\(\int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx\) [408]

Optimal result

Integrand size = 33, antiderivative size = 295 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=-\frac {2 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b^2 (7 A b+11 a B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b B \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d} \] Output:

-2/5*(15*A*a^2*b+3*A*b^3+5*B*a^3+9*B*a*b^2)*cos(d*x+c)^(1/2)*EllipticE(sin 
(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2)/d+2/21*(21*A*a^3+21*A*a*b^2+21*B 
*a^2*b+5*B*b^3)*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*se 
c(d*x+c)^(1/2)/d+2/5*(15*A*a^2*b+3*A*b^3+5*B*a^3+9*B*a*b^2)*sec(d*x+c)^(1/ 
2)*sin(d*x+c)/d+2/21*b*(21*A*a*b+18*B*a^2+5*B*b^2)*sec(d*x+c)^(3/2)*sin(d* 
x+c)/d+2/35*b^2*(7*A*b+11*B*a)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/7*b*B*sec(d 
*x+c)^(3/2)*(a+b*sec(d*x+c))^2*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 3.89 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.76 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {2 \sqrt {\sec (c+d x)} \left (-21 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+21 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \sin (c+d x)+5 b \left (21 a A b+21 a^2 B+5 b^2 B\right ) \tan (c+d x)+21 b^2 (A b+3 a B) \sec (c+d x) \tan (c+d x)+15 b^3 B \sec ^2(c+d x) \tan (c+d x)\right )}{105 d} \] Input:

Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x 
]
 

Output:

(2*Sqrt[Sec[c + d*x]]*(-21*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*Sq 
rt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(21*a^3*A + 21*a*A*b^2 + 21 
*a^2*b*B + 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 21*(15* 
a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*Sin[c + d*x] + 5*b*(21*a*A*b + 21 
*a^2*B + 5*b^2*B)*Tan[c + d*x] + 21*b^2*(A*b + 3*a*B)*Sec[c + d*x]*Tan[c + 
 d*x] + 15*b^3*B*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d)
 

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.94, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.576, Rules used = {3042, 4514, 27, 3042, 4564, 27, 3042, 4535, 3042, 4255, 3042, 4258, 3042, 3119, 4534, 3042, 4258, 3042, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4514

\(\displaystyle \frac {2}{7} \int \frac {1}{2} \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (b (7 A b+11 a B) \sec ^2(c+d x)+\left (5 B b^2+7 a (2 A b+a B)\right ) \sec (c+d x)+a (7 a A+b B)\right )dx+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (b (7 A b+11 a B) \sec ^2(c+d x)+\left (5 B b^2+7 a (2 A b+a B)\right ) \sec (c+d x)+a (7 a A+b B)\right )dx+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b (7 A b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (5 B b^2+7 a (2 A b+a B)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (7 a A+b B)\right )dx+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 4564

\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {\sec (c+d x)} \left (5 (7 a A+b B) a^2+5 b \left (18 B a^2+21 A b a+5 b^2 B\right ) \sec ^2(c+d x)+7 \left (5 B a^3+15 A b a^2+9 b^2 B a+3 A b^3\right ) \sec (c+d x)\right )dx+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {\sec (c+d x)} \left (5 (7 a A+b B) a^2+5 b \left (18 B a^2+21 A b a+5 b^2 B\right ) \sec ^2(c+d x)+7 \left (5 B a^3+15 A b a^2+9 b^2 B a+3 A b^3\right ) \sec (c+d x)\right )dx+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 a A+b B) a^2+5 b \left (18 B a^2+21 A b a+5 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (5 B a^3+15 A b a^2+9 b^2 B a+3 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 4535

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\sec (c+d x)} \left (5 (7 a A+b B) a^2+5 b \left (18 B a^2+21 A b a+5 b^2 B\right ) \sec ^2(c+d x)\right )dx+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \int \sec ^{\frac {3}{2}}(c+d x)dx\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 a A+b B) a^2+5 b \left (18 B a^2+21 A b a+5 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 a A+b B) a^2+5 b \left (18 B a^2+21 A b a+5 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 a A+b B) a^2+5 b \left (18 B a^2+21 A b a+5 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 a A+b B) a^2+5 b \left (18 B a^2+21 A b a+5 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 a A+b B) a^2+5 b \left (18 B a^2+21 A b a+5 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (7 a A+b B) a^2+5 b \left (18 B a^2+21 A b a+5 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 4534

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) \int \sqrt {\sec (c+d x)}dx+\frac {10 b \left (18 a^2 B+21 a A b+5 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 b \left (18 a^2 B+21 a A b+5 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {10 b \left (18 a^2 B+21 a A b+5 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {5}{3} \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 b \left (18 a^2 B+21 a A b+5 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {10 b \left (18 a^2 B+21 a A b+5 b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {10 \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+7 \left (5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b B \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\)

