Integrand size = 43, antiderivative size = 372 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a \left (28 a^2 b B-42 b^3 B+3 a b^2 (13 A-49 C)+a^3 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a^2 \left (54 a A b+21 a^2 B-105 b^2 B-350 a b C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}-\frac {2 a (7 b B-a (A-21 C)) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (3 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)} \] Output:
2/5*(3*B*a^4+30*B*a^2*b^2-5*B*b^4+20*a*b^3*(A-C)+4*a^3*b*(3*A+5*C))*Ellipt icE(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/21*(28*B*a^3*b+84*B*a*b^3+7*b^4*(3*A+C )+42*a^2*b^2*(A+3*C)+a^4*(5*A+7*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)) /d+2/21*a*(28*B*a^2*b-42*B*b^3+3*a*b^2*(13*A-49*C)+a^3*(5*A+7*C))*cos(d*x+ c)^(1/2)*sin(d*x+c)/d+2/105*a^2*(54*A*a*b+21*B*a^2-105*B*b^2-350*C*a*b)*co s(d*x+c)^(3/2)*sin(d*x+c)/d-2/7*a*(7*B*b-a*(A-21*C))*cos(d*x+c)^(1/2)*(b+a *cos(d*x+c))^2*sin(d*x+c)/d+2/3*(3*B*b+8*C*a)*(b+a*cos(d*x+c))^3*sin(d*x+c )/d/cos(d*x+c)^(1/2)+2/3*C*(b+a*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(3/2 )
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 13.86 (sec) , antiderivative size = 3950, normalized size of antiderivative = 10.62 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(Cos[c + d*x]^(13/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(12*a^3*A*b + 20*a*A*b^3 + 3*a^4*B + 30*a^2*b^2*B - 10*b^4* B + 20*a^3*b*C - 40*a*b^3*C + 12*a^3*A*b*Cos[2*c] + 20*a*A*b^3*Cos[2*c] + 3*a^4*B*Cos[2*c] + 30*a^2*b^2*B*Cos[2*c] + 20*a^3*b*C*Cos[2*c])*Csc[c]*Sec [c])/(5*d) + (a^2*(23*a^2*A + 168*A*b^2 + 112*a*b*B + 28*a^2*C)*Cos[d*x]*S in[c])/(21*d) + (2*a^3*(4*A*b + a*B)*Cos[2*d*x]*Sin[2*c])/(5*d) + (a^4*A*C os[3*d*x]*Sin[3*c])/(7*d) + (a^2*(23*a^2*A + 168*A*b^2 + 112*a*b*B + 28*a^ 2*C)*Cos[c]*Sin[d*x])/(21*d) + (4*b^4*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3 *d) + (4*Sec[c]*Sec[c + d*x]*(b^4*C*Sin[c] + 3*b^4*B*Sin[d*x] + 12*a*b^3*C *Sin[d*x]))/(3*d) + (2*a^3*(4*A*b + a*B)*Cos[2*c]*Sin[2*d*x])/(5*d) + (a^4 *A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos [c + d*x] + A*Cos[2*c + 2*d*x])) - (20*a^4*A*Cos[c + d*x]^6*Csc[c]*Hyperge ometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(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]]]])/(21*d*(b + a*C os[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a^2*A*b^2*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/ 2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[ c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x...
Time = 2.57 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.02, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 4600, 3042, 3526, 27, 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.
