\(\int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\) [1049]

Optimal result

Integrand size = 45, antiderivative size = 441 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {2 \left (15 A b^4-56 a^3 b B+56 a b^3 B+10 a^2 b^2 (A-7 C)-5 a^4 (5 A+7 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{105 a d \sqrt {a+b \sec (c+d x)}}+\frac {2 b^3 C \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (15 A b^3+63 a^3 B+161 a b^2 B+5 a^2 b (29 A+49 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{105 a d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 \left (15 A b^2+56 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (5 A b+7 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)} \] Output:

-2/105*(15*A*b^4-56*B*a^3*b+56*B*a*b^3+10*a^2*b^2*(A-7*C)-5*a^4*(5*A+7*C)) 
*((b+a*cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(a/( 
a+b))^(1/2))*sec(d*x+c)^(1/2)/a/d/(a+b*sec(d*x+c))^(1/2)+2*b^3*C*((b+a*cos 
(d*x+c))/(a+b))^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(a/(a+b))^(1 
/2))*sec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)+2/105*(15*A*b^3+63*B*a^3+16 
1*B*a*b^2+5*a^2*b*(29*A+49*C))*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+ 
b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/a/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d 
*x+c)^(1/2)+2/105*(15*A*b^2+56*B*a*b+5*a^2*(5*A+7*C))*(a+b*sec(d*x+c))^(1/ 
2)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/35*(5*A*b+7*B*a)*(a+b*sec(d*x+c))^(3/2) 
*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/7*A*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d/s 
ec(d*x+c)^(5/2)
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \] Input:

Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x] 
^2))/Sec[c + d*x]^(7/2),x]
 

Output:

Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x] 
^2))/Sec[c + d*x]^(7/2), x]
 

Rubi [A] (verified)

Time = 4.82 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.02, number of steps used = 29, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.644, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4582, 27, 3042, 4596, 3042, 4346, 3042, 3286, 3042, 3284, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}

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.

\(\displaystyle \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx\)

\(\Big \downarrow \) 4582

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {\sqrt {a+b \sec (c+d x)} \left (5 (5 A+7 C) a^2+56 b B a+15 A b^2+35 b^2 C \sec ^2(c+d x)+\left (21 B a^2+40 A b a+70 b C a+35 b^2 B\right ) \sec (c+d x)\right )}{2 \sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\sqrt {a+b \sec (c+d x)} \left (5 (5 A+7 C) a^2+56 b B a+15 A b^2+35 b^2 C \sec ^2(c+d x)+\left (21 B a^2+40 A b a+70 b C a+35 b^2 B\right ) \sec (c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (5 A+7 C) a^2+56 b B a+15 A b^2+35 b^2 C \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (21 B a^2+40 A b a+70 b C a+35 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3+105 b^3 C \sec ^2(c+d x)+\left (5 (5 A+7 C) a^3+119 b B a^2+45 b^2 (3 A+7 C) a+105 b^3 B\right ) \sec (c+d x)}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3+105 b^3 C \sec ^2(c+d x)+\left (5 (5 A+7 C) a^3+119 b B a^2+45 b^2 (3 A+7 C) a+105 b^3 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3+105 b^3 C \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (5 (5 A+7 C) a^3+119 b B a^2+45 b^2 (3 A+7 C) a+105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4596

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3+\left (5 (5 A+7 C) a^3+119 b B a^2+45 b^2 (3 A+7 C) a+105 b^3 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx+105 b^3 C \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3+\left (5 (5 A+7 C) a^3+119 b B a^2+45 b^2 (3 A+7 C) a+105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+105 b^3 C \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4346

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3+\left (5 (5 A+7 C) a^3+119 b B a^2+45 b^2 (3 A+7 C) a+105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {105 b^3 C \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3+\left (5 (5 A+7 C) a^3+119 b B a^2+45 b^2 (3 A+7 C) a+105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {105 b^3 C \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3286

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3+\left (5 (5 A+7 C) a^3+119 b B a^2+45 b^2 (3 A+7 C) a+105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {105 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3+\left (5 (5 A+7 C) a^3+119 b B a^2+45 b^2 (3 A+7 C) a+105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {105 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3+\left (5 (5 A+7 C) a^3+119 b B a^2+45 b^2 (3 A+7 C) a+105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4523

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}-\frac {\left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}-\frac {\left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4343

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {\left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {\left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {\left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {\left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {\left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4345

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {\sqrt {\sec (c+d x)} \left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {\sqrt {\sec (c+d x)} \left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {\sqrt {\sec (c+d x)} \left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {\sqrt {\sec (c+d x)} \left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{3} \left (\frac {2 \left (63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {2 \sqrt {\sec (c+d x)} \left (-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}+\frac {210 b^3 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\right )\right )+\frac {2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

Input:

Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/S 
ec[c + d*x]^(7/2),x]
 

Output:

(2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ( 
(2*(5*A*b + 7*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d 
*x]^(3/2)) + (((-2*(15*A*b^4 - 56*a^3*b*B + 56*a*b^3*B + 10*a^2*b^2*(A - 7 
*C) - 5*a^4*(5*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + 
 d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) 
 + (210*b^3*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2 
, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(15 
*A*b^3 + 63*a^3*B + 161*a*b^2*B + 5*a^2*b*(29*A + 49*C))*EllipticE[(c + d* 
x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d* 
x])/(a + b)]*Sqrt[Sec[c + d*x]]))/3 + (2*(15*A*b^2 + 56*a*b*B + 5*a^2*(5*A 
 + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]))/ 
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 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 3284

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

rule 3286

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt 
[c + d*Sin[e + f*x]]   Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + 
 d))*Sin[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] && NeQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]
 

rule 4343

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

rule 4345

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

rule 4346

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_. 
) + (a_)], x_Symbol] :> Simp[d*Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x 
]]/Sqrt[a + b*Csc[e + f*x]])   Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]] 
), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4523

Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d 
_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a   I 
nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) 
/(a*d)   Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ 
[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && 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 4596

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) 
+ (a_)]), x_Symbol] :> Simp[C/d^2   Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*C 
sc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[ 
a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - 
 b^2, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 44.33 (sec) , antiderivative size = 3106, normalized size of antiderivative = 7.04

method	result	size
default	\(\text {Expression too large to display}\)	\(3106\)
parts	\(\text {Expression too large to display}\)	\(3169\)

Input:

int((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2 
),x,method=_RETURNVERBOSE)
 

Output:

2/105/d/((a-b)/(a+b))^(1/2)/a*((63*cos(d*x+c)^2+126*cos(d*x+c)+63)*B*(1/(c 
os(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*b* 
EllipticE(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2 
))+(-63*cos(d*x+c)^2-126*cos(d*x+c)-63)*B*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b 
)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^4*EllipticE(((a-b)/(a+b))^(1/2) 
*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(-25*cos(d*x+c)^2-50*cos(d 
*x+c)-25)*A*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c) 
+1))^(1/2)*a^4*EllipticF(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a 
+b)/(a-b))^(1/2))+(63*cos(d*x+c)^2+126*cos(d*x+c)+63)*B*(1/(cos(d*x+c)+1)) 
^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^4*EllipticF(((a-b 
)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(-35*cos(d*x 
+c)^2-70*cos(d*x+c)-35)*C*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c 
))/(cos(d*x+c)+1))^(1/2)*a^4*EllipticF(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+co 
t(d*x+c)),(-(a+b)/(a-b))^(1/2))+sin(d*x+c)*cos(d*x+c)*(15*cos(d*x+c)^3+15* 
cos(d*x+c)^2+25*cos(d*x+c)+25)*A*((a-b)/(a+b))^(1/2)*a^4+sin(d*x+c)*cos(d* 
x+c)*(21*cos(d*x+c)^2+21*cos(d*x+c)+63)*B*((a-b)/(a+b))^(1/2)*a^4+sin(d*x+ 
c)*cos(d*x+c)*(35*cos(d*x+c)+35)*C*((a-b)/(a+b))^(1/2)*a^4+(161*cos(d*x+c) 
^2+322*cos(d*x+c)+161)*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*( 
1/(cos(d*x+c)+1))^(1/2)*a^2*b^2*EllipticF(((a-b)/(a+b))^(1/2)*(-csc(d*x+c) 
+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(-105*cos(d*x+c)^2-210*cos(d*x+c)-10...
 

Fricas [F]

\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c 
)^(7/2),x, algorithm="fricas")
 

Output:

integral((C*b^2*sec(d*x + c)^4 + (2*C*a*b + B*b^2)*sec(d*x + c)^3 + A*a^2 
+ (C*a^2 + 2*B*a*b + A*b^2)*sec(d*x + c)^2 + (B*a^2 + 2*A*a*b)*sec(d*x + c 
))*sqrt(b*sec(d*x + c) + a)/sec(d*x + c)^(7/2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x 
+c)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c 
)^(7/2),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)/sec(d*x + c)^(7/2), x)
 

Giac [F]

\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c 
)^(7/2),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)/sec(d*x + c)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \] Input:

int(((a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/( 
1/cos(c + d*x))^(7/2),x)
 

Output:

int(((a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/( 
1/cos(c + d*x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )^{4}}d x \right ) a^{3}+3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )^{3}}d x \right ) a^{2} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{2} c +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )^{2}}d x \right ) a \,b^{2}+2 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )}d x \right ) a b c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )}d x \right ) b^{3}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}d x \right ) b^{2} c \] Input:

int((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2 
),x)
 

Output:

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