3.608 \(\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx\)

Optimal. Leaf size=425 \[ \frac{2 \left (a^2-b^2\right ) \left (114 a^2 A b+75 a^3 B+45 a b^2 B-10 A b^3\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{315 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{315 d}+\frac{2 \left (163 a^2 A b+75 a^3 B+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{315 a d}+\frac{2 \left (279 a^2 A b^2+147 a^4 A+435 a^3 b B+45 a b^3 B-10 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a (3 a B+4 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{21 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{9 d} \]

[Out]

(2*(a^2 - b^2)*(114*a^2*A*b - 10*A*b^3 + 75*a^3*B + 45*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(
c + d*x)/2, (2*a)/(a + b)])/(315*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 279*a^2*
A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a +
b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*
a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(49*a^2*A + 75*A*b^2 + 135*a
*b*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(4*A*b + 3*a*B)*Cos[c + d*x]^(5
/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*A*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c
+ d*x])/(9*d)

________________________________________________________________________________________

Rubi [A]  time = 1.71555, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {2955, 4025, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ \frac{2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{315 d}+\frac{2 \left (163 a^2 A b+75 a^3 B+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{315 a d}+\frac{2 \left (a^2-b^2\right ) \left (114 a^2 A b+75 a^3 B+45 a b^2 B-10 A b^3\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (279 a^2 A b^2+147 a^4 A+435 a^3 b B+45 a b^3 B-10 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a (3 a B+4 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{21 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]

[Out]

(2*(a^2 - b^2)*(114*a^2*A*b - 10*A*b^3 + 75*a^3*B + 45*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(
c + d*x)/2, (2*a)/(a + b)])/(315*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 279*a^2*
A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a +
b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*
a*b^2*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(49*a^2*A + 75*A*b^2 + 135*a
*b*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(4*A*b + 3*a*B)*Cos[c + d*x]^(5
/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*A*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c
+ d*x])/(9*d)

Rule 2955

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 4025

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 4094

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] - Dist[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)))*Csc[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 4104

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]

Rule 4035

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 3856

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 2655

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

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

Rule 3858

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(Sqrt[d*
Csc[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 2663

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

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

Rubi steps

\begin{align*} \int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \sec (c+d x))^{5/2} (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) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}-\frac{1}{9} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+b \sec (c+d x)} \left (-\frac{3}{2} a (4 A b+3 a B)-\frac{1}{2} \left (7 a^2 A+9 A b^2+18 a b B\right ) \sec (c+d x)-\frac{1}{2} b (4 a A+9 b B) \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a (4 A b+3 a B) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}-\frac{1}{63} \left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{4} a \left (49 a^2 A+75 A b^2+135 a b B\right )-\frac{1}{4} \left (137 a^2 A b+63 A b^3+45 a^3 B+189 a b^2 B\right ) \sec (c+d x)-\frac{1}{4} b \left (76 a A b+36 a^2 B+63 b^2 B\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac{2 a (4 A b+3 a B) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \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 (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right )+\frac{1}{8} a \left (147 a^3 A+605 a A b^2+585 a^2 b B+315 b^3 B\right ) \sec (c+d x)+\frac{1}{4} a b \left (49 a^2 A+75 A b^2+135 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}\\ &=\frac{2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac{2 a (4 A b+3 a B) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \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+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\right )-\frac{3}{16} a^2 \left (261 a^2 A b+155 A b^3+75 a^3 B+405 a b^2 B\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{945 a^2}\\ &=\frac{2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac{2 a (4 A b+3 a B) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{\left (\left (a^2-b^2\right ) \left (114 a^2 A b-10 A b^3+75 a^3 B+45 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^2}+\frac{\left (\left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 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^2}\\ &=\frac{2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac{2 a (4 A b+3 a B) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{\left (\left (a^2-b^2\right ) \left (114 a^2 A b-10 A b^3+75 a^3 B+45 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^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 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^2 \sqrt{b+a \cos (c+d x)}}\\ &=\frac{2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac{2 a (4 A b+3 a B) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{\left (\left (a^2-b^2\right ) \left (114 a^2 A b-10 A b^3+75 a^3 B+45 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^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 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^2 \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}\\ &=\frac{2 \left (a^2-b^2\right ) \left (114 a^2 A b-10 A b^3+75 a^3 B+45 a b^2 B\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 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^2 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}+\frac{2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac{2 a (4 A b+3 a B) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac{2 a A \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}\\ \end{align*}

