3.1341 \(\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=455 \[ \frac{2 \left (a^2-b^2\right ) \left (a^2 (39 A b+63 b C)+75 a^3 B-18 a b^2 B+8 A b^3\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{315 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{315 a d}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{315 a^2 d}+\frac{2 \sqrt{\cos (c+d x)} \left (3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (3 a B+A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{9 d} \]
[Out]
(2*(a^2 - b^2)*(8*A*b^3 + 75*a^3*B - 18*a*b^2*B + a^2*(39*A*b + 63*b*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*El
lipticF[(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*(8*A*b^4 + 2
46*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)
/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b^3 - 75
*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2
*d) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3
15*a*d) + (2*(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*Cos[c + d*x
]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)
________________________________________________________________________________________
Rubi [A] time = 1.85682, antiderivative size = 455, normalized size of antiderivative
= 1., number of steps used = 12, number of rules used = 10, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\)
= 0.222, Rules used = {4265, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}
\[ \frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{315 a d}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (-2 a^2 b (44 A+63 C)-75 a^3 B-9 a b^2 B+4 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{315 a^2 d}+\frac{2 \left (a^2-b^2\right ) \left (a^2 (39 A b+63 b C)+75 a^3 B-18 a b^2 B+8 A b^3\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \left (3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{315 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (3 a B+A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{9 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(2*(a^2 - b^2)*(8*A*b^3 + 75*a^3*B - 18*a*b^2*B + a^2*(39*A*b + 63*b*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*El
lipticF[(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*(8*A*b^4 + 2
46*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)
/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b^3 - 75
*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2
*d) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3
15*a*d) + (2*(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*Cos[c + d*x
]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)
Rule 4265
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]
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))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 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 b+3 a B)+\frac{1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)+\frac{1}{2} b (4 A+9 C) \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 (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 \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} \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right )+\frac{1}{4} \left (92 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \sec (c+d x)+\frac{1}{4} b (40 A b+36 a B+63 b C) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 (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 \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} \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right )-\frac{1}{8} a \left (396 a b B+21 a^2 (7 A+9 C)+b^2 (209 A+315 C)\right ) \sec (c+d x)-\frac{1}{4} b \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\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 (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 (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 \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} \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\right )+\frac{3}{16} a \left (2 A b^3+75 a^3 B+153 a b^2 B+6 a^2 b (31 A+42 C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{945 a^2}\\ &=-\frac{2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 (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 \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac{\left (\left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\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{\left (\left (a^2-b^2\right ) \left (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\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{2 \left (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 (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 \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 (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\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 (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\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 (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 (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 \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 (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\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 (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\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 (8 A b^3+75 a^3 B-18 a b^2 B+a^2 (39 A b+63 b C)\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^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (8 A b^4+246 a^3 b B-18 a b^3 B+21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)\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 (4 A b^3-75 a^3 B-9 a b^2 B-2 a^2 b (44 A+63 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d}+\frac{2 \left (3 A b^2+72 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac{2 (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 \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 = 24.4856, size = 3703, normalized size = 8.14 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((402*a^2*A*b - 16*A*b
