-C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+21 a b B+6 b^2 (3 A-7 C)\right )}{3 d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{5 d}\right )+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right )+21 \left (3 B a^3+3 b (3 A+5 C) a^2+15 b^2 B a+5 b^3 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+21 a b B+6 b^2 (3 A-7 C)\right )}{3 d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{5 d}\right )+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+21 \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+21 a b B+6 b^2 (3 A-7 C)\right )}{3 d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{5 d}\right )+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+21 \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+21 a b B+6 b^2 (3 A-7 C)\right )}{3 d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{5 d}\right )+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right )}{d}\right )+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+21 a b B+6 b^2 (3 A-7 C)\right )}{3 d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{5 d}\right )+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{5 d}+\frac {1}{5} \left (\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 (5 A+7 C)+21 a b B+6 b^2 (3 A-7 C)\right )}{3 d}+\frac {1}{3} \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right )}{d}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right )}{d}\right )\right )\right )+\frac {2 a (A-7 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\) |
(2*a*(A - 7*C)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7* d) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (( 2*a^2*(11*A*b + 7*a*B - 35*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + ( ((42*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*Elliptic E[(c + d*x)/2, 2])/d + (10*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a ^3*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d)/3 + (10*a*(21*a*b*B + 6*b^2* (3*A - 7*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d))/5)/ 7
3.14.7.3.1 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[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_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -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/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 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)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
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]
Timed out.
hanged
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.36 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, \sqrt {2} {\left (i \, {\left (5 \, A + 7 \, C\right )} a^{3} + 21 i \, B a^{2} b + 21 i \, {\left (A + 3 \, C\right )} a b^{2} + 21 i \, B b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 7 \, C\right )} a^{3} - 21 i \, B a^{2} b - 21 i \, {\left (A + 3 \, C\right )} a b^{2} - 21 i \, B b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-3 i \, B a^{3} - 3 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b - 15 i \, B a b^{2} - 5 i \, {\left (A - C\right )} b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (3 i \, B a^{3} + 3 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b + 15 i \, B a b^{2} + 5 i \, {\left (A - C\right )} b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (15 \, A a^{3} \cos \left (d x + c\right )^{3} + 105 \, C b^{3} + 21 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{3} + 21 \, B a^{2} b + 21 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )} \]
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
-1/105*(5*sqrt(2)*(I*(5*A + 7*C)*a^3 + 21*I*B*a^2*b + 21*I*(A + 3*C)*a*b^2 + 21*I*B*b^3)*cos(d*x + c)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*si n(d*x + c)) + 5*sqrt(2)*(-I*(5*A + 7*C)*a^3 - 21*I*B*a^2*b - 21*I*(A + 3*C )*a*b^2 - 21*I*B*b^3)*cos(d*x + c)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(-3*I*B*a^3 - 3*I*(3*A + 5*C)*a^2*b - 15*I *B*a*b^2 - 5*I*(A - C)*b^3)*cos(d*x + c)*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(3*I*B*a^3 + 3*I*(3*A + 5*C)*a^2*b + 15*I*B*a*b^2 + 5*I*(A - C)*b^3)*cos(d*x + c)*weie rstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c ))) - 2*(15*A*a^3*cos(d*x + c)^3 + 105*C*b^3 + 21*(B*a^3 + 3*A*a^2*b)*cos( d*x + c)^2 + 5*((5*A + 7*C)*a^3 + 21*B*a^2*b + 21*A*a*b^2)*cos(d*x + c))*s qrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c))
Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
\[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3*c os(d*x + c)^(7/2), x)
Time = 20.86 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.44 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,\left (A\,b^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a\,b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {C\,a^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {2\,B\,b^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,B\,a\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,C\,a^2\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {6\,C\,a\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {3\,B\,a^2\,b\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,A\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,b^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,A\,a^2\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
(2*(A*b^3*ellipticE(c/2 + (d*x)/2, 2) + A*a*b^2*ellipticF(c/2 + (d*x)/2, 2 ) + A*a*b^2*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (C*a^3*((2*cos(c + d*x)^ (1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d + (2*B*b^3*e llipticF(c/2 + (d*x)/2, 2))/d + (6*B*a*b^2*ellipticE(c/2 + (d*x)/2, 2))/d + (6*C*a^2*b*ellipticE(c/2 + (d*x)/2, 2))/d + (6*C*a*b^2*ellipticF(c/2 + ( d*x)/2, 2))/d + (3*B*a^2*b*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ell ipticF(c/2 + (d*x)/2, 2))/3))/d - (2*A*a^3*cos(c + d*x)^(9/2)*sin(c + d*x) *hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (2*B*a^3*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, co s(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) + (2*C*b^3*sin(c + d*x)*hyperg eom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x) ^2)^(1/2)) - (6*A*a^2*b*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/ 4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2))