Integrand size = 43, antiderivative size = 384 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (5 a^4 B-30 a^2 b^2 B-3 b^4 B+20 a^3 b (A-C)-4 a b^3 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (84 a^3 b B+28 a b^3 B+42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b \left (413 a^2 b B+63 b^3 B+192 a^3 C+2 a b^2 (175 A+101 C)\right ) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}}-\frac {2 a^2 \left (98 a b B-a^2 (35 A-87 C)+5 b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 \left (35 A b^2+77 a b B+48 a^2 C+25 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (7 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \] Output:
2/5*(5*B*a^4-30*B*a^2*b^2-3*B*b^4+20*a^3*b*(A-C)-4*a*b^3*(5*A+3*C))*Ellipt icE(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/21*(84*B*a^3*b+28*B*a*b^3+42*a^2*b^2*( 3*A+C)+7*a^4*(A+3*C)+b^4*(7*A+5*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)) /d+2/105*b*(413*B*a^2*b+63*B*b^3+192*a^3*C+2*a*b^2*(175*A+101*C))*sin(d*x+ c)/d/cos(d*x+c)^(1/2)-2/105*a^2*(98*B*a*b-a^2*(35*A-87*C)+5*b^2*(7*A+5*C)) *cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/105*(35*A*b^2+77*B*a*b+48*C*a^2+25*C*b^2) *(b+a*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/35*(7*B*b+8*C*a)*(b+a* cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/7*C*(b+a*cos(d*x+c))^4*sin(d *x+c)/d/cos(d*x+c)^(7/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 14.08 (sec) , antiderivative size = 3965, normalized size of antiderivative = 10.33 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Cos[c + d*x]^(3/2)*(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*(20*a^3*A*b - 40*a*A*b^3 + 5*a^4*B - 60*a^2*b^2*B - 6*b^4*B - 40*a^3*b*C - 24*a*b^3*C + 20*a^3*A*b*Cos[2*c] + 5*a^4*B*Cos[2*c])*Csc[c ]*Sec[c])/(5*d) + (4*a^4*A*Cos[d*x]*Sin[c])/(3*d) + (4*a^4*A*Cos[c]*Sin[d* x])/(3*d) + (4*b^4*C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(7*d) + (4*Sec[c]*Sec [c + d*x]^3*(5*b^4*C*Sin[c] + 7*b^4*B*Sin[d*x] + 28*a*b^3*C*Sin[d*x]))/(35 *d) + (4*Sec[c]*Sec[c + d*x]*(35*A*b^4*Sin[c] + 140*a*b^3*B*Sin[c] + 210*a ^2*b^2*C*Sin[c] + 25*b^4*C*Sin[c] + 420*a*A*b^3*Sin[d*x] + 630*a^2*b^2*B*S in[d*x] + 63*b^4*B*Sin[d*x] + 420*a^3*b*C*Sin[d*x] + 252*a*b^3*C*Sin[d*x]) )/(105*d) + (4*Sec[c]*Sec[c + d*x]^2*(21*b^4*B*Sin[c] + 84*a*b^3*C*Sin[c] + 35*A*b^4*Sin[d*x] + 140*a*b^3*B*Sin[d*x] + 210*a^2*b^2*C*Sin[d*x] + 25*b ^4*C*Sin[d*x]))/(105*d)))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d *x] + A*Cos[2*c + 2*d*x])) - (4*a^4*A*Cos[c + d*x]^6*Csc[c]*Hypergeometric PFQ[{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 - Ar cTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[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 ]) - (24*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 + ...
Time = 2.70 (sec) , antiderivative size = 391, 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, 3526, 27, 3042, 3510, 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.