^3 + 345*a^3*B + 36*a*b^2*B + 504*a^2*b*C)*Sin[c + d*x])/(315*a^2) + ((133*a^2*A + 6*A*b^2 + 144*a*b*B + 126*a
^2*C)*Sin[2*(c + d*x)])/(315*a) + ((10*A*b + 9*a*B)*Sin[3*(c + d*x)])/63 + (a*A*Sin[4*(c + d*x)])/18))/(d*(b +
a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*Cos[c + d*x]^(3/2)*((14*a^2*A*Sqrt[Co
s[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (22*A*b^2*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*
Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^4*Sqrt[Cos[c + d*x]])/(315*a^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c
+ d*x]]) + (164*a*b*B*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*b^3*B*Sqrt[C
os[c + d*x]])/(35*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a^2*C*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*
Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*C*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]
]) + (124*a*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*A*b^3*Sqrt[Cos[c +
d*x]]*Sqrt[Sec[c + d*x]])/(315*a*Sqrt[b + a*Cos[c + d*x]]) + (10*a^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/
(21*Sqrt[b + a*Cos[c + d*x]]) + (34*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]])
+ (8*a*b*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]))*(Cos[(c + d*x)/2]^2*Sec[c + d
*x])^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(8*A*b^4 + 246*a^3
*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*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^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*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)] - (8*A
*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(b + a*Cos[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*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c
+ 2*d*x])*Sec[c + d*x]^(7/2)*((-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-
I)*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[I*Arc
Sinh[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^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))
*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)] - (8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(
b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^2*(b + a*Cos[c + d*x])^(3/2)) + (2*Sq
rt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(8*A*b^4 + 246*a^3*b*B - 1
8*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*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^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*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)] - (8*A*b^4 + 2
46*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]
^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^3*Sqrt[b + a*Cos[c + d*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Se
c[c + d*x])^(3/2)*(-((8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(b +
a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2))/2 - I*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A +
9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sq
rt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3
*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*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)]*Tan[(c + d*x)/2] + a*(8
*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(Sec[(c + d*x)/2]^2)^(3/2)*S
in[c + d*x]*Tan[(c + d*x)/2] - (3*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A +
21*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(8*A*b^4 + 246*
a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-
a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec
[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a
*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I
*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b
)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c +
d*x)/2]^2)/(a + b)] - (a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 2
5*B + 63*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c +
d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 2
1*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)
/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(315*a^3*Sqrt[b
+ a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(8*A*b^4 + 246*
a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*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^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*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)] - (
8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(b + a*Cos[c + d*x])*(Sec[(
c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2
*Sec[c + d*x]*Tan[c + d*x]))/(105*a^3*Sqrt[b + a*Cos[c + d*x]])))
________________________________________________________________________________________
Maple [B] time = 1.086, size = 4075, normalized size = 9. \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))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
[Out]
-2/315/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*(cos(d*x+c)+1)^2*(-1+cos(d*x+c))^3*(45*B*((a-b)/
(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^3*a^5*(1/(cos(d*x+c)+1))^(3/2)+63*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^3*a^5*(1
/(cos(d*x+c)+1))^(3/2)*sin(d*x+c)+63*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*a^5*(1/(cos(d*x+c)+1))^(3/2)*sin(d*x+c
)+189*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)*a^5*(1/(cos(d*x+c)+1))^(3/2)*sin(d*x+c)+147*A*((a-b)/(a+b))^(1/2)*sin(d
*x+c)*cos(d*x+c)*a^5*(1/(cos(d*x+c)+1))^(3/2)+35*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^5*a^5*(1/(cos(d*x
+c)+1))^(3/2)+147*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*b*(1/(cos(d*x+c)+1))^(3/2)+75*B*((a-b)/(a+b))^(1/2)*sin
