\(\int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [955]

Optimal result

Integrand size = 43, antiderivative size = 549 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (54 a b B+3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 b d}+\frac {\sqrt {a+b} \left (3 b^2 (11 A+16 (B-C))+4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 d}-\frac {\sqrt {a+b} \left (5 A b^3+8 a^3 B+30 a b^2 B+20 a^2 b (A+2 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a d}+\frac {\left (15 A b^2+42 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {(5 A b+6 a B) \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d} \] Output:

1/24*(a-b)*(a+b)^(1/2)*(54*B*a*b+3*b^2*(11*A-16*C)+8*a^2*(2*A+3*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)/b/d+1/24*(a+b) 
^(1/2)*(3*b^2*(11*A+16*B-16*C)+4*a^2*(4*A+3*B+6*C)+2*a*b*(13*A+27*B+72*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)/d-1/8*( 
a+b)^(1/2)*(5*A*b^3+8*B*a^3+30*B*a*b^2+20*a^2*b*(A+2*C))*cot(d*x+c)*Ellipt 
icPi((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/d+1/24*(15*A*b 
^2+42*B*a*b+8*a^2*(2*A+3*C))*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/d+1/12*(5*A 
*b+6*B*a)*cos(d*x+c)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/3*A*cos(d*x+c)^ 
2*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(5347\) vs. \(2(549)=1098\).

Time = 24.32 (sec) , antiderivative size = 5347, normalized size of antiderivative = 9.74 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:

Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + 
C*Sec[c + d*x]^2),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.70 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.395, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4582, 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.

\(\displaystyle \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\)

\(\Big \downarrow \) 4582

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {1}{2} \cos (c+d x) \sqrt {a+b \sec (c+d x)} \left (8 (2 A+3 C) a^2+42 b B a+15 A b^2-3 b (3 A b-8 C b+2 a B) \sec ^2(c+d x)+2 \left (6 B a^2+b (11 A+24 C) a+12 b^2 B\right ) \sec (c+d x)\right )dx+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \cos (c+d x) \sqrt {a+b \sec (c+d x)} \left (8 (2 A+3 C) a^2+42 b B a+15 A b^2-3 b (3 A b-8 C b+2 a B) \sec ^2(c+d x)+2 \left (6 B a^2+b (11 A+24 C) a+12 b^2 B\right ) \sec (c+d x)\right )dx+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (8 (2 A+3 C) a^2+42 b B a+15 A b^2-3 b (3 A b-8 C b+2 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (6 B a^2+b (11 A+24 C) a+12 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\int \frac {-b \left (8 (2 A+3 C) a^2+54 b B a+3 b^2 (11 A-16 C)\right ) \sec ^2(c+d x)+2 b \left (6 B a^2+b (13 A+72 C) a+24 b^2 B\right ) \sec (c+d x)+3 \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {-b \left (8 (2 A+3 C) a^2+54 b B a+3 b^2 (11 A-16 C)\right ) \sec ^2(c+d x)+2 b \left (6 B a^2+b (13 A+72 C) a+24 b^2 B\right ) \sec (c+d x)+3 \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {-b \left (8 (2 A+3 C) a^2+54 b B a+3 b^2 (11 A-16 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (6 B a^2+b (13 A+72 C) a+24 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\int \frac {3 \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )+\left (b \left (8 (2 A+3 C) a^2+54 b B a+3 b^2 (11 A-16 C)\right )+2 b \left (6 B a^2+b (13 A+72 C) a+24 b^2 B\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-b \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx\right )+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (\int \frac {3 \left (8 B a^3+20 b (A+2 C) a^2+30 b^2 B a+5 A b^3\right )+\left (b \left (8 (2 A+3 C) a^2+54 b B a+3 b^2 (11 A-16 C)\right )+2 b \left (6 B a^2+b (13 A+72 C) a+24 b^2 B\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (-b \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)+3 b^2 (11 A+16 (B-C))\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+3 \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx\right )+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (b \left (4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)+3 b^2 (11 A+16 (B-C))\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (b \left (4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)+3 b^2 (11 A+16 (B-C))\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}\right )+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {1}{2} \left (-b \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \cot (c+d x) \left (4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)+3 b^2 (11 A+16 (B-C))\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}\right )+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{d}+\frac {1}{2} \left (\frac {2 \sqrt {a+b} \cot (c+d x) \left (4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)+3 b^2 (11 A+16 (B-C))\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}\right )\right )+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}\)