| \(\displaystyle \int \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{3/2} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+b)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {2}{7} \int \frac {(b+a \cos (c+d x))^3 \left (a (7 A-3 C) \cos ^2(c+d x)+(7 A b+5 C b+7 a B) \cos (c+d x)+7 b B+8 a C\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {(b+a \cos (c+d x))^3 \left (a (7 A-3 C) \cos ^2(c+d x)+(7 A b+5 C b+7 a B) \cos (c+d x)+7 b B+8 a C\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (a (7 A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(7 A b+5 C b+7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+7 b B+8 a C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {(b+a \cos (c+d x))^2 \left (48 C a^2+(35 a A-21 b B-39 a C) \cos ^2(c+d x) a+77 b B a+35 A b^2+25 b^2 C+\left (35 B a^2+70 A b a+34 b C a+21 b^2 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {(b+a \cos (c+d x))^2 \left (48 C a^2+(35 a A-21 b B-39 a C) \cos ^2(c+d x) a+77 b B a+35 A b^2+25 b^2 C+\left (35 B a^2+70 A b a+34 b C a+21 b^2 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (48 C a^2+(35 a A-21 b B-39 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+77 b B a+35 A b^2+25 b^2 C+\left (35 B a^2+70 A b a+34 b C a+21 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {(b+a \cos (c+d x)) \left (192 C a^3+413 b B a^2-3 \left (-\left ((35 A-87 C) a^2\right )+98 b B a+5 b^2 (7 A+5 C)\right ) \cos ^2(c+d x) a+2 b^2 (175 A+101 C) a+63 b^3 B+\left (105 B a^3+(315 A b+33 C b) a^2+77 b^2 B a+5 b^3 (7 A+5 C)\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {(b+a \cos (c+d x)) \left (192 C a^3+413 b B a^2-3 \left (-\left ((35 A-87 C) a^2\right )+98 b B a+5 b^2 (7 A+5 C)\right ) \cos ^2(c+d x) a+2 b^2 (175 A+101 C) a+63 b^3 B+\left (105 B a^3+(315 A b+33 C b) a^2+77 b^2 B a+5 b^3 (7 A+5 C)\right ) \cos (c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (192 C a^3+413 b B a^2-3 \left (-\left ((35 A-87 C) a^2\right )+98 b B a+5 b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+2 b^2 (175 A+101 C) a+63 b^3 B+\left (105 B a^3+(315 A b+33 C b) a^2+77 b^2 B a+5 b^3 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 b \sin (c+d x) \left (192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{d \sqrt {\cos (c+d x)}}-2 \int -\frac {192 C a^4+518 b B a^3-3 \left (-\left ((35 A-87 C) a^2\right )+98 b B a+5 b^2 (7 A+5 C)\right ) \cos ^2(c+d x) a^2+5 b^2 (133 A+47 C) a^2+140 b^3 B a+5 b^4 (7 A+5 C)+21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {192 C a^4+518 b B a^3-3 \left (-\left ((35 A-87 C) a^2\right )+98 b B a+5 b^2 (7 A+5 C)\right ) \cos ^2(c+d x) a^2+5 b^2 (133 A+47 C) a^2+140 b^3 B a+5 b^4 (7 A+5 C)+21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) \left (192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {192 C a^4+518 b B a^3-3 \left (-\left ((35 A-87 C) a^2\right )+98 b B a+5 b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^2+5 b^2 (133 A+47 C) a^2+140 b^3 B a+5 b^4 (7 A+5 C)+21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) \left (192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2}{3} \int \frac {3 \left (5 \left (7 (A+3 C) a^4+84 b B a^3+42 b^2 (3 A+C) a^2+28 b^3 B a+b^4 (7 A+5 C)\right )+21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 a^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^2 (35 A-87 C)\right )+98 a b B+5 b^2 (7 A+5 C)\right )}{d}+\frac {2 b \sin (c+d x) \left (192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left (7 (A+3 C) a^4+84 b B a^3+42 b^2 (3 A+C) a^2+28 b^3 B a+b^4 (7 A+5 C)\right )+21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {2 a^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^2 (35 A-87 C)\right )+98 a b B+5 b^2 (7 A+5 C)\right )}{d}+\frac {2 b \sin (c+d x) \left (192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left (7 (A+3 C) a^4+84 b B a^3+42 b^2 (3 A+C) a^2+28 b^3 B a+b^4 (7 A+5 C)\right )+21 \left (5 B a^4+20 b (A-C) a^3-30 b^2 B a^2-4 b^3 (5 A+3 C) a-3 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^2 (35 A-87 C)\right )+98 a b B+5 b^2 (7 A+5 C)\right )}{d}+\frac {2 b \sin (c+d x) \left (192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+21 \left (5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx-\frac {2 a^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^2 (35 A-87 C)\right )+98 a b B+5 b^2 (7 A+5 C)\right )}{d}+\frac {2 b \sin (c+d x) \left (192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+21 \left (5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 a^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^2 (35 A-87 