\(\int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1310]

Optimal result

Integrand size = 43, antiderivative size = 294 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (15 a^2 b B+3 b^3 B-5 a^3 (A-C)+3 a b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (21 a^3 B+21 a b^2 B+21 a^2 b (3 A+C)+b^3 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b \left (35 A b^2+63 a b B+24 a^2 C+25 b^2 C\right ) \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (98 a^2 b B+21 b^3 B+24 a^3 C+21 a b^2 (5 A+3 C)\right ) \sin (c+d x)}{35 d \sqrt {\cos (c+d x)}}+\frac {2 (7 b B+6 a C) (b+a \cos (c+d x))^2 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 C (b+a \cos (c+d x))^3 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)} \] Output:

-2/5*(15*B*a^2*b+3*B*b^3-5*a^3*(A-C)+3*a*b^2*(5*A+3*C))*EllipticE(sin(1/2* 
d*x+1/2*c),2^(1/2))/d+2/21*(21*B*a^3+21*B*a*b^2+21*a^2*b*(3*A+C)+b^3*(7*A+ 
5*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+2/105*b*(35*A*b^2+63*B*a*b+ 
24*C*a^2+25*C*b^2)*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/35*(98*B*a^2*b+21*B*b^3 
+24*a^3*C+21*a*b^2*(5*A+3*C))*sin(d*x+c)/d/cos(d*x+c)^(1/2)+2/35*(7*B*b+6* 
C*a)*(b+a*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/7*C*(b+a*cos(d*x+c 
))^3*sin(d*x+c)/d/cos(d*x+c)^(7/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

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

Time = 11.05 (sec) , antiderivative size = 3225, normalized size of antiderivative = 10.97 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:

Integrate[Sqrt[Cos[c + d*x]]*(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)*((-2*(5*a^3*A - 30*a*A*b^2 - 30*a^2*b*B - 6*b^3*B - 10*a^3*C - 1 
8*a*b^2*C + 5*a^3*A*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (4*b^3*C*Sec[c]*Sec[c 
 + d*x]^4*Sin[d*x])/(7*d) + (4*Sec[c]*Sec[c + d*x]^3*(5*b^3*C*Sin[c] + 7*b 
^3*B*Sin[d*x] + 21*a*b^2*C*Sin[d*x]))/(35*d) + (4*Sec[c]*Sec[c + d*x]*(35* 
A*b^3*Sin[c] + 105*a*b^2*B*Sin[c] + 105*a^2*b*C*Sin[c] + 25*b^3*C*Sin[c] + 
 315*a*A*b^2*Sin[d*x] + 315*a^2*b*B*Sin[d*x] + 63*b^3*B*Sin[d*x] + 105*a^3 
*C*Sin[d*x] + 189*a*b^2*C*Sin[d*x]))/(105*d) + (4*Sec[c]*Sec[c + d*x]^2*(2 
1*b^3*B*Sin[c] + 63*a*b^2*C*Sin[c] + 35*A*b^3*Sin[d*x] + 105*a*b^2*B*Sin[d 
*x] + 105*a^2*b*C*Sin[d*x] + 25*b^3*C*Sin[d*x]))/(105*d)))/((b + a*Cos[c + 
 d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (12*a^2*A*b* 
Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTa 
n[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[-(Sq 
rt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - Arc 
Tan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*C 
os[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^3*Cos[c + d*x]^5*Csc[c]*Hype 
rgeometricPFQ[{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 - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]...
 

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.03, 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, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4600

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

\(\displaystyle \frac {2}{7} \int \frac {(b+a \cos (c+d x))^2 \left (a (7 A-C) \cos ^2(c+d x)+(7 A b+5 C b+7 a B) \cos (c+d x)+7 b B+6 a C\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \int \frac {(b+a \cos (c+d x))^2 \left (a (7 A-C) \cos ^2(c+d x)+(7 A b+5 C b+7 a B) \cos (c+d x)+7 b B+6 a C\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (a (7 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(7 A b+5 C b+7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+7 b B+6 a C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3526