Input:

Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[ 
c + d*x]^2),x]
 

Output:

(A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d) + (((5*A* 
b + 6*a*B)*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d) + ( 
((2*(a - b)*Sqrt[a + b]*(54*a*b*B + 3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C 
))*Cot[c + d*x]*EllipticE[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))])/(b*d) + (2*Sqrt[a + b]*(3*b^2*(11*A + 16*(B - C)) + 4*a 
^2*(4*A + 3*B + 6*C) + 2*a*b*(13*A + 27*B + 72*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]*(5*A*b^3 + 8*a^3*B + 30*a*b^2*B + 20*a^2*b*(A + 2*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))/2 + ((15*A*b^2 + 42*a*b*B + 8*a^2*(2*A + 3*C))* 
Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d)/4)/6
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2855\) vs. \(2(503)=1006\).

Time = 216.50 (sec) , antiderivative size = 2856, normalized size of antiderivative = 5.20

method	result	size
default	\(\text {Expression too large to display}\)	\(2856\)

Input:

int(cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, 
method=_RETURNVERBOSE)
 

Output:

-1/24/d*(-48*C*a*b^2*cos(d*x+c)*sin(d*x+c)+48*(cos(d*x+c)^2+2*cos(d*x+c)+1 
)*B*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c 
)+1))^(1/2)*a^3*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+ 
24*(cos(d*x+c)^2+2*cos(d*x+c)+1)*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a 
+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*EllipticE(-csc(d*x+c)+cot(d 
*x+c),((a-b)/(a+b))^(1/2))+48*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*C*(cos(d*x+c) 
/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^3 
*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+48*(cos(d*x+c)^2+2* 
cos(d*x+c)+1)*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c) 
)/(cos(d*x+c)+1))^(1/2)*b^3*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b)) 
^(1/2))+sin(d*x+c)*cos(d*x+c)*(-59*cos(d*x+c)-26)*a*A*b^2+12*(cos(d*x+c)^2 
+2*cos(d*x+c)+1)*B*(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^2*b*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/( 
a+b))^(1/2))+30*(cos(d*x+c)^2+2*cos(d*x+c)+1)*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)*b^3*EllipticPi(-cs 
c(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+2*sin(d*x+c)*cos(d*x+c)*(-8-17 
*cos(d*x+c)^2-17*cos(d*x+c))*A*a^2*b+6*sin(d*x+c)*cos(d*x+c)*(-2-11*cos(d* 
x+c))*B*a^2*b-24*C*a^2*b*cos(d*x+c)*sin(d*x+c)+33*(cos(d*x+c)^2+2*cos(d*x+ 
c)+1)*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*b^2*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1...
 

Fricas [F]

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

integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**3*(a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+ 
c)**2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(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)^(5/ 
2)*cos(d*x + c)^3, x)
 

Giac [F]

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

integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(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)^(5/ 
2)*cos(d*x + c)^3, x)
 

Mupad [F(-1)]

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

int(cos(c + d*x)^3*(a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos( 
c + d*x)^2),x)
                                                                                    
                                                                                    
 

Output:

int(cos(c + d*x)^3*(a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos( 
c + d*x)^2), x)
 

Reduce [F]

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

int(cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(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)**3*sec(c + d*x)**4,x)*b**2*c + 2 
*int(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**3,x)*a*b*c + i 
nt(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**3,x)*b**3 + int( 
sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**2,x)*a**2*c + 3*int 
(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**2,x)*a*b**2 + 3*in 
t(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x),x)*a**2*b + int(sq 
rt(sec(c + d*x)*b + a)*cos(c + d*x)**3,x)*a**3