3.14.17 \(\int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1317]

3.14.17.1 Optimal result
3.14.17.2 Mathematica [C] (warning: unable to verify)
3.14.17.3 Rubi [A] (verified)
3.14.17.4 Maple [B] (verified)
3.14.17.5 Fricas [C] (verification not implemented)
3.14.17.6 Sympy [F(-1)]
3.14.17.7 Maxima [F(-1)]
3.14.17.8 Giac [F]
3.14.17.9 Mupad [B] (verification not implemented)

3.14.17.1 Optimal result

Integrand size = 43, antiderivative size = 401 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b \left (261 a^2 b B+75 b^3 B+64 a^3 C+2 a b^2 (147 A+101 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (1098 a^3 b B+756 a b^3 B+192 a^4 C+21 b^4 (9 A+7 C)+7 a^2 b^2 (261 A+155 C)\right ) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)}}+\frac {2 \left (63 A b^2+117 a b B+48 a^2 C+49 b^2 C\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (9 b B+8 a C) (b+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (b+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]

output
-2/15*(60*B*a^3*b+36*B*a*b^3-15*a^4*(A-C)+18*a^2*b^2*(5*A+3*C)+b^4*(9*A+7* 
C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+ 
1/2*c),2^(1/2))/d+2/21*(21*B*a^4+42*B*a^2*b^2+5*B*b^4+28*a^3*b*(3*A+C)+4*a 
*b^3*(7*A+5*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF( 
sin(1/2*d*x+1/2*c),2^(1/2))/d+2/315*b*(261*B*a^2*b+75*B*b^3+64*a^3*C+2*a*b 
^2*(147*A+101*C))*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/315*(63*A*b^2+117*B*a*b+ 
48*C*a^2+49*C*b^2)*(b+a*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/63*( 
9*B*b+8*C*a)*(b+a*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(7/2)+2/9*C*(b+a*c 
os(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(9/2)+2/315*(1098*B*a^3*b+756*B*a*b^3 
+192*a^4*C+21*b^4*(9*A+7*C)+7*a^2*b^2*(261*A+155*C))*sin(d*x+c)/d/cos(d*x+ 
c)^(1/2)
 
3.14.17.2 Mathematica [C] (warning: unable to verify)

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

Time = 16.46 (sec) , antiderivative size = 4150, normalized size of antiderivative = 10.35 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]

input
Integrate[Sqrt[Cos[c + d*x]]*(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*(15*a^4*A - 180*a^2*A*b^2 - 18*A*b^4 - 120*a^3*b*B - 72*a*b 
^3*B - 30*a^4*C - 108*a^2*b^2*C - 14*b^4*C + 15*a^4*A*Cos[2*c])*Csc[c]*Sec 
[c])/(15*d) + (4*b^4*C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(9*d) + (4*Sec[c]*S 
ec[c + d*x]^4*(7*b^4*C*Sin[c] + 9*b^4*B*Sin[d*x] + 36*a*b^3*C*Sin[d*x]))/( 
63*d) + (4*Sec[c]*Sec[c + d*x]^2*(63*A*b^4*Sin[c] + 252*a*b^3*B*Sin[c] + 3 
78*a^2*b^2*C*Sin[c] + 49*b^4*C*Sin[c] + 420*a*A*b^3*Sin[d*x] + 630*a^2*b^2 
*B*Sin[d*x] + 75*b^4*B*Sin[d*x] + 420*a^3*b*C*Sin[d*x] + 300*a*b^3*C*Sin[d 
*x]))/(315*d) + (4*Sec[c]*Sec[c + d*x]^3*(45*b^4*B*Sin[c] + 180*a*b^3*C*Si 
n[c] + 63*A*b^4*Sin[d*x] + 252*a*b^3*B*Sin[d*x] + 378*a^2*b^2*C*Sin[d*x] + 
 49*b^4*C*Sin[d*x]))/(315*d) + (4*Sec[c]*Sec[c + d*x]*(140*a*A*b^3*Sin[c] 
+ 210*a^2*b^2*B*Sin[c] + 25*b^4*B*Sin[c] + 140*a^3*b*C*Sin[c] + 100*a*b^3* 
C*Sin[c] + 630*a^2*A*b^2*Sin[d*x] + 63*A*b^4*Sin[d*x] + 420*a^3*b*B*Sin[d* 
x] + 252*a*b^3*B*Sin[d*x] + 105*a^4*C*Sin[d*x] + 378*a^2*b^2*C*Sin[d*x] + 
49*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])) - (16*a^3*A*b*Cos[c + d*x]^6*Csc[c]*Hyperge 
ometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + 
d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]* 
Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d 
*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*C...
 
