\(\int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx\) [600]

Optimal result

Integrand size = 35, antiderivative size = 427 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\frac {2 \left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d} \]

[Out]

2/315*(a^2-b^2)*(39*A*a^2*b+8*A*b^3+75*B*a^3-18*B*a*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellip
ticF(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)/a^3/d/cos(d*x+c)^(1/2)/(a+b*se
c(d*x+c))^(1/2)+2/315*(49*A*a^2+3*A*b^2+72*B*a*b)*cos(d*x+c)^(3/2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d+2/63*
(10*A*b+9*B*a)*cos(d*x+c)^(5/2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d+2/9*a*A*cos(d*x+c)^(7/2)*sin(d*x+c)*(a+b*s
ec(d*x+c))^(1/2)/d+2/315*(88*A*a^2*b-4*A*b^3+75*B*a^3+9*B*a*b^2)*sin(d*x+c)*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^
(1/2)/a^2/d+2/315*(147*A*a^4+33*A*a^2*b^2+8*A*b^4+246*B*a^3*b-18*B*a*b^3)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2
*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/a^3/
d/((b+a*cos(d*x+c))/(a+b))^(1/2)

Rubi [A] (verified)

Time = 1.96 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3034, 4110, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{315 a d}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}{315 a^2 d}+\frac {2 \left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (9 a B+10 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{63 d}+\frac {2 a A \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d} \]

[In]

Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]

[Out]

(2*(a^2 - b^2)*(39*a^2*A*b + 8*A*b^3 + 75*a^3*B - 18*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c
+ d*x)/2, (2*a)/(a + b)])/(315*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 33*a^2*A*b
^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*S
ec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B
)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*C
os[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(10*A*b + 9*a*B)*Cos[c + d*x]^(5/2)*Sq
rt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])
/(9*d)

Rule 2732

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 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/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 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 3034

Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.)*((g_.)*sin[(e_.
) + (f_.)*(x_)])^(p_.), x_Symbol] :> Dist[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p, Int[(a + b*Csc[e + f*x])^m*((
c + d*Csc[e + f*x])^n/(g*Csc[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d
, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] && IntegerQ[n])

Rule 3941

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

Rule 3943

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

Rule 4110

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

Rule 4120

Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(
b_.) + (a_)]), x_Symbol] :> Dist[A/a, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Dist[(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] && Ne
Q[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]

Rule 4189

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

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}-\frac {1}{9} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{2} a (10 A b+9 a B)-\frac {1}{2} \left (7 a^2 A+9 A b^2+18 a b B\right ) \sec (c+d x)-\frac {3}{2} b (2 a A+3 b B) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} a \left (49 a^2 A+3 A b^2+72 a b B\right )+\frac {1}{4} a \left (92 a A b+45 a^2 B+63 b^2 B\right ) \sec (c+d x)+a b (10 A b+9 a B) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{63 a} \\ & = \frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}-\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3}{8} a \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right )-\frac {1}{8} a^2 \left (147 a^2 A+209 A b^2+396 a b B\right ) \sec (c+d x)-\frac {1}{4} a b \left (49 a^2 A+3 A b^2+72 a b B\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{315 a^2} \\ & = \frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{16} a \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right )+\frac {3}{16} a^2 \left (186 a^2 A b+2 A b^3+75 a^3 B+153 a b^2 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{945 a^3} \\ & = \frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}+\frac {\left (\left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 a^3}+\frac {\left (\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{315 a^3} \\ & = \frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}+\frac {\left (\left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{315 a^3 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{315 a^3 \sqrt {b+a \cos (c+d x)}} \\ & = \frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d}+\frac {\left (\left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{315 a^3 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{315 a^3 \sqrt {\frac {b+a \cos (c+d x)}{a+b}}} \\ & = \frac {2 \left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\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 )}{315 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 (10 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 21.92 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.26 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\frac {\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {\left (402 a^2 A b-16 A b^3+345 a^3 B+36 a b^2 B\right ) \sin (c+d x)}{630 a^2}+\frac {\left (133 a^2 A+6 A b^2+144 a b B\right ) \sin (2 (c+d x))}{630 a}+\frac {1}{126} (10 A b+9 a B) \sin (3 (c+d x))+\frac {1}{36} a A \sin (4 (c+d x))\right )}{d (b+a \cos (c+d x))}-\frac {2 \cos ^{\frac {3}{2}}(c+d x) \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} (a+b \sec (c+d x))^{3/2} \left (-i (a+b) \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) E\left (i \text {arcsinh}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+i a (a+b) \left (8 A b^3-6 a b^2 (A+3 B)+3 a^3 (49 A+25 B)+3 a^2 b (13 A+57 B)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )^{3/2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 a^3 d (b+a \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x)} \]

[In]

Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]

[Out]

(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(((402*a^2*A*b - 16*A*b^3 + 345*a^3*B + 36*a*b^2*B)*Sin[c + d*x
])/(630*a^2) + ((133*a^2*A + 6*A*b^2 + 144*a*b*B)*Sin[2*(c + d*x)])/(630*a) + ((10*A*b + 9*a*B)*Sin[3*(c + d*x
)])/126 + (a*A*Sin[4*(c + d*x)])/36))/(d*(b + a*Cos[c + d*x])) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec
[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B -
18*a*b^3*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d
*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^3*(49*A + 25*B) + 3*a^2*b*(
13*A + 57*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c +
d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*(b + a*C
os[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^3*d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2
))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4704\) vs. \(2(445)=890\).

Time = 13.19 (sec) , antiderivative size = 4705, normalized size of antiderivative = 11.02

method	result	size
default	\(\text {Expression too large to display}\)	\(4705\)

[In]

int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

2/315/d*(-18*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a*b^4*sin(d*x+c)+147*A*(1/(a+b)*(b+a*cos(d*x+c))/(
cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*b-33*A*(1
/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a
-b))^(1/2))*a^3*b^2+33*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*
x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b^3-8*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((
(a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^4-186*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*
x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*b+33*A*(1/(a+b)
*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(
1/2))*a^3*b^2-2*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-cs
c(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b^3+8*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(
a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^4-246*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1)
)^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*b+246*B*(1/(a+b)*(b+a*
cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*
a^3*b^2+18*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x
+c)),(-(a+b)/(a-b))^(1/2))*a^2*b^3-18*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b)
)^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^4+246*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1
/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*b-153*B*(1/(a+b)*(b+a*cos(
d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3*
b^2-18*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c))
,(-(a+b)/(a-b))^(1/2))*a^2*b^3+8*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*b^5*sin(d*x+c)-147*A*(1/(a+b)*
(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1
/2))*a^5*cos(d*x+c)+8*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x
+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b^5*cos(d*x+c)+147*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ell
ipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^5*cos(d*x+c)-75*B*(1/(a+b)*(b+a*cos
(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^5
*cos(d*x+c)-9*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^2*b^3*cos(d*x+c)*sin(d*x+c)+85*A*((a-b)/(a+b))^
(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^4*b*cos(d*x+c)^4*sin(d*x+c)+85*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)
*a^4*b*cos(d*x+c)^3*sin(d*x+c)+117*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^4*b*cos(d*x+c)^3*sin(d*x+c
)+137*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^4*b*cos(d*x+c)^2*sin(d*x+c)+53*A*((a-b)/(a+b))^(1/2)*(1
/(cos(d*x+c)+1))^(1/2)*a^3*b^2*cos(d*x+c)^2*sin(d*x+c)-A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^2*b^3*
cos(d*x+c)^2*sin(d*x+c)+117*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^4*b*cos(d*x+c)^2*sin(d*x+c)+81*B*
((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^3*b^2*cos(d*x+c)^2*sin(d*x+c)+137*A*((a-b)/(a+b))^(1/2)*(1/(cos
(d*x+c)+1))^(1/2)*a^4*b*cos(d*x+c)*sin(d*x+c)+121*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^3*b^2*cos(d
*x+c)*sin(d*x+c)-A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^2*b^3*cos(d*x+c)*sin(d*x+c)+4*A*((a-b)/(a+b)
)^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a*b^4*cos(d*x+c)*sin(d*x+c)+321*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2
)*a^4*b*cos(d*x+c)*sin(d*x+c)+81*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^3*b^2*cos(d*x+c)*sin(d*x+c)-
147*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-
(a+b)/(a-b))^(1/2))*a^5+8*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot
(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*b^5+147*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(
((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^5-75*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+
c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^5+53*A*((a-b)/(a+b)
)^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^3*b^2*cos(d*x+c)^3*sin(d*x+c)+147*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1
))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*b*cos(d*x+c)-33*A*(1/
(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-
b))^(1/2))*a^3*b^2*cos(d*x+c)+33*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/
2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b^3*cos(d*x+c)-8*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+
1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^4*cos(d*x+c)-186*A*(
1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(
a-b))^(1/2))*a^4*b*cos(d*x+c)+33*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/
2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3*b^2*cos(d*x+c)-2*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+
1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b^3*cos(d*x+c)+8*A*(
1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(
a-b))^(1/2))*a*b^4*cos(d*x+c)-246*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1
/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*b*cos(d*x+c)+246*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)
+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3*b^2*cos(d*x+c)+18*B
*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)
/(a-b))^(1/2))*a^2*b^3*cos(d*x+c)-18*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))
^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^4*cos(d*x+c)+246*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x
+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*b*cos(d*x+c)-153
*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+
b)/(a-b))^(1/2))*a^3*b^2*cos(d*x+c)-18*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b
))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b^3*cos(d*x+c)+35*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x
+c)+1))^(1/2)*a^5*cos(d*x+c)^5*sin(d*x+c)+35*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^5*cos(d*x+c)^4*s
in(d*x+c)+45*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^5*cos(d*x+c)^4*sin(d*x+c)+49*A*((a-b)/(a+b))^(1/
2)*(1/(cos(d*x+c)+1))^(1/2)*a^5*cos(d*x+c)^3*sin(d*x+c)+45*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^5*
cos(d*x+c)^3*sin(d*x+c)+49*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^5*cos(d*x+c)^2*sin(d*x+c)+75*B*((a
-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^5*cos(d*x+c)^2*sin(d*x+c)+147*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)
+1))^(1/2)*a^5*cos(d*x+c)*sin(d*x+c)+75*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^5*cos(d*x+c)*sin(d*x+
c)+147*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^4*b*sin(d*x+c)+88*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)
+1))^(1/2)*a^3*b^2*sin(d*x+c)+33*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^2*b^3*sin(d*x+c)-4*A*((a-b)/
(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a*b^4*sin(d*x+c)+75*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^4*b
*sin(d*x+c)+246*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^3*b^2*sin(d*x+c)+9*B*((a-b)/(a+b))^(1/2)*(1/(
cos(d*x+c)+1))^(1/2)*a^2*b^3*sin(d*x+c))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/(1/(cos(d*x+c)+1))^(1/2)/(b+a
*cos(d*x+c))/a^3/((a-b)/(a+b))^(1/2)/(cos(d*x+c)+1)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.54 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\frac {6 \, {\left (35 \, A a^{5} \cos \left (d x + c\right )^{3} + 75 \, B a^{5} + 88 \, A a^{4} b + 9 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3} + 5 \, {\left (9 \, B a^{5} + 10 \, A a^{4} b\right )} \cos \left (d x + c\right )^{2} + {\left (49 \, A a^{5} + 72 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-225 i \, B a^{5} - 264 i \, A a^{4} b + 33 i \, B a^{3} b^{2} + 60 i \, A a^{2} b^{3} - 36 i \, B a b^{4} + 16 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (225 i \, B a^{5} + 264 i \, A a^{4} b - 33 i \, B a^{3} b^{2} - 60 i \, A a^{2} b^{3} + 36 i \, B a b^{4} - 16 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-147 i \, A a^{5} - 246 i \, B a^{4} b - 33 i \, A a^{3} b^{2} + 18 i \, B a^{2} b^{3} - 8 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (147 i \, A a^{5} + 246 i \, B a^{4} b + 33 i \, A a^{3} b^{2} - 18 i \, B a^{2} b^{3} + 8 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right )}{945 \, a^{4} d} \]

