\(\int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1203]

Optimal result

Integrand size = 43, antiderivative size = 231 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {4 a^3 (17 A+21 B+27 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^3 (11 A+13 B+21 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^3 (32 A+41 B+42 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 A \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 (2 A+3 B) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (73 A+99 B+63 C) \sqrt {\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d} \] Output:

4/15*a^3*(17*A+21*B+27*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+4/21*a^3 
*(11*A+13*B+21*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+4/105*a^3*(32*A 
+41*B+42*C)*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/9*A*cos(d*x+c)^(1/2)*(a+a*cos( 
d*x+c))^3*sin(d*x+c)/d+2/21*(2*A+3*B)*cos(d*x+c)^(1/2)*(a^2+a^2*cos(d*x+c) 
)^2*sin(d*x+c)/a/d+2/315*(73*A+99*B+63*C)*cos(d*x+c)^(1/2)*(a^3+a^3*cos(d* 
x+c))*sin(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 8.94 (sec) , antiderivative size = 1697, normalized size of antiderivative = 7.35 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + 
C*Sec[c + d*x]^2),x]
 

Output:

(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Se 
c[c + d*x] + C*Sec[c + d*x]^2)*(-1/15*((17*A + 21*B + 27*C)*Cot[c])/d + (( 
97*A + 107*B + 84*C)*Cos[d*x]*Sin[c])/(168*d) + ((73*A + 54*B + 18*C)*Cos[ 
2*d*x]*Sin[2*c])/(360*d) + ((3*A + B)*Cos[3*d*x]*Sin[3*c])/(56*d) + (A*Cos 
[4*d*x]*Sin[4*c])/(144*d) + ((97*A + 107*B + 84*C)*Cos[c]*Sin[d*x])/(168*d 
) + ((73*A + 54*B + 18*C)*Cos[2*c]*Sin[2*d*x])/(360*d) + ((3*A + B)*Cos[3* 
c]*Sin[3*d*x])/(56*d) + (A*Cos[4*c]*Sin[4*d*x])/(144*d)))/(A + 2*C + 2*B*C 
os[c + d*x] + A*Cos[2*c + 2*d*x]) - (11*A*Cos[c + d*x]^5*Csc[c]*Hypergeome 
tricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2] 
^6*(a + a*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]] 
]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c 
]^2]) - (13*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, S 
in[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*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 - S 
in[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTa 
n[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos 
[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (C*Cos[c + d*x]^5*Cs 
c[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*...
 

Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.04, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 4600, 3042, 3524, 27, 3042, 3455, 27, 3042, 3455, 27, 3042, 3447, 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.

\(\displaystyle \int \cos ^{\frac {9}{2}}(c+d x) (a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^{9/2} (a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\)

\(\Big \downarrow \) 4600

\(\displaystyle \int \frac {(a \cos (c+d x)+a)^3 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\sqrt {\cos (c+d x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 3524

\(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^3 (a (A+9 C)+3 a (2 A+3 B) \cos (c+d x))}{2 \sqrt {\cos (c+d x)}}dx}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^3 (a (A+9 C)+3 a (2 A+3 B) \cos (c+d x))}{\sqrt {\cos (c+d x)}}dx}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (A+9 C)+3 a (2 A+3 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {\frac {2}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((13 A+9 B+63 C) a^2+(73 A+99 B+63 C) \cos (c+d x) a^2\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((13 A+9 B+63 C) a^2+(73 A+99 B+63 C) \cos (c+d x) a^2\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((13 A+9 B+63 C) a^2+(73 A+99 B+63 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {3 (\cos (c+d x) a+a) \left ((23 A+24 B+63 C) a^3+3 (32 A+41 B+42 C) \cos (c+d x) a^3\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {(\cos (c+d x) a+a) \left ((23 A+24 B+63 C) a^3+3 (32 A+41 B+42 C) \cos (c+d x) a^3\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((23 A+24 B+63 C) a^3+3 (32 A+41 B+42 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3447

\(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {3 (32 A+41 B+42 C) \cos ^2(c+d x) a^4+(23 A+24 B+63 C) a^4+\left (3 (32 A+41 B+42 C) a^4+(23 A+24 B+63 C) a^4\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {3 (32 A+41 B+42 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+(23 A+24 B+63 C) a^4+\left (3 (32 A+41 B+42 C) a^4+(23 A+24 B+63 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\frac {2}{3} \int \frac {3 \left (5 (11 A+13 B+21 C) a^4+7 (17 A+21 B+27 C) \cos (c+d x) a^4\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \frac {5 (11 A+13 B+21 C) a^4+7 (17 A+21 B+27 C) \cos (c+d x) a^4}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \frac {5 (11 A+13 B+21 C) a^4+7 (17 A+21 B+27 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+13 B+21 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 a^4 (17 A+21 B+27 C) \int \sqrt {\cos (c+d x)}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+13 B+21 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a^4 (17 A+21 B+27 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+13 B+21 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 a^4 (17 A+21 B+27 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {1}{7} \left (\frac {2 (73 A+99 B+63 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}+\frac {6}{5} \left (\frac {10 a^4 (11 A+13 B+21 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {14 a^4 (17 A+21 B+27 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )\right )+\frac {6 (2 A+3 B) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\)

Input:

Int[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[ 
c + d*x]^2),x]
 

Output:

(2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + ((6*( 
2*A + 3*B)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7* 
d) + ((2*(73*A + 99*B + 63*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])* 
Sin[c + d*x])/(5*d) + (6*((14*a^4*(17*A + 21*B + 27*C)*EllipticE[(c + d*x) 
/2, 2])/d + (10*a^4*(11*A + 13*B + 21*C)*EllipticF[(c + d*x)/2, 2])/d + (2 
*a^4*(32*A + 41*B + 42*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d))/5)/7)/(9*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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]
 

rule 3120

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]
 

rule 3227

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]
 

rule 3447

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a 
 + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), 
 x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
 

rule 3455

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[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n 
 + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a + b*Sin[e + 
f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 
) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + 
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 
 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1 
] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

rule 3502

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]
 

rule 3524

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/(b*d*(m + 
 n + 2))   Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m 
+ n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} 
, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !Lt 
Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
 

rule 4600

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(513\) vs. \(2(214)=428\).

