\(\int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1306]

Optimal result

Integrand size = 43, antiderivative size = 296 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 (2 A b+3 a B) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac {2 A \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{9 d} \] Output:

2/15*(27*B*a^2*b+15*B*b^3+9*a*b^2*(3*A+5*C)+a^3*(7*A+9*C))*EllipticE(sin(1 
/2*d*x+1/2*c),2^(1/2))/d+2/21*(5*B*a^3+21*B*a*b^2+7*b^3*(A+3*C)+3*a^2*b*(5 
*A+7*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+2/63*(8*A*b^3+15*B*a^3+5 
4*B*a*b^2+9*a^2*b*(5*A+7*C))*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/315*a*(24*A*b 
^2+99*B*a*b+7*a^2*(7*A+9*C))*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/21*(2*A*b+3*B 
*a)*cos(d*x+c)^(1/2)*(b+a*cos(d*x+c))^2*sin(d*x+c)/d+2/9*A*cos(d*x+c)^(1/2 
)*(b+a*cos(d*x+c))^3*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 = 9.21 (sec) , antiderivative size = 3237, normalized size of antiderivative = 10.94 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:

Integrate[Cos[c + d*x]^(9/2)*(a + b*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)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c 
+ d*x]^2)*((-4*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B + 9*a^3*C + 4 
5*a*b^2*C)*Cot[c])/(15*d) + ((69*a^2*A*b + 28*A*b^3 + 23*a^3*B + 84*a*b^2* 
B + 84*a^2*b*C)*Cos[d*x]*Sin[c])/(21*d) + (a*(19*a^2*A + 54*A*b^2 + 54*a*b 
*B + 18*a^2*C)*Cos[2*d*x]*Sin[2*c])/(45*d) + (a^2*(3*A*b + a*B)*Cos[3*d*x] 
*Sin[3*c])/(7*d) + (a^3*A*Cos[4*d*x]*Sin[4*c])/(18*d) + ((69*a^2*A*b + 28* 
A*b^3 + 23*a^3*B + 84*a*b^2*B + 84*a^2*b*C)*Cos[c]*Sin[d*x])/(21*d) + (a*( 
19*a^2*A + 54*A*b^2 + 54*a*b*B + 18*a^2*C)*Cos[2*c]*Sin[2*d*x])/(45*d) + ( 
a^2*(3*A*b + a*B)*Cos[3*c]*Sin[3*d*x])/(7*d) + (a^3*A*Cos[4*c]*Sin[4*d*x]) 
/(18*d)))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c 
+ 2*d*x])) - (20*a^2*A*b*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2 
}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c 
 + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - A 
rcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]] 
)]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(b + a*Cos[c + d*x])^3*(A + 2 
*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^3 
*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcT 
an[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x] 
^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(S 
qrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x -...
 

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.04, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {3042, 4600, 3042, 3528, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4600

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

\(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(b+a \cos (c+d x)) \left (\left (7 (7 A+9 C) a^2+99 b B a+24 A b^2\right ) \cos ^2(c+d x)+\left (45 B a^2+86 A b a+126 b C a+63 b^2 B\right ) \cos (c+d x)+b (13 A b+63 C b+9 a B)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(b+a \cos (c+d x)) \left (\left (7 (7 A+9 C) a^2+99 b B a+24 A b^2\right ) \cos ^2(c+d x)+\left (45 B a^2+86 A b a+126 b C a+63 b^2 B\right ) \cos (c+d x)+b (13 A b+63 C b+9 a B)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (7 (7 A+9 C) a^2+99 b B a+24 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 B a^2+86 A b a+126 b C a+63 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+b (13 A b+63 C b+9 a B)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 3512

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 (13 A b+63 C b+9 a B) b^2+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right ) \cos ^2(c+d x)+21 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 (13 A b+63 C b+9 a B) b^2+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right ) \cos ^2(c+d x)+21 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 (13 A b+63 C b+9 a B) b^2+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+21 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (5 \left (5 B a^3+3 b (5 A+7 C) a^2+21 b^2 B a+7 b^3 (A+3 C)\right )+7 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (5 B a^3+3 b (5 A+7 C) a^2+21 b^2 B a+7 b^3 (A+3 C)\right )+7 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (5 B a^3+3 b (5 A+7 C) a^2+21 b^2 B a+7 b^3 (A+3 C)\right )+7 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right )}{d}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{d}\right )\right )\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}\right )+\frac {2 A \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}\)

Input:

Int[Cos[c + d*x]^(9/2)*(a + b*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]]*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + ((6*( 
2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7* 
d) + ((2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Si 
n[c + d*x])/(5*d) + (3*((14*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + 
 a^3*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/d + (10*(5*a^3*B + 21*a*b^2*B 
 + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d) + 
(10*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d 
*x]]*Sin[c + d*x])/d)/5)/7)/9
 