(d*x+c)*a^4*b*(1/(cos(d*x+c)+1))^(3/2)+246*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b^2*(1/(cos(d*x+c)+1))^(3/2)+9
*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^3*(1/(cos(d*x+c)+1))^(3/2)-18*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a*b^4*(
1/(cos(d*x+c)+1))^(3/2)+189*C*((a-b)/(a+b))^(1/2)*a^4*b*(1/(cos(d*x+c)+1))^(3/2)*sin(d*x+c)+126*C*((a-b)/(a+b)
)^(1/2)*a^3*b^2*(1/(cos(d*x+c)+1))^(3/2)*sin(d*x+c)+63*C*((a-b)/(a+b))^(1/2)*a^2*b^3*(1/(cos(d*x+c)+1))^(3/2)*
sin(d*x+c)+33*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^3*(1/(cos(d*x+c)+1))^(3/2)-4*A*((a-b)/(a+b))^(1/2)*sin(d*
x+c)*a*b^4*(1/(cos(d*x+c)+1))^(3/2)+49*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*a^5*(1/(cos(d*x+c)+1))^(3
/2)+49*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^3*a^5*(1/(cos(d*x+c)+1))^(3/2)+45*B*((a-b)/(a+b))^(1/2)*sin
(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)*cos(d*x+c)^4*a^5+75*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)
*cos(d*x+c)^2*a^5+75*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)*cos(d*x+c)*a^5+35*A*((a-b)/(a+b
))^(1/2)*sin(d*x+c)*cos(d*x+c)^4*a^5*(1/(cos(d*x+c)+1))^(3/2)+8*A*(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+147*A*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*(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^5-75*B*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+189*C*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/s
in(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^5-189*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
+189*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*a^4*b*(1/(cos(d*x+c)+1))^(3/2)*sin(d*x+c)+137*A*((a-b)/(a+b))^(1/2)*si
n(d*x+c)*cos(d*x+c)*a^4*b*(1/(cos(d*x+c)+1))^(3/2)+137*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*a^4*b*(1/
(cos(d*x+c)+1))^(3/2)+53*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*a^3*b^2*(1/(cos(d*x+c)+1))^(3/2)-A*((a-
b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*a^2*b^3*(1/(cos(d*x+c)+1))^(3/2)+85*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*c
os(d*x+c)^3*a^4*b*(1/(cos(d*x+c)+1))^(3/2)+53*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^3*a^3*b^2*(1/(cos(d*
x+c)+1))^(3/2)+85*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^4*a^4*b*(1/(cos(d*x+c)+1))^(3/2)+81*B*((a-b)/(a+
b))^(1/2)*sin(d*x+c)*cos(d*x+c)*a^3*b^2*(1/(cos(d*x+c)+1))^(3/2)+4*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)
*a*b^4*(1/(cos(d*x+c)+1))^(3/2)+117*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^3*a^4*b*(1/(cos(d*x+c)+1))^(3/
2)+121*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)*a^3*b^2*(1/(cos(d*x+c)+1))^(3/2)-A*((a-b)/(a+b))^(1/2)*sin(
d*x+c)*cos(d*x+c)*a^2*b^3*(1/(cos(d*x+c)+1))^(3/2)-9*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)*a^2*b^3*(1/(c
os(d*x+c)+1))^(3/2)+117*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*a^4*b*(1/(cos(d*x+c)+1))^(3/2)+81*B*((a-
b)/(a+b))^(1/2)*sin(d*x+c)*cos(d*x+c)^2*a^3*b^2*(1/(cos(d*x+c)+1))^(3/2)+321*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*
cos(d*x+c)*a^4*b*(1/(cos(d*x+c)+1))^(3/2)+189*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)*a^4*b*(1/(cos(d*x+c)+1))^(3/2)*
sin(d*x+c)+189*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)*a^3*b^2*(1/(cos(d*x+c)+1))^(3/2)*sin(d*x+c)+88*A*((a-b)/(a+b))
^(1/2)*sin(d*x+c)*a^3*b^2*(1/(cos(d*x+c)+1))^(3/2)-186*A*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+246*B*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-1
53*B*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^2-18*B*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^3-246*B*(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+246*B*(1/(a+b)*(b+a*c
os(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+18*B*(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)/s
in(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((-1+cos
(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^4-252*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+63*C*Elliptic
F((-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+189*C*EllipticE((-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-63*C*EllipticE((-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+63*C*EllipticE((-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+8*A*
((a-b)/(a+b))^(1/2)*sin(d*x+c)*b^5*(1/(cos(d*x+c)+1))^(3/2)+33*A*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^2-2*A*EllipticF((-1+co
s(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+8*A*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*(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-33*A*(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+33*A*(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^3-8*A*(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)/a^3/((a-b)/(a+b))^(1/2)/(b+a*cos(d*x+c))/sin(d*x+c)^6/(1/(cos(d*x+c
)+1))^(3/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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{3}{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))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
[Out]
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/2)*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 (C b \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} +{\left (C a + B b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + A a \cos \left (d x + c\right )^{4} +{\left (B 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))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
integral((C*b*cos(d*x + c)^4*sec(d*x + c)^3 + (C*a + B*b)*cos(d*x + c)^4*sec(d*x + c)^2 + A*a*cos(d*x + c)^4 +
(B*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))**(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),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))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")
[Out]
Timed out