Time = 159.33 (sec) , antiderivative size = 514, normalized size of antiderivative = 2.23

method	result	size
default	\(-\frac {4 \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}}\, a^{3} \left (-560 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (2200 A +360 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-3412 A -1296 B -252 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (2702 A +1806 B +882 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-738 A -624 B -378 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+165 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )-357 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )+195 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )-441 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )+315 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )-567 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \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 )\right )}{315 \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}\)	\(514\)

Input:

int(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, 
method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

-4/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-560*A 
*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+(2200*A+360*B)*sin(1/2*d*x+1/2*c 
)^8*cos(1/2*d*x+1/2*c)+(-3412*A-1296*B-252*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2 
*d*x+1/2*c)+(2702*A+1806*B+882*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+ 
(-738*A-624*B-378*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+165*A*(sin(1/ 
2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d 
*x+1/2*c),2^(1/2))-357*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c 
)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+195*B*(sin(1/2*d*x+1/2* 
c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c), 
2^(1/2))-441*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/ 
2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+315*C*(sin(1/2*d*x+1/2*c)^2)^(1/2 
)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-5 
67*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellipti 
cE(cos(1/2*d*x+1/2*c),2^(1/2)))/(-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
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.04 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (11 \, A + 13 \, B + 21 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (11 \, A + 13 \, B + 21 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (17 \, A + 21 \, B + 27 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (17 \, A + 21 \, B + 27 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, A a^{3} \cos \left (d x + c\right )^{3} + 45 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + 7 \, {\left (34 \, A + 27 \, B + 9 \, C\right )} a^{3} \cos \left (d x + c\right ) + 15 \, {\left (22 \, A + 26 \, B + 21 \, C\right )} a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d} \] Input:

integrate(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="fricas")
 

Output:

-2/315*(15*I*sqrt(2)*(11*A + 13*B + 21*C)*a^3*weierstrassPInverse(-4, 0, c 
os(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(11*A + 13*B + 21*C)*a^3*weie 
rstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*I*sqrt(2)*(17*A 
 + 21*B + 27*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos( 
d*x + c) + I*sin(d*x + c))) + 21*I*sqrt(2)*(17*A + 21*B + 27*C)*a^3*weiers 
trassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) 
) - (35*A*a^3*cos(d*x + c)^3 + 45*(3*A + B)*a^3*cos(d*x + c)^2 + 7*(34*A + 
 27*B + 9*C)*a^3*cos(d*x + c) + 15*(22*A + 26*B + 21*C)*a^3)*sqrt(cos(d*x 
+ c))*sin(d*x + c))/d
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**(9/2)*(a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+ 
c)**2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:

integrate(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^3*(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)*(a*sec(d*x + c) + a)^3*c 
os(d*x + c)^(9/2), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 14.29 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.86 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,\left (B\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+B\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+B\,a^3\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {A\,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 {6\,C\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {4\,C\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,C\,a^3\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{d}-\frac {6\,A\,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\,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 )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,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\,B\,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\,C\,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}} \] Input:

int(cos(c + d*x)^(9/2)*(a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos( 
c + d*x)^2),x)
 

Output:

(2*(B*a^3*ellipticE(c/2 + (d*x)/2, 2) + B*a^3*ellipticF(c/2 + (d*x)/2, 2) 
+ B*a^3*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (A*a^3*((2*cos(c + d*x)^(1/2 
)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d + (6*C*a^3*ellip 
ticE(c/2 + (d*x)/2, 2))/d + (4*C*a^3*ellipticF(c/2 + (d*x)/2, 2))/d + (2*C 
*a^3*cos(c + d*x)^(1/2)*sin(c + d*x))/d - (6*A*a^3*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*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))/(3*d*(sin(c + d*x)^2)^(1/2)) - (2*A*a^3*cos(c + d*x 
)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d* 
(sin(c + d*x)^2)^(1/2)) - (6*B*a^3*cos(c + d*x)^(7/2)*sin(c + d*x)*hyperge 
om([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*B* 
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*C*a^3*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))
 

Reduce [F]

\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{4}d x \right ) b +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{3}d x \right ) a +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{3}d x \right ) b +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{3}d x \right ) c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) a +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a \right ) \] Input:

int(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
 

Output:

a**3*(int(sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**5,x)*c + int(sq 
rt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**4,x)*b + 3*int(sqrt(cos(c + 
 d*x))*cos(c + d*x)**4*sec(c + d*x)**4,x)*c + int(sqrt(cos(c + d*x))*cos(c 
 + d*x)**4*sec(c + d*x)**3,x)*a + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**4 
*sec(c + d*x)**3,x)*b + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d 
*x)**3,x)*c + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**2,x)* 
a + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**2,x)*b + int(sq 
rt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**2,x)*c + 3*int(sqrt(cos(c + 
 d*x))*cos(c + d*x)**4*sec(c + d*x),x)*a + int(sqrt(cos(c + d*x))*cos(c + 
d*x)**4*sec(c + d*x),x)*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*a)