\(\int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1214]

Optimal result

Integrand size = 43, antiderivative size = 271 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {8 a^4 (7 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {8 a^4 (35 A+28 B+17 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {4 a^4 (175 A+287 B+253 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (7 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (35 A+77 B+73 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 (175 A+238 B+197 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}} \] Output:

-8/5*a^4*(7*B+8*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+8/21*a^4*(35*A+ 
28*B+17*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d-4/105*a^4*(175*A+287*B 
+253*C)*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/35*a*(7*B+8*C)*(a+a*cos(d*x+c))^3* 
sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/7*C*(a+a*cos(d*x+c))^4*sin(d*x+c)/d/cos(d* 
x+c)^(7/2)+2/105*(35*A+77*B+73*C)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d/cos( 
d*x+c)^(3/2)+4/105*(175*A+238*B+197*C)*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d/c 
os(d*x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

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

Time = 12.79 (sec) , antiderivative size = 1454, normalized size of antiderivative = 5.37 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

Integrate[Cos[c + d*x]^(3/2)*(a + a*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)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Se 
c[c + d*x] + C*Sec[c + d*x]^2)*(-1/40*((-20*A - 61*B - 64*C + 20*A*Cos[2*c 
] + 5*B*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Cos[d*x]*Sin[c])/(12*d) + (A*Cos[c 
]*Sin[d*x])/(12*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(28*d) + (Sec[c]*S 
ec[c + d*x]^3*(5*C*Sin[c] + 7*B*Sin[d*x] + 28*C*Sin[d*x]))/(140*d) + (Sec[ 
c]*Sec[c + d*x]^2*(21*B*Sin[c] + 84*C*Sin[c] + 35*A*Sin[d*x] + 140*B*Sin[d 
*x] + 235*C*Sin[d*x]))/(420*d) + (Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 140*B 
*Sin[c] + 235*C*Sin[c] + 420*A*Sin[d*x] + 693*B*Sin[d*x] + 672*C*Sin[d*x]) 
)/(420*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (5*A*Cos[c 
 + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot 
[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*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]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2* 
c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c + d*x]^6*Csc[c]*Hypergeometri 
cPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8* 
(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - A 
rcTan[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]]]]) 
/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]...
 

Rubi [A] (verified)

Time = 2.11 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.08, 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, 3522, 27, 3042, 3454, 27, 3042, 3454, 3042, 3454, 27, 3042, 3447, 3042, 3502, 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.

\(\displaystyle \int \cos ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^{3/2} (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\)

\(\Big \downarrow \) 4600

\(\displaystyle \int \frac {(a \cos (c+d x)+a)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\)

\(\Big \downarrow \) 3522

\(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^4 (a (7 B+8 C)+a (7 A-3 C) \cos (c+d x))}{2 \cos ^{\frac {7}{2}}(c+d x)}dx}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^4 (a (7 B+8 C)+a (7 A-3 C) \cos (c+d x))}{\cos ^{\frac {7}{2}}(c+d x)}dx}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (7 B+8 C)+a (7 A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3454

\(\displaystyle \frac {\frac {2}{5} \int \frac {(\cos (c+d x) a+a)^3 \left ((35 A+77 B+73 C) a^2+(35 A-21 B-39 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \int \frac {(\cos (c+d x) a+a)^3 \left ((35 A+77 B+73 C) a^2+(35 A-21 B-39 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((35 A+77 B+73 C) a^2+(35 A-21 B-39 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3454

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \int \frac {(\cos (c+d x) a+a)^2 \left (a^3 (175 A+238 B+197 C)-21 a^3 (7 B+8 C) \cos (c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a^3 (175 A+238 B+197 C)-21 a^3 (7 B+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3454

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (2 \int \frac {3 (\cos (c+d x) a+a) \left (a^4 (175 A+189 B+141 C)-a^4 (175 A+287 B+253 C) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \int \frac {(\cos (c+d x) a+a) \left (a^4 (175 A+189 B+141 C)-a^4 (175 A+287 B+253 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (a^4 (175 A+189 B+141 C)-a^4 (175 A+287 B+253 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3447

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \int \frac {-\left ((175 A+287 B+253 C) \cos ^2(c+d x) a^5\right )+(175 A+189 B+141 C) a^5+\left (a^5 (175 A+189 B+141 C)-a^5 (175 A+287 B+253 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \int \frac {-\left ((175 A+287 B+253 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5\right )+(175 A+189 B+141 C) a^5+\left (a^5 (175 A+189 B+141 C)-a^5 (175 A+287 B+253 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \left (\frac {2}{3} \int \frac {5 a^5 (35 A+28 B+17 C)-21 a^5 (7 B+8 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \left (\frac {2}{3} \int \frac {5 a^5 (35 A+28 B+17 C)-21 a^5 (7 B+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \left (\frac {2}{3} \left (5 a^5 (35 A+28 B+17 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-21 a^5 (7 B+8 C) \int \sqrt {\cos (c+d x)}dx\right )-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \left (\frac {2}{3} \left (5 a^5 (35 A+28 B+17 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-21 a^5 (7 B+8 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \left (\frac {2}{3} \left (5 a^5 (35 A+28 B+17 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {42 a^5 (7 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\frac {2}{3} \left (\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}+3 \left (\frac {2}{3} \left (\frac {10 a^5 (35 A+28 B+17 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {42 a^5 (7 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

Input:

Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[ 
c + d*x]^2),x]
 

Output:

(2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*a 
^2*(7*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2 
)) + ((2*a^3*(35*A + 77*B + 73*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3* 
d*Cos[c + d*x]^(3/2)) + (2*((2*(175*A + 238*B + 197*C)*(a^5 + a^5*Cos[c + 
d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + 3*((2*((-42*a^5*(7*B + 8*C)*E 
llipticE[(c + d*x)/2, 2])/d + (10*a^5*(35*A + 28*B + 17*C)*EllipticF[(c + 
d*x)/2, 2])/d))/3 - (2*a^5*(175*A + 287*B + 253*C)*Sqrt[Cos[c + d*x]]*Sin[ 
c + d*x])/(3*d))))/3)/5)/(7*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 3454

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^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ 
e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + 
 a*d))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp 
[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B 
*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f 
, A, B}, 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 3522

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/(b*d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^m* 
(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*( 
a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2* 
(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, 
 x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ 
[m, -2^(-1)] && (LtQ[n, -1] || EqQ[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. \(1534\) vs. \(2(250)=500\).

Time = 8.29 (sec) , antiderivative size = 1535, normalized size of antiderivative = 5.66

method	result	size
default	\(\text {Expression too large to display}\)	\(1535\)

Input:

int(cos(d*x+c)^(3/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, 
method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

8/105*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4/sin(1/ 
2*d*x+1/2*c)^3/(16*sin(1/2*d*x+1/2*c)^8-32*sin(1/2*d*x+1/2*c)^6+24*sin(1/2 
*d*x+1/2*c)^4-8*sin(1/2*d*x+1/2*c)^2+1)*(-1470*A*cos(1/2*d*x+1/2*c)*sin(1/ 
2*d*x+1/2*c)^4+245*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)*A+503*sin(1/2*d 
*x+1/2*c)^2*cos(1/2*d*x+1/2*c)*C-2240*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2 
*c)^8+280*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10-147*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))+882*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c) 
,2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2+4502*C*cos 
(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-2570*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x 
+1/2*c)^4-2380*B*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+427*B*cos(1/2*d*x 
+1/2*c)*sin(1/2*d*x+1/2*c)^2-2100*A*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin( 
1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+ 
1/2*c)^4-1020*C*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1 
/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^4-2016*C*(2*s 
in(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/ 
2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^4+1050*A*(2*sin(1/2*d*x+1/2*c)^2- 
1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2) 
)*sin(1/2*d*x+1/2*c)^2+510*C*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x 
+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.07 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (10 i \, \sqrt {2} {\left (35 \, A + 28 \, B + 17 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 10 i \, \sqrt {2} {\left (35 \, A + 28 \, B + 17 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 42 i \, \sqrt {2} {\left (7 \, B + 8 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 42 i \, \sqrt {2} {\left (7 \, B + 8 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, A a^{4} \cos \left (d x + c\right )^{4} + 21 \, {\left (20 \, A + 33 \, B + 32 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 5 \, {\left (7 \, A + 28 \, B + 47 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 21 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 15 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{105 \, d \cos \left (d x + c\right )^{4}} \] Input:

integrate(cos(d*x+c)^(3/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="fricas")
 

Output:

-2/105*(10*I*sqrt(2)*(35*A + 28*B + 17*C)*a^4*cos(d*x + c)^4*weierstrassPI 
nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 10*I*sqrt(2)*(35*A + 28*B + 
 17*C)*a^4*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( 
d*x + c)) + 42*I*sqrt(2)*(7*B + 8*C)*a^4*cos(d*x + c)^4*weierstrassZeta(-4 
, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 42*I*sqr 
t(2)*(7*B + 8*C)*a^4*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInv 
erse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (35*A*a^4*cos(d*x + c)^4 + 2 
1*(20*A + 33*B + 32*C)*a^4*cos(d*x + c)^3 + 5*(7*A + 28*B + 47*C)*a^4*cos( 
d*x + c)^2 + 21*(B + 4*C)*a^4*cos(d*x + c) + 15*C*a^4)*sqrt(cos(d*x + c))* 
sin(d*x + c))/(d*cos(d*x + c)^4)
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**(3/2)*(a+a*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+ 
c)**2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^(3/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

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

integrate(cos(d*x+c)^(3/2)*(a+a*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)*(a*sec(d*x + c) + a)^4*c 
os(d*x + c)^(3/2), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 18.26 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.07 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

int(cos(c + d*x)^(3/2)*(a + a/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos( 
c + d*x)^2),x)
 

Output:

(2*(12*A*a^4*ellipticE(c/2 + (d*x)/2, 2) + 19*A*a^4*ellipticF(c/2 + (d*x)/ 
2, 2) + A*a^4*cos(c + d*x)^(1/2)*sin(c + d*x)))/(3*d) + (2*(B*a^4*elliptic 
E(c/2 + (d*x)/2, 2) + 4*B*a^4*ellipticF(c/2 + (d*x)/2, 2)))/d + (2*((34*B* 
a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (B*a^4*sin 
(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)))*hypergeom([-1/4, 1 
/2], 3/4, cos(c + d*x)^2))/(5*d) + (2*C*a^4*ellipticF(c/2 + (d*x)/2, 2))/d 
 + (8*A*a^4*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)) + (2*A*a^4*sin(c + d*x)*hypergeo 
m([-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*B*a^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*B*a^4*sin(c + d 
*x)*hypergeom([-1/4, 1/2], 7/4, cos(c + d*x)^2))/(15*d*cos(c + d*x)^(1/2)* 
(sin(c + d*x)^2)^(1/2)) + (8*C*a^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)) + (4*C*a^ 
4*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)) + (8*C*a^4*sin(c + d*x)*hypergeom([-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*a^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))
 

Reduce [F]

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

int(cos(d*x+c)^(3/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
 

Output:

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