\(\int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) [406]

Optimal result

Integrand size = 33, antiderivative size = 254 \[ \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (7 a^2 A+9 A b^2+18 a b B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (7 b^2 B+5 a (2 A b+a B)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a (2 A b+a B) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 a^2 A+9 A b^2+18 a b B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 b^2 B+5 a (2 A b+a B)\right ) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}} \] Output:

2/15*(7*A*a^2+9*A*b^2+18*B*a*b)*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*(7*b^2*B+5*a*(2*A*b+B*a))*cos(d*x+c)^ 
(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+2/9*a^2*A* 
sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/7*a*(2*A*b+B*a)*sin(d*x+c)/d/sec(d*x+c)^(5 
/2)+2/45*(7*A*a^2+9*A*b^2+18*B*a*b)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/21*(7* 
b^2*B+5*a*(2*A*b+B*a))*sin(d*x+c)/d/sec(d*x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 4.78 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {\sqrt {\sec (c+d x)} \left (168 \left (7 a^2 A+9 A b^2+18 a b B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+120 \left (10 a A b+5 a^2 B+7 b^2 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (7 \left (43 a^2 A+36 A b^2+72 a b B\right ) \cos (c+d x)+5 \left (156 a A b+78 a^2 B+84 b^2 B+18 a (2 A b+a B) \cos (2 (c+d x))+7 a^2 A \cos (3 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{1260 d} \] Input:

Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2) 
,x]
 

Output:

(Sqrt[Sec[c + d*x]]*(168*(7*a^2*A + 9*A*b^2 + 18*a*b*B)*Sqrt[Cos[c + d*x]] 
*EllipticE[(c + d*x)/2, 2] + 120*(10*a*A*b + 5*a^2*B + 7*b^2*B)*Sqrt[Cos[c 
 + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*(43*a^2*A + 36*A*b^2 + 72*a*b*B)*C 
os[c + d*x] + 5*(156*a*A*b + 78*a^2*B + 84*b^2*B + 18*a*(2*A*b + a*B)*Cos[ 
2*(c + d*x)] + 7*a^2*A*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.91, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 4512, 27, 3042, 4535, 3042, 4256, 3042, 4258, 3042, 3119, 4533, 3042, 4256, 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.

\(\displaystyle \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4512

\(\displaystyle \frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {2}{9} \int -\frac {9 b^2 B \sec ^2(c+d x)+\left (7 A a^2+18 b B a+9 A b^2\right ) \sec (c+d x)+9 a (2 A b+a B)}{2 \sec ^{\frac {7}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4535

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4256

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{9} \left (\int \frac {9 b^2 B \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a (2 A b+a B)}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\left (7 a^2 A+18 a b B+9 A b^2\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4533

\(\displaystyle \frac {1}{9} \left (\frac {9}{7} \left (5 a (a B+2 A b)+7 b^2 B\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx+\left (7 a^2 A+18 a b B+9 A b^2\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {9}{7} \left (5 a (a B+2 A b)+7 b^2 B\right ) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\left (7 a^2 A+18 a b B+9 A b^2\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {1}{9} \left (\frac {9}{7} \left (5 a (a B+2 A b)+7 b^2 B\right ) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\left (7 a^2 A+18 a b B+9 A b^2\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {9}{7} \left (5 a (a B+2 A b)+7 b^2 B\right ) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\left (7 a^2 A+18 a b B+9 A b^2\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{9} \left (\frac {9}{7} \left (5 a (a B+2 A b)+7 b^2 B\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\left (7 a^2 A+18 a b B+9 A b^2\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {9}{7} \left (5 a (a B+2 A b)+7 b^2 B\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\left (7 a^2 A+18 a b B+9 A b^2\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {18 a (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{9} \left (\left (7 a^2 A+18 a b B+9 A b^2\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )+\frac {9}{7} \left (5 a (a B+2 A b)+7 b^2 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {18 a (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 A \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

Input:

Int[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]
 

Output:

(2*a^2*A*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((18*a*(2*A*b + a*B)*Sin 
[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (7*a^2*A + 9*A*b^2 + 18*a*b*B)*((6*S 
qrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2 
*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))) + (9*(7*b^2*B + 5*a*(2*A*b + a*B) 
)*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3* 
d) + (2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])))/7)/9
 

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 4256

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n)   Int[(b*Csc[c 
+ d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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 4512

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

rule 4533

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs. \(2(229)=458\).