Input:

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

Output:

(2*b*B*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + ((2 
*b^2*(7*A*b + 11*a*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + ((10*(21*a^ 
3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + 
 d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (10*b*(21*a*A*b + 18*a^2*B + 5*b^2 
*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + 7*(15*a^2*A*b + 3*A*b^3 + 5*a 
^3*B + 9*a*b^2*B)*((-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[S 
ec[c + d*x]])/d + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d))/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 4255

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) 
  Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] 
&& IntegerQ[2*n]
 

rule 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] 
)^n*Sin[c + d*x]^n   Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && 
 EqQ[n^2, 1/4]
 

rule 4514

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

rule 4534

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

rule 4535

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* 
(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b   Int[(b*Cs 
c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) 
, x] /; FreeQ[{b, e, f, A, B, C, m}, x]
 

rule 4564

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_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ 
n/(f*(n + 2))), x] + Simp[1/(n + 2)   Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 
2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( 
n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] 
&&  !LtQ[n, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(916\) vs. \(2(270)=540\).

Time = 12.72 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.11

method	result	size
default	\(\text {Expression too large to display}\)	\(917\)
parts	\(\text {Expression too large to display}\)	\(1175\)

Input:

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

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*a^3*A*(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*B*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^2*(3*A*b+B*a)/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*cos(1/2*d*x+1/ 
2*c)*sin(1/2*d*x+1/2*c)^2-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d 
*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2))+2/5*b^2*(A*b+3*B*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)/s 
in(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*Ellipti 
cE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x 
+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+ 
1/2*c)^4+12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^( 
1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2+8*cos(1/2*d*x+1 
/2*c)*sin(1/2*d*x+1/2*c)^2-3*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/ 
2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2))*(-2*sin(1/2*d*x...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.23 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=-\frac {5 \, \sqrt {2} {\left (21 i \, A a^{3} + 21 i \, B a^{2} b + 21 i \, A a b^{2} + 5 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-21 i \, A a^{3} - 21 i \, B a^{2} b - 21 i \, A a b^{2} - 5 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, B a^{3} + 15 i \, A a^{2} b + 9 i \, B a b^{2} + 3 i \, A b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-5 i \, B a^{3} - 15 i \, A a^{2} b - 9 i \, B a b^{2} - 3 i \, A b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (15 \, B b^{3} + 21 \, {\left (5 \, B a^{3} + 15 \, A a^{2} b + 9 \, B a b^{2} + 3 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (21 \, B a^{2} b + 21 \, A a b^{2} + 5 \, B b^{3}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \] Input:

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

Output:

-1/105*(5*sqrt(2)*(21*I*A*a^3 + 21*I*B*a^2*b + 21*I*A*a*b^2 + 5*I*B*b^3)*c 
os(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 
5*sqrt(2)*(-21*I*A*a^3 - 21*I*B*a^2*b - 21*I*A*a*b^2 - 5*I*B*b^3)*cos(d*x 
+ c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt 
(2)*(5*I*B*a^3 + 15*I*A*a^2*b + 9*I*B*a*b^2 + 3*I*A*b^3)*cos(d*x + c)^3*we 
ierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + 
 c))) + 21*sqrt(2)*(-5*I*B*a^3 - 15*I*A*a^2*b - 9*I*B*a*b^2 - 3*I*A*b^3)*c 
os(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + 
c) - I*sin(d*x + c))) - 2*(15*B*b^3 + 21*(5*B*a^3 + 15*A*a^2*b + 9*B*a*b^2 
 + 3*A*b^3)*cos(d*x + c)^3 + 5*(21*B*a^2*b + 21*A*a*b^2 + 5*B*b^3)*cos(d*x 
 + c)^2 + 21*(3*B*a*b^2 + A*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + 
 c)))/(d*cos(d*x + c)^3)
 

Sympy [F(-1)]

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

integrate(sec(d*x+c)**(1/2)*(a+b*sec(d*x+c))**3*(A+B*sec(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorith 
m="maxima")
 

Output:

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3*sqrt(sec(d*x + c)), 
x)
 

Giac [F]

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

integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorith 
m="giac")
 

Output:

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3*sqrt(sec(d*x + c)), 
x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(sec(c + d*x)),x)*a**4 + int(sqrt(sec(c + d*x))*sec(c + d*x)**4,x) 
*b**4 + 4*int(sqrt(sec(c + d*x))*sec(c + d*x)**3,x)*a*b**3 + 6*int(sqrt(se 
c(c + d*x))*sec(c + d*x)**2,x)*a**2*b**2 + 4*int(sqrt(sec(c + d*x))*sec(c 
+ d*x),x)*a**3*b