3.14.17.3 Rubi [A] (verified)

Time = 2.65 (sec) , antiderivative size = 411, 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, 3500, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4600

\(\displaystyle \int \frac {(a \cos (c+d x)+b)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\)

\(\Big \downarrow \) 3526

\(\displaystyle \frac {2}{9} \int \frac {(b+a \cos (c+d x))^3 \left (a (9 A-C) \cos ^2(c+d x)+(9 A b+7 C b+9 a B) \cos (c+d x)+9 b B+8 a C\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int \frac {(b+a \cos (c+d x))^3 \left (a (9 A-C) \cos ^2(c+d x)+(9 A b+7 C b+9 a B) \cos (c+d x)+9 b B+8 a C\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (a (9 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(9 A b+7 C b+9 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+9 b B+8 a C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3526

\(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(b+a \cos (c+d x))^2 \left (48 C a^2+3 (21 a A-3 b B-5 a C) \cos ^2(c+d x) a+117 b B a+63 A b^2+49 b^2 C+\left (63 B a^2+126 A b a+82 b C a+45 b^2 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(b+a \cos (c+d x))^2 \left (48 C a^2+3 (21 a A-3 b B-5 a C) \cos ^2(c+d x) a+117 b B a+63 A b^2+49 b^2 C+\left (63 B a^2+126 A b a+82 b C a+45 b^2 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (48 C a^2+3 (21 a A-3 b B-5 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+117 b B a+63 A b^2+49 b^2 C+\left (63 B a^2+126 A b a+82 b C a+45 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3526

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {(b+a \cos (c+d x)) \left (-a \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)+\left (315 B a^3+b (945 A+479 C) a^2+531 b^2 B a+21 b^3 (9 A+7 C)\right ) \cos (c+d x)+3 \left (64 C a^3+261 b B a^2+2 b^2 (147 A+101 C) a+75 b^3 B\right )\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {(b+a \cos (c+d x)) \left (-a \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)+\left (315 B a^3+b (945 A+479 C) a^2+531 b^2 B a+21 b^3 (9 A+7 C)\right ) \cos (c+d x)+3 \left (64 C a^3+261 b B a^2+2 b^2 (147 A+101 C) a+75 b^3 B\right )\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-a \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (315 B a^3+b (945 A+479 C) a^2+531 b^2 B a+21 b^3 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (64 C a^3+261 b B a^2+2 b^2 (147 A+101 C) a+75 b^3 B\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3510

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int -\frac {3 \left (192 C a^4+1098 b B a^3-\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x) a^2+7 b^2 (261 A+155 C) a^2+756 b^3 B a+21 b^4 (9 A+7 C)+15 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {192 C a^4+1098 b B a^3-\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x) a^2+7 b^2 (261 A+155 C) a^2+756 b^3 B a+21 b^4 (9 A+7 C)+15 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {192 C a^4+1098 b B a^3-\left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^2+7 b^2 (261 A+155 C) a^2+756 b^3 B a+21 b^4 (9 A+7 C)+15 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3500

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (2 \int \frac {3 \left (5 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right )-7 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right )-7 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right )-7 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-7 \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-7 \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right ) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\frac {2 b \sin (c+d x) \left (64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right )}{d \sqrt {\cos (c+d x)}}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{d}-\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right )}{d}\right )\right )\right )+\frac {2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\)

input
Int[Sqrt[Cos[c + d*x]]*(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])/(9*d*Cos[c + d*x]^(9/2)) + ((2*( 
9*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2 
)) + ((2*(63*A*b^2 + 117*a*b*B + 48*a^2*C + 49*b^2*C)*(b + a*Cos[c + d*x]) 
^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (3*((-14*(60*a^3*b*B + 36*a*b^ 
3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*EllipticE 
[(c + d*x)/2, 2])/d + (10*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3 
*A + C) + 4*a*b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/d) + (2*b*(261*a 
^2*b*B + 75*b^3*B + 64*a^3*C + 2*a*b^2*(147*A + 101*C))*Sin[c + d*x])/(d*C 
os[c + d*x]^(3/2)) + (2*(1098*a^3*b*B + 756*a*b^3*B + 192*a^4*C + 21*b^4*( 
9*A + 7*C) + 7*a^2*b^2*(261*A + 155*C))*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x] 
]))/5)/7)/9
 

3.14.17.3.1 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 3500
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 
 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* 
(a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x 
])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A 
*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, 
B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
 

rule 3510
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]
 

rule 3526
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]
 

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]
 
3.14.17.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1522\) vs. \(2(429)=858\).

Time = 6.32 (sec) , antiderivative size = 1523, normalized size of antiderivative = 3.80

method result size
default \(\text {Expression too large to display}\) \(1523\)

input
int((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2),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*B*a^4*(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*a^4*(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))+8*A*a^3*b*(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)*Elliptic 
F(cos(1/2*d*x+1/2*c),2^(1/2))+2*A*a^4*(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*(-1/144*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)^5-7/180*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)^3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d* 
x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(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))-7/15*(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+si 
n(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*b^3*(B*b+4*C*a)*(-1/56*cos(1/2*d*x+1/2...
 