Mathematica [C]  time = 18.3075, size = 542, normalized size = 1.28 \[ \frac{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac{1}{630} \left (133 a^2 A+270 a b B+150 A b^2\right ) \sin (2 (c+d x))+\frac{\left (747 a^2 A b+345 a^3 B+540 a b^2 B+20 A b^3\right ) \sin (c+d x)}{630 a}+\frac{1}{36} a^2 A \sin (4 (c+d x))+\frac{1}{126} a (9 a B+19 A b) \sin (3 (c+d x))\right )}{d (a \cos (c+d x)+b)^2}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left (\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} (a+b \sec (c+d x))^{5/2} \left (i a (a+b) \left (6 a^2 b (19 A+60 B)+3 a^3 (49 A+25 B)+15 a b^2 (11 A+3 B)-10 A b^3\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} \text{EllipticF}\left (i \sinh ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{b-a}{a+b}\right )-\left (279 a^2 A b^2+147 a^4 A+435 a^3 b B+45 a b^3 B-10 A b^4\right ) \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )^{3/2} (a \cos (c+d x)+b)-i (a+b) \left (279 a^2 A b^2+147 a^4 A+435 a^3 b B+45 a b^3 B-10 A b^4\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} E\left (i \sinh ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{b-a}{a+b}\right )\right )}{315 a^2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]

[Out]

(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(((747*a^2*A*b + 20*A*b^3 + 345*a^3*B + 540*a*b^2*B)*Sin[c + d*
x])/(630*a) + ((133*a^2*A + 150*A*b^2 + 270*a*b*B)*Sin[2*(c + d*x)])/630 + (a*(19*A*b + 9*a*B)*Sin[3*(c + d*x)
])/126 + (a^2*A*Sin[4*(c + d*x)])/36))/(d*(b + a*Cos[c + d*x])^2) - (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])^(5/2)*((-I)*(a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b
*B + 45*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)*(-10*A*b^3 + 15*a*b^2*(11*A + 3*B) + 3*a^3*(49*A + 25*B)
+ 6*a^2*b*(19*A + 60*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 + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^
3*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^2*d*(b + a*Cos[c + d*x])^3*Sec[
c + d*x]^(5/2))

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Maple [B]  time = 0.689, size = 3069, normalized size = 7.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x)

[Out]