Time = 27.25 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.40

method	result	size
default	\(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-1120 A \,a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (2240 A \,a^{2}+1440 A a b +720 B \,a^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2072 A \,a^{2}-2160 A a b -504 A \,b^{2}-1080 B \,a^{2}-1008 B a b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (952 A \,a^{2}+1680 A a b +504 A \,b^{2}+840 B \,a^{2}+1008 B a b +420 b^{2} B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-168 A \,a^{2}-480 A a b -126 A \,b^{2}-240 B \,a^{2}-252 B a b -210 b^{2} B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+150 A a b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}-189 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}+75 B \,a^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+105 b^{2} B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-378 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a b \right )}{315 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(610\)
parts	\(\text {Expression too large to display}\)	\(820\)

Input:

int((a+b*sec(d*x+c))^2*(A+B*sec(d*x+c))/sec(d*x+c)^(9/2),x,method=_RETURNV 
ERBOSE)
 

Output:

-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*A*a^ 
2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+(2240*A*a^2+1440*A*a*b+720*B*a^ 
2)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-2072*A*a^2-2160*A*a*b-504*A*b 
^2-1080*B*a^2-1008*B*a*b)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(952*A*a 
^2+1680*A*a*b+504*A*b^2+840*B*a^2+1008*B*a*b+420*B*b^2)*sin(1/2*d*x+1/2*c) 
^4*cos(1/2*d*x+1/2*c)+(-168*A*a^2-480*A*a*b-126*A*b^2-240*B*a^2-252*B*a*b- 
210*B*b^2)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+150*A*a*b*(sin(1/2*d*x+ 
1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2 
*c),2^(1/2))-147*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1) 
^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2-189*A*(sin(1/2*d*x+1/2*c) 
^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^ 
(1/2))*b^2+75*B*a^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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+105*b^2*B*(sin(1/2*d*x+1/2*c 
)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2 
^(1/2))-378*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2 
)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b)/(-2*sin(1/2*d*x+1/2*c)^4+sin( 
1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2 
)/d
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, B a^{2} + 10 i \, A a b + 7 i \, B b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, B a^{2} - 10 i \, A a b - 7 i \, B b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-7 i \, A a^{2} - 18 i \, B a b - 9 i \, A b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (7 i \, A a^{2} + 18 i \, B a b + 9 i \, A b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, A a^{2} \cos \left (d x + c\right )^{4} + 45 \, {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (7 \, A a^{2} + 18 \, B a b + 9 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, B a^{2} + 10 \, A a b + 7 \, B b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \] Input:

integrate((a+b*sec(d*x+c))^2*(A+B*sec(d*x+c))/sec(d*x+c)^(9/2),x, algorith 
m="fricas")
 

Output:

-1/315*(15*sqrt(2)*(5*I*B*a^2 + 10*I*A*a*b + 7*I*B*b^2)*weierstrassPInvers 
e(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-5*I*B*a^2 - 10*I*A* 
a*b - 7*I*B*b^2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) 
 + 21*sqrt(2)*(-7*I*A*a^2 - 18*I*B*a*b - 9*I*A*b^2)*weierstrassZeta(-4, 0, 
 weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*( 
7*I*A*a^2 + 18*I*B*a*b + 9*I*A*b^2)*weierstrassZeta(-4, 0, weierstrassPInv 
erse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(35*A*a^2*cos(d*x + c)^4 + 
 45*(B*a^2 + 2*A*a*b)*cos(d*x + c)^3 + 7*(7*A*a^2 + 18*B*a*b + 9*A*b^2)*co 
s(d*x + c)^2 + 15*(5*B*a^2 + 10*A*a*b + 7*B*b^2)*cos(d*x + c))*sin(d*x + c 
)/sqrt(cos(d*x + c)))/d
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((a+b*sec(d*x+c))**2*(A+B*sec(d*x+c))/sec(d*x+c)**(9/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((a+b*sec(d*x+c))^2*(A+B*sec(d*x+c))/sec(d*x+c)^(9/2),x, algorith 
m="maxima")
 

Output:

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^2/sec(d*x + c)^(9/2), 
x)
 

Giac [F]

\[ \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((a+b*sec(d*x+c))^2*(A+B*sec(d*x+c))/sec(d*x+c)^(9/2),x, algorith 
m="giac")
 

Output:

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^2/sec(d*x + c)^(9/2), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \] Input:

int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^2)/(1/cos(c + d*x))^(9/2),x 
)
 

Output:

int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^2)/(1/cos(c + d*x))^(9/2), 
x)
 

Reduce [F]

\[ \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) a^{3}+3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) a^{2} b +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) a \,b^{2}+\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) b^{3} \] Input:

int((a+b*sec(d*x+c))^2*(A+B*sec(d*x+c))/sec(d*x+c)^(9/2),x)
 

Output:

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