3.14.17.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.18 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.31 \[ \int \sqrt {\cos (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 {15 \, \sqrt {2} {\left (21 i \, B a^{4} + 28 i \, {\left (3 \, A + C\right )} a^{3} b + 42 i \, B a^{2} b^{2} + 4 i \, {\left (7 \, A + 5 \, C\right )} a b^{3} + 5 i \, B b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-21 i \, B a^{4} - 28 i \, {\left (3 \, A + C\right )} a^{3} b - 42 i \, B a^{2} b^{2} - 4 i \, {\left (7 \, A + 5 \, C\right )} a b^{3} - 5 i \, B b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-15 i \, {\left (A - C\right )} a^{4} + 60 i \, B a^{3} b + 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} + 36 i \, B a b^{3} + i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{5} {\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 (15 i \, {\left (A - C\right )} a^{4} - 60 i \, B a^{3} b - 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} - 36 i \, B a b^{3} - i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{5} {\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 \, C b^{4} + 21 \, {\left (15 \, C a^{4} + 60 \, B a^{3} b + 18 \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} + 36 \, B a b^{3} + {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (28 \, C a^{3} b + 42 \, B a^{2} b^{2} + 4 \, {\left (7 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (54 \, C a^{2} b^{2} + 36 \, B a b^{3} + {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\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 )}{315 \, d \cos \left (d x + c\right )^{5}} \]

input
integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1 
/2),x, algorithm="fricas")
 
output
-1/315*(15*sqrt(2)*(21*I*B*a^4 + 28*I*(3*A + C)*a^3*b + 42*I*B*a^2*b^2 + 4 
*I*(7*A + 5*C)*a*b^3 + 5*I*B*b^4)*cos(d*x + c)^5*weierstrassPInverse(-4, 0 
, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-21*I*B*a^4 - 28*I*(3*A + C 
)*a^3*b - 42*I*B*a^2*b^2 - 4*I*(7*A + 5*C)*a*b^3 - 5*I*B*b^4)*cos(d*x + c) 
^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)* 
(-15*I*(A - C)*a^4 + 60*I*B*a^3*b + 18*I*(5*A + 3*C)*a^2*b^2 + 36*I*B*a*b^ 
3 + I*(9*A + 7*C)*b^4)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPI 
nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(15*I*(A - C)*a 
^4 - 60*I*B*a^3*b - 18*I*(5*A + 3*C)*a^2*b^2 - 36*I*B*a*b^3 - I*(9*A + 7*C 
)*b^4)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, co 
s(d*x + c) - I*sin(d*x + c))) - 2*(35*C*b^4 + 21*(15*C*a^4 + 60*B*a^3*b + 
18*(5*A + 3*C)*a^2*b^2 + 36*B*a*b^3 + (9*A + 7*C)*b^4)*cos(d*x + c)^4 + 15 
*(28*C*a^3*b + 42*B*a^2*b^2 + 4*(7*A + 5*C)*a*b^3 + 5*B*b^4)*cos(d*x + c)^ 
3 + 7*(54*C*a^2*b^2 + 36*B*a*b^3 + (9*A + 7*C)*b^4)*cos(d*x + c)^2 + 45*(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)^5)
 
3.14.17.6 Sympy [F(-1)]

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

input
integrate((a+b*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)*cos(d*x+c)* 
*(1/2),x)
 
output
Timed out
 
3.14.17.7 Maxima [F(-1)]

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

input
integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1 
/2),x, algorithm="maxima")
 
output
Timed out
 
3.14.17.8 Giac [F]

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

input
integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1 
/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*s 
qrt(cos(d*x + c)), x)
 
3.14.17.9 Mupad [B] (verification not implemented)

Time = 30.44 (sec) , antiderivative size = 866, normalized size of antiderivative = 2.16 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

input
int(cos(c + d*x)^(1/2)*(a + b/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos( 
c + d*x)^2),x)
 
output
(8*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)*((4*C*a*b^3*sin(c + d*x))/( 
cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (7*C*a^3*b*sin(c + d*x))/(cos 
(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (3*C*a*b^3*sin(c + d*x))/(cos(c 
+ d*x)^(7/2)*(sin(c + d*x)^2)^(1/2))))/(21*d) - (8*hypergeom([-1/4, 1/2], 
7/4, cos(c + d*x)^2)*((7*C*b^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + 
d*x)^2)^(1/2)) + (5*C*b^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^ 
2)^(1/2)) + (54*C*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^ 
2)^(1/2))))/(135*d) + (2*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((45* 
C*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (28*C*b^ 
4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (12*C*b^4*si 
n(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (5*C*b^4*sin(c + 
 d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2)) + (216*C*a^2*b^2*sin(c 
+ d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (54*C*a^2*b^2*sin(c 
+ d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2))))/(45*d) + (2*A*a^4*el 
lipticE(c/2 + (d*x)/2, 2))/d + (2*B*a^4*ellipticF(c/2 + (d*x)/2, 2))/d + ( 
8*A*a^3*b*ellipticF(c/2 + (d*x)/2, 2))/d + (2*A*b^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*B*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*sin(c 
+ d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(3*d*cos(c + d*x)^(...