| \(\displaystyle \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{7/2} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+b)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {2}{3} \int \frac {(b+a \cos (c+d x))^3 \left (a (3 A-7 C) \cos ^2(c+d x)+(3 A b+C b+3 a B) \cos (c+d x)+3 b B+8 a C\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {(b+a \cos (c+d x))^3 \left (a (3 A-7 C) \cos ^2(c+d x)+(3 A b+C b+3 a B) \cos (c+d x)+3 b B+8 a C\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (a (3 A-7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(3 A b+C b+3 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+3 b B+8 a C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{3} \left (2 \int \frac {(b+a \cos (c+d x))^2 \left (48 C a^2+3 (a A-7 b B-21 a C) \cos ^2(c+d x) a+21 b B a+3 A b^2+b^2 C+\left (3 B a^2+6 A b a-14 b C a-3 b^2 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {(b+a \cos (c+d x))^2 \left (48 C a^2-3 (7 b B-a (A-21 C)) \cos ^2(c+d x) a+21 b B a+3 A b^2+b^2 C+\left (3 B a^2+6 A b a-14 b C a-3 b^2 B\right ) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (48 C a^2-3 (7 b B-a (A-21 C)) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+21 b B a+3 A b^2+b^2 C+\left (3 B a^2+6 A b a-14 b C a-3 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{7} \int \frac {(b+a \cos (c+d x)) \left (a \left (21 B a^2+54 A b a-350 b C a-105 b^2 B\right ) \cos ^2(c+d x)+\left (3 (5 A+7 C) a^3+63 b B a^2+7 b^2 (9 A-13 C) a-21 b^3 B\right ) \cos (c+d x)+b \left (3 (A+91 C) a^2+126 b B a+7 b^2 (3 A+C)\right )\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \int \frac {(b+a \cos (c+d x)) \left (a \left (21 B a^2+54 A b a-350 b C a-105 b^2 B\right ) \cos ^2(c+d x)+\left (3 (5 A+7 C) a^3+63 b B a^2+7 b^2 (9 A-13 C) a-21 b^3 B\right ) \cos (c+d x)+b \left (3 (A+91 C) a^2+126 b B a+7 b^2 (3 A+C)\right )\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a \left (21 B a^2+54 A b a-350 b C a-105 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (3 (5 A+7 C) a^3+63 b B a^2+7 b^2 (9 A-13 C) a-21 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+b \left (3 (A+91 C) a^2+126 b B a+7 b^2 (3 A+C)\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 \left (3 (A+91 C) a^2+126 b B a+7 b^2 (3 A+C)\right ) b^2+15 a \left ((5 A+7 C) a^3+28 b B a^2+3 b^2 (13 A-49 C) a-42 b^3 B\right ) \cos ^2(c+d x)+21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (21 a^2 B+54 a A b-350 a b C-105 b^2 B\right )}{5 d}\right )+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (3 (A+91 C) a^2+126 b B a+7 b^2 (3 A+C)\right ) b^2+15 a \left ((5 A+7 C) a^3+28 b B a^2+3 b^2 (13 A-49 C) a-42 b^3 B\right ) \cos ^2(c+d x)+21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (21 a^2 B+54 a A b-350 a b C-105 b^2 B\right )}{5 d}\right )+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (3 (A+91 C) a^2+126 b B a+7 b^2 (3 A+C)\right ) b^2+15 a \left ((5 A+7 C) a^3+28 b B a^2+3 b^2 (13 A-49 C) a-42 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (21 a^2 B+54 a A b-350 a b C-105 b^2 B\right )}{5 d}\right )+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {3 \left (5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right )+21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^3 (5 A+7 C)+28 a^2 b B+3 a b^2 (13 A-49 C)-42 b^3 B\right )}{d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (21 a^2 B+54 a A b-350 a b C-105 b^2 B\right )}{5 d}\right )+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right )+21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^3 (5 A+7 C)+28 a^2 b B+3 a b^2 (13 A-49 C)-42 b^3 B\right )}{d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (21 a^2 B+54 a A b-350 a b C-105 b^2 B\right )}{5 d}\right )+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right )+21 \left (3 B a^4+4 b (3 A+5 C) a^3+30 b^2 B a^2+20 b^3 (A-C) a-5 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^3 (5 A+7 C)+28 a^2 b B+3 a b^2 (13 A-49 C)-42 b^3 B\right )}{d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (21 a^2 B+54 a A b-350 a b C-105 b^2 B\right )}{5 