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 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 3512

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]
 

rule 3528

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])))
 

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 [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.27 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (35 \, A a^{3} \cos \left (d x + c\right )^{3} + 75 \, B a^{3} + 45 \, {\left (5 \, A + 7 \, C\right )} a^{2} b + 315 \, B a b^{2} + 105 \, A b^{3} + 45 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{2} + 7 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{3} + 27 \, B a^{2} b + 27 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (5 i \, B a^{3} + 3 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b + 21 i \, B a b^{2} + 7 i \, {\left (A + 3 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-5 i \, B a^{3} - 3 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b - 21 i \, B a b^{2} - 7 i \, {\left (A + 3 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{3} - 27 i \, B a^{2} b - 9 i \, {\left (3 \, A + 5 \, C\right )} a b^{2} - 15 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (i \, {\left (7 \, A + 9 \, C\right )} a^{3} + 27 i \, B a^{2} b + 9 i \, {\left (3 \, A + 5 \, C\right )} a b^{2} + 15 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{315 \, d} \] Input:

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

Output:

1/315*(2*(35*A*a^3*cos(d*x + c)^3 + 75*B*a^3 + 45*(5*A + 7*C)*a^2*b + 315* 
B*a*b^2 + 105*A*b^3 + 45*(B*a^3 + 3*A*a^2*b)*cos(d*x + c)^2 + 7*((7*A + 9* 
C)*a^3 + 27*B*a^2*b + 27*A*a*b^2)*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x 
 + c) - 15*sqrt(2)*(5*I*B*a^3 + 3*I*(5*A + 7*C)*a^2*b + 21*I*B*a*b^2 + 7*I 
*(A + 3*C)*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) 
- 15*sqrt(2)*(-5*I*B*a^3 - 3*I*(5*A + 7*C)*a^2*b - 21*I*B*a*b^2 - 7*I*(A + 
 3*C)*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21* 
sqrt(2)*(-I*(7*A + 9*C)*a^3 - 27*I*B*a^2*b - 9*I*(3*A + 5*C)*a*b^2 - 15*I* 
B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I* 
sin(d*x + c))) - 21*sqrt(2)*(I*(7*A + 9*C)*a^3 + 27*I*B*a^2*b + 9*I*(3*A + 
 5*C)*a*b^2 + 15*I*B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0 
, cos(d*x + c) - I*sin(d*x + c))))/d
 

Sympy [F(-1)]

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

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

integrate(cos(d*x+c)^(9/2)*(a+b*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]

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

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

Mupad [B] (verification not implemented)

Time = 13.98 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.53 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(B*b^3*ellipticE(c/2 + (d*x)/2, 2) + B*a*b^2*ellipticF(c/2 + (d*x)/2, 2 
) + B*a*b^2*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (A*b^3*((2*cos(c + d*x)^ 
(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d + (2*C*b^3*e 
llipticF(c/2 + (d*x)/2, 2))/d + (6*C*a*b^2*ellipticE(c/2 + (d*x)/2, 2))/d 
+ (3*C*a^2*b*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + ( 
d*x)/2, 2))/3))/d - (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)) - (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) 
) - (6*A*a*b^2*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^2*b*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)) - (6*B*a^2*b*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([ 
1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2))
 

Reduce [F]

\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{5}d x \right ) b^{3} c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{4}d x \right ) a \,b^{2} c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{4}d x \right ) b^{4}+3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{3}d x \right ) a^{2} b c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{3}d x \right ) a \,b^{3}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) a^{3} c +6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) a^{2} b^{2}+4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) a^{3} b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a^{4} \] Input:

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

Output:

int(sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**5,x)*b**3*c + 3*int(s 
qrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**4,x)*a*b**2*c + int(sqrt(c 
os(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**4,x)*b**4 + 3*int(sqrt(cos(c + 
d*x))*cos(c + d*x)**4*sec(c + d*x)**3,x)*a**2*b*c + 4*int(sqrt(cos(c + d*x 
))*cos(c + d*x)**4*sec(c + d*x)**3,x)*a*b**3 + int(sqrt(cos(c + d*x))*cos( 
c + d*x)**4*sec(c + d*x)**2,x)*a**3*c + 6*int(sqrt(cos(c + d*x))*cos(c + d 
*x)**4*sec(c + d*x)**2,x)*a**2*b**2 + 4*int(sqrt(cos(c + d*x))*cos(c + d*x 
)**4*sec(c + d*x),x)*a**3*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*a* 
*4