[In]

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/945*(6*(35*A*a^5*cos(d*x + c)^3 + 75*B*a^5 + 88*A*a^4*b + 9*B*a^3*b^2 - 4*A*a^2*b^3 + 5*(9*B*a^5 + 10*A*a^4*
b)*cos(d*x + c)^2 + (49*A*a^5 + 72*B*a^4*b + 3*A*a^3*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)
)*sqrt(cos(d*x + c))*sin(d*x + c) + sqrt(2)*(-225*I*B*a^5 - 264*I*A*a^4*b + 33*I*B*a^3*b^2 + 60*I*A*a^2*b^3 -
36*I*B*a*b^4 + 16*I*A*b^5)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1
/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a) + sqrt(2)*(225*I*B*a^5 + 264*I*A*a^4*b - 33*I*B*a^3*b^2 -
60*I*A*a^2*b^3 + 36*I*B*a*b^4 - 16*I*A*b^5)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*
b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqrt(2)*(-147*I*A*a^5 - 246*I*B*a^4*b
- 33*I*A*a^3*b^2 + 18*I*B*a^2*b^3 - 8*I*A*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^
2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x +
c) + 3*I*a*sin(d*x + c) + 2*b)/a)) - 3*sqrt(2)*(147*I*A*a^5 + 246*I*B*a^4*b + 33*I*A*a^3*b^2 - 18*I*B*a^2*b^3
+ 8*I*A*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInve
rse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a)
))/(a^4*d)

Sympy [F(-1)]

Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**(9/2)*(a+b*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \]

[In]

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/2)*cos(d*x + c)^(9/2), x)

Giac [F]

\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \]

[In]

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/2)*cos(d*x + c)^(9/2), x)

Mupad [F(-1)]

Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^{9/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]

[In]

int(cos(c + d*x)^(9/2)*(A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(3/2),x)

[Out]

int(cos(c + d*x)^(9/2)*(A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(3/2), x)