2/315/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*(-1+cos(d*x+c))*(cos(d*x+c)+1)*(-147*A*sin(d*x+c)
*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos
(d*x+c)+1))^(1/2)*a^5+147*A*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c
))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^5+10*A*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x
+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*b^5+98*A*cos(d*x+
c)^2*((a-b)/(a+b))^(1/2)*a^5*(1/(cos(d*x+c)+1))^(1/2)-147*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^5*(1/(cos(d*x+c)+
1))^(1/2)-10*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*b^5*(1/(cos(d*x+c)+1))^(1/2)-147*A*((a-b)/(a+b))^(1/2)*a^4*b*(1/
(cos(d*x+c)+1))^(1/2)-163*A*((a-b)/(a+b))^(1/2)*a^3*b^2*(1/(cos(d*x+c)+1))^(1/2)-279*A*((a-b)/(a+b))^(1/2)*a^2
*b^3*(1/(cos(d*x+c)+1))^(1/2)-5*A*((a-b)/(a+b))^(1/2)*a*b^4*(1/(cos(d*x+c)+1))^(1/2)-75*B*((a-b)/(a+b))^(1/2)*
a^4*b*(1/(cos(d*x+c)+1))^(1/2)-435*B*((a-b)/(a+b))^(1/2)*a^3*b^2*(1/(cos(d*x+c)+1))^(1/2)-279*A*sin(d*x+c)*Ell
ipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x
+c)+1))^(1/2)*a^3*b^2+155*A*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))
^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^3+10*A*((a-b)/(a+b))^(1/2)*b^5*(1/(cos(d*x+c)+1)
)^(1/2)+10*A*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b
)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^4-147*A*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2
)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*b+279*A*sin(d*x+c)*(1/(a+
b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-
b))^(1/2))*a^3*b^2-279*A*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*
((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^3-10*A*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*
x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^4+170*A*cos(
d*x+c)^4*((a-b)/(a+b))^(1/2)*a^3*b^2*(1/(cos(d*x+c)+1))^(1/2)+180*B*cos(d*x+c)^4*((a-b)/(a+b))^(1/2)*a^4*b*(1/
(cos(d*x+c)+1))^(1/2)+82*A*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^4*b*(1/(cos(d*x+c)+1))^(1/2)+80*A*cos(d*x+c)^3*(
(a-b)/(a+b))^(1/2)*a^2*b^3*(1/(cos(d*x+c)+1))^(1/2)+270*B*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^3*b^2*(1/(cos(d*x
+c)+1))^(1/2)+130*A*cos(d*x+c)^5*((a-b)/(a+b))^(1/2)*a^4*b*(1/(cos(d*x+c)+1))^(1/2)+272*A*cos(d*x+c)^2*((a-b)/
(a+b))^(1/2)*a^3*b^2*(1/(cos(d*x+c)+1))^(1/2)-5*A*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a*b^4*(1/(cos(d*x+c)+1))^(1
/2)+330*B*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^4*b*(1/(cos(d*x+c)+1))^(1/2)+180*B*cos(d*x+c)^2*((a-b)/(a+b))^(1/
2)*a^2*b^3*(1/(cos(d*x+c)+1))^(1/2)-65*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^4*b*(1/(cos(d*x+c)+1))^(1/2)-279*A*c
os(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b^2*(1/(cos(d*x+c)+1))^(1/2)+199*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^2*b^3*(1
/(cos(d*x+c)+1))^(1/2)+10*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^4*(1/(cos(d*x+c)+1))^(1/2)-435*B*cos(d*x+c)*((a
-b)/(a+b))^(1/2)*a^4*b*(1/(cos(d*x+c)+1))^(1/2)+165*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b^2*(1/(cos(d*x+c)+1)
)^(1/2)-45*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^2*b^3*(1/(cos(d*x+c)+1))^(1/2)+45*B*cos(d*x+c)*((a-b)/(a+b))^(1/
2)*a*b^4*(1/(cos(d*x+c)+1))^(1/2)-435*B*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-
(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^4*b+405*B*sin(d*x+c)*EllipticF((-1+cos(d
*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^
3*b^2-45*B*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*
(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^3+435*B*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2
)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*b-435*B*sin(d*x+c)*(1/(a+
b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-
b))^(1/2))*a^3*b^2+45*B*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*(
(a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^3-45*B*sin(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x
+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^4+261*A*sin(d
*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))
/(cos(d*x+c)+1))^(1/2)*a^4*b-135*B*((a-b)/(a+b))^(1/2)*a^2*b^3*(1/(cos(d*x+c)+1))^(1/2)-45*B*((a-b)/(a+b))^(1/
2)*a*b^4*(1/(cos(d*x+c)+1))^(1/2)+45*B*cos(d*x+c)^5*((a-b)/(a+b))^(1/2)*a^5*(1/(cos(d*x+c)+1))^(1/2)+30*B*cos(
d*x+c)^3*((a-b)/(a+b))^(1/2)*a^5*(1/(cos(d*x+c)+1))^(1/2)+75*B*sin(d*x+c)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+
b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^5-75*B*cos(d*x+c)
*((a-b)/(a+b))^(1/2)*a^5*(1/(cos(d*x+c)+1))^(1/2)+35*A*cos(d*x+c)^6*((a-b)/(a+b))^(1/2)*a^5*(1/(cos(d*x+c)+1))
^(1/2)+14*A*cos(d*x+c)^4*((a-b)/(a+b))^(1/2)*a^5*(1/(cos(d*x+c)+1))^(1/2))/a^2/((a-b)/(a+b))^(1/2)/(b+a*cos(d*
x+c))/(1/(cos(d*x+c)+1))^(1/2)/sin(d*x+c)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/2)*cos(d*x + c)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{2} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right )^{4} +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )\right )} \sqrt{b \sec \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**(9/2)*(a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

Timed out