d}\right )+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+21 \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^3 (5 A+7 C)+28 a^2 b B+3 a b^2 (13 A-49 C)-42 b^3 B\right )}{d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (21 a^2 B+54 a A b-350 a b C-105 b^2 B\right )}{5 d}\right )+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+21 \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^3 (5 A+7 C)+28 a^2 b B+3 a b^2 (13 A-49 C)-42 b^3 B\right )}{d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (21 a^2 B+54 a A b-350 a b C-105 b^2 B\right )}{5 d}\right )+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {1}{5} \left (5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^3 (5 A+7 C)+28 a^2 b B+3 a b^2 (13 A-49 C)-42 b^3 B\right )}{d}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right )}{d}\right )+\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (21 a^2 B+54 a A b-350 a b C-105 b^2 B\right )}{5 d}\right )+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{7} \left (\frac {2 a^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (21 a^2 B+54 a A b-350 a b C-105 b^2 B\right )}{5 d}+\frac {1}{5} \left (\frac {10 a \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^3 (5 A+7 C)+28 a^2 b B+3 a b^2 (13 A-49 C)-42 b^3 B\right )}{d}+\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right )}{d}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right )}{d}\right )\right )+\frac {6 a \sin (c+d x) \sqrt {\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac {2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
Input:
Int[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[ c + d*x]^2),x]
Output:
(2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + ((6*a *(a*A - 7*b*B - 21*a*C)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(3*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*S qrt[Cos[c + d*x]]) + ((2*a^2*(54*a*A*b + 21*a^2*B - 105*b^2*B - 350*a*b*C) *Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + ((42*(3*a^4*B + 30*a^2*b^2*B - 5 *b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2] )/d + (10*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C ) + a^4*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d + (10*a*(28*a^2*b*B - 42 *b^3*B + 3*a*b^2*(13*A - 49*C) + a^3*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d)/5)/7)/3
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
Input:
int(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
int(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.22 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^4 \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^{4} + 28 i \, B a^{3} b + 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} + 84 i \, B a b^{3} + 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} {\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^{4} - 28 i \, B a^{3} b - 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} - 84 i \, B a b^{3} - 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} {\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^{4} - 4 i \, {\left (3 \, A + 5 \, C\right )} a^{3} b - 30 i \, B a^{2} b^{2} - 20 i \, {\left (A - C\right )} a b^{3} + 5 i \, B b^{4}\right )} \cos \left (d x + c\right )^{2} {\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^{4} + 4 i \, {\left (3 \, A + 5 \, C\right )} a^{3} b + 30 i \, B a^{2} b^{2} + 20 i \, {\left (A - C\right )} a b^{3} - 5 i \, B b^{4}\right )} \cos \left (d x + c\right )^{2} {\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^{4} \cos \left (d x + c\right )^{4} + 35 \, C b^{4} + 21 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{4} + 28 \, B a^{3} b + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (4 \, C a b^{3} + B b^{4}\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 )^{2}} \] Input:
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
Output:
-1/105*(5*sqrt(2)*(I*(5*A + 7*C)*a^4 + 28*I*B*a^3*b + 42*I*(A + 3*C)*a^2*b ^2 + 84*I*B*a*b^3 + 7*I*(3*A + C)*b^4)*cos(d*x + c)^2*weierstrassPInverse( -4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-I*(5*A + 7*C)*a^4 - 28 *I*B*a^3*b - 42*I*(A + 3*C)*a^2*b^2 - 84*I*B*a*b^3 - 7*I*(3*A + C)*b^4)*co s(d*x + c)^2*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2 1*sqrt(2)*(-3*I*B*a^4 - 4*I*(3*A + 5*C)*a^3*b - 30*I*B*a^2*b^2 - 20*I*(A - C)*a*b^3 + 5*I*B*b^4)*cos(d*x + c)^2*weierstrassZeta(-4, 0, weierstrassPI nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(3*I*B*a^4 + 4* I*(3*A + 5*C)*a^3*b + 30*I*B*a^2*b^2 + 20*I*(A - C)*a*b^3 - 5*I*B*b^4)*cos (d*x + c)^2*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(15*A*a^4*cos(d*x + c)^4 + 35*C*b^4 + 21*(B*a^4 + 4*A*a^3*b)*cos(d*x + c)^3 + 5*((5*A + 7*C)*a^4 + 28*B*a^3*b + 42*A*a^2*b^2 )*cos(d*x + c)^2 + 105*(4*C*a*b^3 + B*b^4)*cos(d*x + c))*sqrt(cos(d*x + c) )*sin(d*x + c))/(d*cos(d*x + c)^2)
Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(7/2)*(a+b*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+ c)**2),x)
Output:
Timed out
Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
Output:
Timed out
\[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {7}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^4*c os(d*x + c)^(7/2), x)
Time = 14.94 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.39 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^(7/2)*(a + b/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos( c + d*x)^2),x)
Output:
(2*(A*b^4*ellipticF(c/2 + (d*x)/2, 2) + 4*A*a*b^3*ellipticE(c/2 + (d*x)/2, 2) + 2*A*a^2*b^2*ellipticF(c/2 + (d*x)/2, 2) + 2*A*a^2*b^2*cos(c + d*x)^( 1/2)*sin(c + d*x)))/d + (2*(C*a^4*ellipticF(c/2 + (d*x)/2, 2) + 12*C*a^3*b *ellipticE(c/2 + (d*x)/2, 2) + C*a^4*cos(c + d*x)^(1/2)*sin(c + d*x) + 18* C*a^2*b^2*ellipticF(c/2 + (d*x)/2, 2)))/(3*d) + (8*B*a*b^3*ellipticF(c/2 + (d*x)/2, 2))/d + (4*B*a^3*b*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*e llipticF(c/2 + (d*x)/2, 2))/3))/d + (12*B*a^2*b^2*ellipticE(c/2 + (d*x)/2, 2))/d - (2*A*a^4*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^4*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)) + (2*B*b^4*sin(c + d*x)*hypergeom([-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)) + (2*C*b^4*sin (c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(3*d*cos(c + d*x)^( 3/2)*(sin(c + d*x)^2)^(1/2)) - (8*A*a^3*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)) + (8*C*a*b^3*sin(c + d*x)*hypergeom([-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))
\[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{6}d x \right ) b^{4} c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{5}d x \right ) a \,b^{3} c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{5}d x \right ) b^{5}+6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4}d x \right ) a^{2} b^{2} c +5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4}d x \right ) a \,b^{4}+4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}d x \right ) a^{3} b c +10 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}d x \right ) a^{2} b^{3}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{2}d x \right ) a^{4} c +10 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{2}d x \right ) a^{3} b^{2}+5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )d x \right ) a^{4} b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a^{5} \] Input:
int(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
int(sqrt(cos(c + d*x))*cos(c + d*x)**3*sec(c + d*x)**6,x)*b**4*c + 4*int(s qrt(cos(c + d*x))*cos(c + d*x)**3*sec(c + d*x)**5,x)*a*b**3*c + int(sqrt(c os(c + d*x))*cos(c + d*x)**3*sec(c + d*x)**5,x)*b**5 + 6*int(sqrt(cos(c + d*x))*cos(c + d*x)**3*sec(c + d*x)**4,x)*a**2*b**2*c + 5*int(sqrt(cos(c + d*x))*cos(c + d*x)**3*sec(c + d*x)**4,x)*a*b**4 + 4*int(sqrt(cos(c + d*x)) *cos(c + d*x)**3*sec(c + d*x)**3,x)*a**3*b*c + 10*int(sqrt(cos(c + d*x))*c os(c + d*x)**3*sec(c + d*x)**3,x)*a**2*b**3 + int(sqrt(cos(c + d*x))*cos(c + d*x)**3*sec(c + d*x)**2,x)*a**4*c + 10*int(sqrt(cos(c + d*x))*cos(c + d *x)**3*sec(c + d*x)**2,x)*a**3*b**2 + 5*int(sqrt(cos(c + d*x))*cos(c + d*x )**3*sec(c + d*x),x)*a**4*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a* *5