C)\right )+98 a b B+5 b^2 (7 A+5 C)\right )}{d}+\frac {2 b \sin (c+d x) \left (192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^2 (35 A-87 C)\right )+98 a b B+5 b^2 (7 A+5 C)\right )}{d}+\frac {2 b \sin (c+d x) \left (192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{d \sqrt {\cos (c+d x)}}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right ) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{3} \left (-\frac {2 a^2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^2 (35 A-87 C)\right )+98 a b B+5 b^2 (7 A+5 C)\right )}{d}+\frac {2 b \sin (c+d x) \left (192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right )}{d \sqrt {\cos (c+d x)}}+\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right )}{d}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right )}{d}\right )\right )+\frac {2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
Input:
Int[Cos[c + d*x]^(3/2)*(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])/(7*d*Cos[c + d*x]^(7/2)) + ((2*( 7*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2 )) + ((2*(35*A*b^2 + 77*a*b*B + 48*a^2*C + 25*b^2*C)*(b + a*Cos[c + d*x])^ 2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + ((42*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2 ])/d + (10*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3* C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/d + (2*b*(413*a^2*b*B + 6 3*b^3*B + 192*a^3*C + 2*a*b^2*(175*A + 101*C))*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*a^2*(98*a*b*B - a^2*(35*A - 87*C) + 5*b^2*(7*A + 5*C))*Sqrt [Cos[c + d*x]]*Sin[c + d*x])/d)/3)/5)/7
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[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, 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[(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. \(1593\) vs. \(2(363)=726\).
Time = 10.31 (sec) , antiderivative size = 1594, normalized size of antiderivative = 4.15
Input:
int(cos(d*x+c)^(3/2)*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
Output:
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*a^4*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)/(-2*sin(1/2*d*x+1 /2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+ 2*a^4*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)/(-2 *sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/ 2*c),2^(1/2))-2*a^3*(2*A*a-4*A*b-B*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)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^ (1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c), 2^(1/2)))+2*C*b^4*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1 /2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c )*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c) ^2-1/2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1 /2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2 *d*x+1/2*c),2^(1/2)))+2*b^2*(A*b^2+4*B*a*b+6*C*a^2)*(-1/6*cos(1/2*d*x+1/2* c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c )^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1 /2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2 *d*x+1/2*c),2^(1/2)))+2/5*b^3*(B*b+4*C*a)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1 /2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(24*sin(1/2 *d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*El...
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.24 \[ \int \cos ^{\frac {3}{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 (7 i \, {\left (A + 3 \, C\right )} a^{4} + 84 i \, B a^{3} b + 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} + 28 i \, B a b^{3} + i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, {\left (A + 3 \, C\right )} a^{4} - 84 i \, B a^{3} b - 42 i \, {\left (3 \, A + C\right )} a^{2} b^{2} - 28 i \, B a b^{3} - i \, {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-5 i \, B a^{4} - 20 i \, {\left (A - C\right )} a^{3} b + 30 i \, B a^{2} b^{2} + 4 i \, {\left (5 \, A + 3 \, C\right )} a b^{3} + 3 i \, B b^{4}\right )} \cos \left (d x + c\right )^{4} {\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 (5 i \, B a^{4} + 20 i \, {\left (A - C\right )} a^{3} b - 30 i \, B a^{2} b^{2} - 4 i \, {\left (5 \, A + 3 \, C\right )} a b^{3} - 3 i \, B b^{4}\right )} \cos \left (d x + c\right )^{4} {\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 (35 \, A a^{4} \cos \left (d x + c\right )^{4} + 15 \, C b^{4} + 21 \, {\left (20 \, C a^{3} b + 30 \, B a^{2} b^{2} + 4 \, {\left (5 \, A + 3 \, C\right )} a b^{3} + 3 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (42 \, C a^{2} b^{2} + 28 \, B a b^{3} + {\left (7 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\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 )^{4}} \] Input:
integrate(cos(d*x+c)^(3/2)*(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)*(7*I*(A + 3*C)*a^4 + 84*I*B*a^3*b + 42*I*(3*A + C)*a^2*b ^2 + 28*I*B*a*b^3 + I*(7*A + 5*C)*b^4)*cos(d*x + c)^4*weierstrassPInverse( -4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-7*I*(A + 3*C)*a^4 - 84 *I*B*a^3*b - 42*I*(3*A + C)*a^2*b^2 - 28*I*B*a*b^3 - I*(7*A + 5*C)*b^4)*co s(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2 1*sqrt(2)*(-5*I*B*a^4 - 20*I*(A - C)*a^3*b + 30*I*B*a^2*b^2 + 4*I*(5*A + 3 *C)*a*b^3 + 3*I*B*b^4)*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPI nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(5*I*B*a^4 + 20 *I*(A - C)*a^3*b - 30*I*B*a^2*b^2 - 4*I*(5*A + 3*C)*a*b^3 - 3*I*B*b^4)*cos (d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(35*A*a^4*cos(d*x + c)^4 + 15*C*b^4 + 21*(20*C*a^3 *b + 30*B*a^2*b^2 + 4*(5*A + 3*C)*a*b^3 + 3*B*b^4)*cos(d*x + c)^3 + 5*(42* C*a^2*b^2 + 28*B*a*b^3 + (7*A + 5*C)*b^4)*cos(d*x + c)^2 + 21*(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)^4)
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(3/2)*(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. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^(3/2)*(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
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(3/2)*(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)^(3/2), x)
Time = 19.99 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.46 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^(3/2)*(a + b/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos( c + d*x)^2),x)
Output:
(2*(A*a^4*ellipticF(c/2 + (d*x)/2, 2) + 12*A*a^3*b*ellipticE(c/2 + (d*x)/2 , 2) + A*a^4*cos(c + d*x)^(1/2)*sin(c + d*x) + 18*A*a^2*b^2*ellipticF(c/2 + (d*x)/2, 2)))/(3*d) + (2*B*a^4*ellipticE(c/2 + (d*x)/2, 2))/d + (2*C*a^4 *ellipticF(c/2 + (d*x)/2, 2))/d + (8*B*a^3*b*ellipticF(c/2 + (d*x)/2, 2))/ d + (2*A*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)) + (2*B*b^4*sin(c + d*x)*hyper geom([-5/4, 1/2], -1/4, cos(c + d*x)^2))/(5*d*cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (2*C*b^4*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x)^2))/(7*d*cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)) + (8*A*a*b^3*si n(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)) + (8*B*a*b^3*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*C*a^3*b*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*c os(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (8*C*a*b^3*sin(c + d*x)*hyperg eom([-5/4, 1/2], -1/4, cos(c + d*x)^2))/(5*d*cos(c + d*x)^(5/2)*(sin(c + d *x)^2)^(1/2)) + (12*B*a^2*b^2*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)) + (4*C*a^2*b^2 *sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(d*cos(c + d*x) ^(3/2)*(sin(c + d*x)^2)^(1/2))
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{6}d x \right ) b^{4} c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{5}d x \right ) a \,b^{3} c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{5}d x \right ) b^{5}+6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4}d x \right ) a^{2} b^{2} c +5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4}d x \right ) a \,b^{4}+4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3}d x \right ) a^{3} b c +10 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3}d x \right ) a^{2} b^{3}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{2}d x \right ) a^{4} c +10 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{2}d x \right ) a^{3} b^{2}+5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )d x \right ) a^{4} b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{5} \] Input:
int(cos(d*x+c)^(3/2)*(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)*sec(c + d*x)**6,x)*b**4*c + 4*int(sqrt (cos(c + d*x))*cos(c + d*x)*sec(c + d*x)**5,x)*a*b**3*c + int(sqrt(cos(c + d*x))*cos(c + d*x)*sec(c + d*x)**5,x)*b**5 + 6*int(sqrt(cos(c + d*x))*cos (c + d*x)*sec(c + d*x)**4,x)*a**2*b**2*c + 5*int(sqrt(cos(c + d*x))*cos(c + d*x)*sec(c + d*x)**4,x)*a*b**4 + 4*int(sqrt(cos(c + d*x))*cos(c + d*x)*s ec(c + d*x)**3,x)*a**3*b*c + 10*int(sqrt(cos(c + d*x))*cos(c + d*x)*sec(c + d*x)**3,x)*a**2*b**3 + int(sqrt(cos(c + d*x))*cos(c + d*x)*sec(c + d*x)* *2,x)*a**4*c + 10*int(sqrt(cos(c + d*x))*cos(c + d*x)*sec(c + d*x)**2,x)*a **3*b**2 + 5*int(sqrt(cos(c + d*x))*cos(c + d*x)*sec(c + d*x),x)*a**4*b + int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a**5