\(\int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [949]

Optimal result

Integrand size = 43, antiderivative size = 650 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a^2 b d}-\frac {\sqrt {a+b} \left (9 A b^3-6 a b^2 (A+4 B)-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a^2 d}-\frac {\sqrt {a+b} \left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^3 d}-\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a^2 d}+\frac {\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{96 a d}+\frac {(3 A b+8 a B) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d} \] Output:

-1/192*(a-b)*(a+b)^(1/2)*(9*A*b^3-128*B*a^3-24*B*a*b^2-12*a^2*b*(13*A+20*C 
))*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^( 
1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/b 
/d-1/192*(a+b)^(1/2)*(9*A*b^3-6*a*b^2*(A+4*B)-8*a^3*(9*A+16*B+12*C)-4*a^2* 
b*(39*A+28*B+60*C))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2 
),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/( 
a-b))^(1/2)/a^2/d-1/64*(a+b)^(1/2)*(3*A*b^4+96*B*a^3*b-8*B*a*b^3+24*a^2*b^ 
2*(A+2*C)+16*a^4*(3*A+4*C))*cot(d*x+c)*EllipticPi((a+b*sec(d*x+c))^(1/2)/( 
a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b 
*(1+sec(d*x+c))/(a-b))^(1/2)/a^3/d-1/192*(9*A*b^3-128*B*a^3-24*B*a*b^2-12* 
a^2*b*(13*A+20*C))*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a^2/d+1/96*(3*A*b^2+5 
6*B*a*b+12*a^2*(3*A+4*C))*cos(d*x+c)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a/d 
+1/24*(3*A*b+8*B*a)*cos(d*x+c)^2*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/d+1/4*A 
*cos(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 16.57 (sec) , antiderivative size = 759, normalized size of antiderivative = 1.17 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{48} (9 A b+8 a B) \sin (c+d x)+\frac {\left (48 a^2 A+3 A b^2+56 a b B+48 a^2 C\right ) \sin (2 (c+d x))}{96 a}+\frac {1}{48} (9 A b+8 a B) \sin (3 (c+d x))+\frac {1}{16} a A \sin (4 (c+d x))\right )}{d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}-\frac {\cos ^5(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-a (a+b) \left (-9 A b^3+128 a^3 B+24 a b^2 B+12 a^2 b (13 A+20 C)\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+b (a+b) \left (9 A b^3-6 a b^2 (3 A+4 B)+8 a^3 (9 A+16 B+12 C)+12 a^2 b (7 A+4 (B+3 C))\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-3 \left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \left ((-a+b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )+2 a \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-a \left (-9 A b^3+128 a^3 B+24 a b^2 B+12 a^2 b (13 A+20 C)\right ) (b+a \cos (c+d x)) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sec (c+d x) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{96 a^3 d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2}} \] Input:

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

Output:

(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + 
 d*x]^2)*(((9*A*b + 8*a*B)*Sin[c + d*x])/48 + ((48*a^2*A + 3*A*b^2 + 56*a* 
b*B + 48*a^2*C)*Sin[2*(c + d*x)])/(96*a) + ((9*A*b + 8*a*B)*Sin[3*(c + d*x 
)])/48 + (a*A*Sin[4*(c + d*x)])/16))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2* 
B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (Cos[c + d*x]^5*(a + b*Sec[c + d*x 
])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-(a*(a + b)*(-9*A*b^3 + 
128*a^3*B + 24*a*b^2*B + 12*a^2*b*(13*A + 20*C))*EllipticE[ArcSin[Tan[(c + 
 d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])* 
Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(9*A*b^3 - 6*a*b^2*(3*A + 4*B) + 
 8*a^3*(9*A + 16*B + 12*C) + 12*a^2*b*(7*A + 4*(B + 3*C)))*EllipticF[ArcSi 
n[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[ 
c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 3*(3*A*b^4 + 96*a^3*b*B - 8*a*b^3 
*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*((-a + b)*EllipticF[ArcSin 
[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + 
d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])* 
Sec[(c + d*x)/2]^2)/(a + b)] - a*(-9*A*b^3 + 128*a^3*B + 24*a*b^2*B + 12*a 
^2*b*(13*A + 20*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2) 
^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(96*a^3*d*(b + a*Cos[c + d*x])^2*(A 
 + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c + d*x]*Sec[(c + d*x 
)/2]^2)^(3/2))
 

Rubi [A] (verified)

Time = 3.52 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.02, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\)

\(\Big \downarrow \) 4582

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{8} \left (\frac {1}{3} \int \frac {\cos ^2(c+d x) \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2+3 b (9 A b+16 C b+8 a B) \sec ^2(c+d x)+2 \left (16 B a^2+33 A b a+48 b C a+24 b^2 B\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {\cos ^2(c+d x) \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2+3 b (9 A b+16 C b+8 a B) \sec ^2(c+d x)+2 \left (16 B a^2+33 A b a+48 b C a+24 b^2 B\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \int \frac {12 (3 A+4 C) a^2+56 b B a+3 A b^2+3 b (9 A b+16 C b+8 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (16 B a^2+33 A b a+48 b C a+24 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 4592

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a-2 \left (12 (3 A+4 C) a^2+104 b B a+3 b^2 (19 A+32 C)\right ) \sec (c+d x) a+9 A b^3-b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \sec ^2(c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a-2 \left (12 (3 A+4 C) a^2+104 b B a+3 b^2 (19 A+32 C)\right ) \sec (c+d x) a+9 A b^3-b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int \frac {-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a-2 \left (12 (3 A+4 C) a^2+104 b B a+3 b^2 (19 A+32 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+9 A b^3-b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 4592