\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {(b+a \cos (c+d x)) \left (24 C a^2+(35 a A-7 b B-11 a C) \cos ^2(c+d x) a+63 b B a+35 A b^2+25 b^2 C+\left (35 B a^2+70 A b a+38 b C a+21 b^2 B\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {(b+a \cos (c+d x)) \left (24 C a^2+(35 a A-7 b B-11 a C) \cos ^2(c+d x) a+63 b B a+35 A b^2+25 b^2 C+\left (35 B a^2+70 A b a+38 b C a+21 b^2 B\right ) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (24 C a^2+(35 a A-7 b B-11 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+63 b B a+35 A b^2+25 b^2 C+\left (35 B a^2+70 A b a+38 b C a+21 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3510

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b \sin (c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int -\frac {3 a^2 (35 a A-7 b B-11 a C) \cos ^2(c+d x)+5 \left (21 B a^3+21 b (3 A+C) a^2+21 b^2 B a+b^3 (7 A+5 C)\right ) \cos (c+d x)+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {3 a^2 (35 a A-7 b B-11 a C) \cos ^2(c+d x)+5 \left (21 B a^3+21 b (3 A+C) a^2+21 b^2 B a+b^3 (7 A+5 C)\right ) \cos (c+d x)+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 b \sin (c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {3 a^2 (35 a A-7 b B-11 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+5 \left (21 B a^3+21 b (3 A+C) a^2+21 b^2 B a+b^3 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (24 C a^3+98 b B a^2+21 b^2 (5 A+3 C) a+21 b^3 B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 b \sin (c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3500

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (2 \int \frac {5 \left (21 B a^3+21 b (3 A+C) a^2+21 b^2 B a+b^3 (7 A+5 C)\right )-21 \left (-5 (A-C) a^3+15 b B a^2+3 b^2 (5 A+3 C) a+3 b^3 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 \sin (c+d x) \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 b \sin (c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left (21 B a^3+21 b (3 A+C) a^2+21 b^2 B a+b^3 (7 A+5 C)\right )-21 \left (-5 (A-C) a^3+15 b B a^2+3 b^2 (5 A+3 C) a+3 b^3 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {6 \sin (c+d x) \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 b \sin (c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {5 \left (21 B a^3+21 b (3 A+C) a^2+21 b^2 B a+b^3 (7 A+5 C)\right )-21 \left (-5 (A-C) a^3+15 b B a^2+3 b^2 (5 A+3 C) a+3 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sin (c+d x) \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 b \sin (c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-21 \left (-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx+\frac {6 \sin (c+d x) \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 b \sin (c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-21 \left (-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {6 \sin (c+d x) \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 b \sin (c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right )}{d}+\frac {6 \sin (c+d x) \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 b \sin (c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b \sin (c+d x) \left (24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right )}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{3} \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right )}{d}-\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right )}{d}+\frac {6 \sin (c+d x) \left (24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right )}{d \sqrt {\cos (c+d x)}}\right )\right )+\frac {2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

Input:

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

Output:

(2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*( 
7*b*B + 6*a*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2 
)) + ((2*b*(35*A*b^2 + 63*a*b*B + 24*a^2*C + 25*b^2*C)*Sin[c + d*x])/(3*d* 
Cos[c + d*x]^(3/2)) + ((-42*(15*a^2*b*B + 3*b^3*B - 5*a^3*(A - C) + 3*a*b^ 
2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/d + (10*(21*a^3*B + 21*a*b^2*B + 
 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/d + (6*( 
98*a^2*b*B + 21*b^3*B + 24*a^3*C + 21*a*b^2*(5*A + 3*C))*Sin[c + d*x])/(d* 
Sqrt[Cos[c + d*x]]))/3)/5)/7
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1177\) vs. \(2(277)=554\).