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a+9 A b^3\right ) \sec ^2(c+d x)+2 a b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \sec (c+d x)+3 \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a+9 A b^3\right ) \sec ^2(c+d x)+2 a b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \sec (c+d x)+3 \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int \frac {b \left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a+9 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {3 \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )+\left (2 a b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right )-b \left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a+9 A b^3\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {3 \left (16 (3 A+4 C) a^4+96 b B a^3+24 b^2 (A+2 C) a^2-8 b^3 B a+3 A b^4\right )+\left (2 a b \left (12 (3 A+4 C) a^2+56 b B a+3 A b^2\right )-b \left (-128 B a^3-12 b (13 A+20 C) a^2-24 b^2 B a+9 A b^3\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 C)-6 a b^2 (A+4 B)+9 A b^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+3 \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-b \left (-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 C)-6 a b^2 (A+4 B)+9 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-b \left (-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 C)-6 a b^2 (A+4 B)+9 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {b \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \cot (c+d x) \left (-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 C)-6 a b^2 (A+4 B)+9 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {1}{8} \left (\frac {1}{6} \left (\frac {\sin (c+d x) \cos (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\frac {\sin (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {-\frac {2 \sqrt {a+b} \cot (c+d x) \left (-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 C)-6 a b^2 (A+4 B)+9 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}}{4 a}\right )+\frac {(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}\)

Input:

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

Output:

(A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + (((3*A* 
b + 8*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + ( 
((3*A*b^2 + 56*a*b*B + 12*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + 
 d*x]]*Sin[c + d*x])/(2*a*d) - (-1/2*((-2*(a - b)*Sqrt[a + b]*(9*A*b^3 - 1 
28*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Cot[c + d*x]*EllipticE[Arc 
Sin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - S 
ec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2 
*Sqrt[a + b]*(9*A*b^3 - 6*a*b^2*(A + 4*B) - 8*a^3*(9*A + 16*B + 12*C) - 4* 
a^2*b*(39*A + 28*B + 60*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c 
 + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b 
)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (6*Sqrt[a + b]*(3*A*b^4 + 
96*a^3*b*B - 8*a*b^3*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*Cot[c 
+ d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], 
 (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[ 
c + d*x]))/(a - b))])/(a*d))/a + ((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a 
^2*b*(13*A + 20*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d))/(4*a))/6 
)/8
 

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 4271

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a 
 + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) 
*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ 
c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[a^2 - b^2, 0]
 

rule 4319

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* 
x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt 
[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4409

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ 
.) + (a_)], x_Symbol] :> Simp[c   Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + 
Simp[d   Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, 
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
 

rule 4492

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a 
 + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e 
+ f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + 
 f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, 
 f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
 

rule 4546

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C 
)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C   Int[Csc[e + f*x]*(( 
1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A 
, B, C}, x] && NeQ[a^2 - b^2, 0]
 

rule 4582

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e 
 + f*x])^n/(f*n)), x] - Simp[1/(d*n)   Int[(a + b*Csc[e + f*x])^(m - 1)*(d* 
Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs 
c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a 
, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
 

rule 4592

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d 
*Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n)   Int[(a + b*Csc[e + f*x])^m 
*(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* 
Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d 
, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3106\) vs. \(2(601)=1202\).

Time = 15.03 (sec) , antiderivative size = 3107, normalized size of antiderivative = 4.78

method	result	size
default	\(\text {Expression too large to display}\)	\(3107\)

Input:

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

Output:

1/192/d/a^2*((192*cos(d*x+c)^2+384*cos(d*x+c)+192)*C*(1/(a+b)*(b+a*cos(d*x 
+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^4*EllipticF 
(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(-288*cos(d*x+c)^2-576*cos(d* 
x+c)-288)*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(c 
os(d*x+c)+1))^(1/2)*a^4*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b)) 
^(1/2))+(-384*cos(d*x+c)^2-768*cos(d*x+c)-384)*C*(1/(a+b)*(b+a*cos(d*x+c)) 
/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^4*EllipticPi(-c 
sc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+(-72*cos(d*x+c)^2-144*cos(d*x 
+c)-72)*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos 
(d*x+c)+1))^(1/2)*a^3*b*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/ 
2))+(-96*cos(d*x+c)^2-192*cos(d*x+c)-96)*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos( 
d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^3*b*EllipticF(-csc(d* 
x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(144*cos(d*x+c)^2+288*cos(d*x+c)+144) 
*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c) 
+1))^(1/2)*a^4*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))-9*A*b 
^4*cos(d*x+c)*sin(d*x+c)+240*C*a^2*b^2*cos(d*x+c)*sin(d*x+c)+sin(d*x+c)*co 
s(d*x+c)^2*(48*cos(d*x+c)^3+48*cos(d*x+c)^2+72*cos(d*x+c)+72)*a^4*A+sin(d* 
x+c)*cos(d*x+c)^2*(96*cos(d*x+c)+96)*a^4*C+sin(d*x+c)*cos(d*x+c)*(120*cos( 
d*x+c)^3+120*cos(d*x+c)^2+228*cos(d*x+c)+72)*A*a^3*b+sin(d*x+c)*cos(d*x+c) 
*(78*cos(d*x+c)^2+78*cos(d*x+c)+156)*A*a^2*b^2+sin(d*x+c)*cos(d*x+c)*(-...
 

Fricas [F]

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

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

Output:

integral((C*b*cos(d*x + c)^4*sec(d*x + c)^3 + (C*a + B*b)*cos(d*x + c)^4*s 
ec(d*x + c)^2 + A*a*cos(d*x + c)^4 + (B*a + A*b)*cos(d*x + c)^4*sec(d*x + 
c))*sqrt(b*sec(d*x + c) + a), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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