Time = 9.26 (sec) , antiderivative size = 1178, normalized size of antiderivative = 4.01

method	result	size
default	\(\text {Expression too large to display}\)	\(1178\)

Input:

int(cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, 
method=_RETURNVERBOSE)
 

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*B*a^3*(sin(1 
/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1 
/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+ 
2*C*b^3*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2 
*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin( 
1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+ 
5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*si 
n(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c 
),2^(1/2)))+2*a*(3*A*b^2+3*B*a*b+C*a^2)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d* 
x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin( 
1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellip 
ticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2))+2*b*(A*b^2+ 
3*B*a*b+3*C*a^2)*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2 
*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^ 
2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/ 
2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2/5*b^2*(B*b+ 
3*C*a)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c 
)^2-1)/sin(1/2*d*x+1/2*c)^2*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12 
*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(s 
in(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c)*s...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.41 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, \sqrt {2} {\left (21 i \, B a^{3} + 21 i \, {\left (3 \, A + C\right )} a^{2} b + 21 i \, B a b^{2} + i \, {\left (7 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-21 i \, B a^{3} - 21 i \, {\left (3 \, A + C\right )} a^{2} b - 21 i \, B a b^{2} - i \, {\left (7 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-5 i \, {\left (A - C\right )} a^{3} + 15 i \, B a^{2} b + 3 i \, {\left (5 \, A + 3 \, C\right )} a b^{2} + 3 i \, B b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, {\left (A - C\right )} a^{3} - 15 i \, B a^{2} b - 3 i \, {\left (5 \, A + 3 \, C\right )} a b^{2} - 3 i \, B b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (15 \, C b^{3} + 21 \, {\left (5 \, C a^{3} + 15 \, B a^{2} b + 3 \, {\left (5 \, A + 3 \, C\right )} a b^{2} + 3 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (21 \, C a^{2} b + 21 \, B a b^{2} + {\left (7 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{4}} \] Input:

integrate(cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

integrate(cos(d*x+c)**(1/2)*(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. \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^(1/2)*(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]

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

integrate(cos(d*x+c)^(1/2)*(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*s 
qrt(cos(d*x + c)), x)
 

Mupad [B] (verification not implemented)

Time = 17.97 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.50 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

int(cos(c + d*x)^(1/2)*(a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos( 
c + d*x)^2),x)
 

Output:

(2*(A*a^3*ellipticE(c/2 + (d*x)/2, 2) + 3*A*a^2*b*ellipticF(c/2 + (d*x)/2, 
 2)))/d + ((2*C*b^3*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x) 
^2))/7 + 2*C*a^3*cos(c + d*x)^3*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, c 
os(c + d*x)^2) + 2*C*a^2*b*cos(c + d*x)^2*sin(c + d*x)*hypergeom([-3/4, 1/ 
2], 1/4, cos(c + d*x)^2) + (6*C*a*b^2*cos(c + d*x)*sin(c + d*x)*hypergeom( 
[-5/4, 1/2], -1/4, cos(c + d*x)^2))/5)/(d*cos(c + d*x)^(7/2)*(1 - cos(c + 
d*x)^2)^(1/2)) + (2*B*a^3*ellipticF(c/2 + (d*x)/2, 2))/d + (2*A*b^3*sin(c 
+ d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(3*d*cos(c + d*x)^(3/2 
)*(sin(c + d*x)^2)^(1/2)) + (2*B*b^3*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)) + (6 
*A*a*b^2*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos( 
c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (6*B*a^2*b*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)) + (2*B*a*b^2*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x) 
^2))/(d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2))
 

Reduce [F]

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

int(cos(d*x+c)^(1/2)*(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)),x)*a**4 + int(sqrt(cos(c + d*x))*sec(c + d*x)**5,x) 
*b**3*c + 3*int(sqrt(cos(c + d*x))*sec(c + d*x)**4,x)*a*b**2*c + int(sqrt( 
cos(c + d*x))*sec(c + d*x)**4,x)*b**4 + 3*int(sqrt(cos(c + d*x))*sec(c + d 
*x)**3,x)*a**2*b*c + 4*int(sqrt(cos(c + d*x))*sec(c + d*x)**3,x)*a*b**3 + 
int(sqrt(cos(c + d*x))*sec(c + d*x)**2,x)*a**3*c + 6*int(sqrt(cos(c + d*x) 
)*sec(c + d*x)**2,x)*a**2*b**2 + 4*int(sqrt(cos(c + d*x))*sec(c + d*x),x)* 
a**3*b