3.1306 \(\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=296 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (3 a^2 b (5 A+7 C)+5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)\right )}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{315 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (9 a^2 b (5 A+7 C)+15 a^3 B+54 a b^2 B+8 A b^3\right )}{63 d}+\frac{2 (3 a B+2 A b) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^2}{21 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d} \]
[Out]
(2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(5*a
^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(8*A*b^3 + 1
5*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a*(24*A*b^2 + 99*a*b*
B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*(b + a
*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)
________________________________________________________________________________________
Rubi [A] time = 0.910354, antiderivative size = 296, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used =
{4112, 3049, 3033, 3023, 2748, 2641, 2639} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 b (5 A+7 C)+5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)\right )}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{315 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (9 a^2 b (5 A+7 C)+15 a^3 B+54 a b^2 B+8 A b^3\right )}{63 d}+\frac{2 (3 a B+2 A b) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^2}{21 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(5*a
^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(8*A*b^3 + 1
5*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a*(24*A*b^2 + 99*a*b*
B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*(b + a
*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)
Rule 4112
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sec[(e_.)
+ (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[d^(m + 2), Int[(b + a*Cos[e + f*x])^m*(d*
Cos[e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}
, x] && !IntegerQ[n] && IntegerQ[m]
Rule 3049
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e +
f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Rule 3033
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b
*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*
(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e +
f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]
Rule 3023
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Rule 2748
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Rule 2641
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]
Rule 2639
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]
Rubi steps
\begin{align*} \int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \frac{(b+a \cos (c+d x))^3 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 A \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac{2}{9} \int \frac{(b+a \cos (c+d x))^2 \left (\frac{1}{2} b (A+9 C)+\frac{1}{2} (7 a A+9 b B+9 a C) \cos (c+d x)+\frac{3}{2} (2 A b+3 a B) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (2 A b+3 a B) \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 A \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac{4}{63} \int \frac{(b+a \cos (c+d x)) \left (\frac{1}{4} b (13 A b+9 a B+63 b C)+\frac{1}{4} \left (86 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \cos (c+d x)+\frac{1}{4} \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (2 A b+3 a B) \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 A \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac{8}{315} \int \frac{\frac{5}{8} b^2 (13 A b+9 a B+63 b C)+\frac{21}{8} \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \cos (c+d x)+\frac{15}{8} \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{63 d}+\frac{2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (2 A b+3 a B) \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 A \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac{16}{945} \int \frac{\frac{45}{16} \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right )+\frac{63}{16} \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{63 d}+\frac{2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (2 A b+3 a B) \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 A \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac{1}{21} \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{15} \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{63 d}+\frac{2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (2 A b+3 a B) \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 A \sqrt{\cos (c+d x)} (b+a \cos (c+d x))^3 \sin (c+d x)}{9 d}\\ \end{align*}
Mathematica [C] time = 7.27416, size = 3237, normalized size = 10.94 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(7*a^3*A + 27*a*A*b^2
+ 27*a^2*b*B + 15*b^3*B + 9*a^3*C + 45*a*b^2*C)*Cot[c])/(15*d) + ((69*a^2*A*b + 28*A*b^3 + 23*a^3*B + 84*a*b^
2*B + 84*a^2*b*C)*Cos[d*x]*Sin[c])/(21*d) + (a*(19*a^2*A + 54*A*b^2 + 54*a*b*B + 18*a^2*C)*Cos[2*d*x]*Sin[2*c]
)/(45*d) + (a^2*(3*A*b + a*B)*Cos[3*d*x]*Sin[3*c])/(7*d) + (a^3*A*Cos[4*d*x]*Sin[4*c])/(18*d) + ((69*a^2*A*b +
28*A*b^3 + 23*a^3*B + 84*a*b^2*B + 84*a^2*b*C)*Cos[c]*Sin[d*x])/(21*d) + (a*(19*a^2*A + 54*A*b^2 + 54*a*b*B +
18*a^2*C)*Cos[2*c]*Sin[2*d*x])/(45*d) + (a^2*(3*A*b + a*B)*Cos[3*c]*Sin[3*d*x])/(7*d) + (a^3*A*Cos[4*c]*Sin[4
*d*x])/(18*d)))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (20*a^2*A*b*Cos[c
+ d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A +
B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[
1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(b + a*Cos[c + d*x]
)^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^3*Cos[c + d*x]^5*Csc[c]*Hyp
ergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*
Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]
*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*C
os[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (20*a^3*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4
, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec
[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[C
ot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Co
s[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a*b^2*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Si
n[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot
[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1
+ Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt
[1 + Cot[c]^2]) - (4*a^2*b*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c
]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d
*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[C
ot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4
*b^3*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c +
d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*
Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Co
s[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (14*a^3*A*Cos[c + d*x]^5
*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}
, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 +
Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((
Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2
])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(b + a*Cos[c + d*x
])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (18*a*A*b^2*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x
])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]
]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]
]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*
Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/
Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c +
d*x] + A*Cos[2*c + 2*d*x])) - (18*a^2*b*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] +
C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c
]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + Arc
Tan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2]
+ (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTa
n[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])
) - (2*b^3*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Hypergeome
tricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x
+ ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2
]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan
[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]
))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (6*a^3*C*Cos[c + d*x]^5*Csc[
c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos
[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[
d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d
*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(C
os[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^3*(
A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (6*a*b^2*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A
+ B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Si
n[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[
Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])
/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Co
s[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A
*Cos[2*c + 2*d*x]))
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Maple [B] time = 2.827, size = 975, normalized size = 3.3 \begin{align*} \text{result 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*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
[Out]
-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^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2
*c)^10+(2240*A*a^3+2160*A*a^2*b+720*B*a^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-2072*A*a^3-3240*A*a^2*b-1
512*A*a*b^2-1080*B*a^3-1512*B*a^2*b-504*C*a^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(952*A*a^3+2520*A*a^2*b
+1512*A*a*b^2+420*A*b^3+840*B*a^3+1512*B*a^2*b+1260*B*a*b^2+504*C*a^3+1260*C*a^2*b)*sin(1/2*d*x+1/2*c)^4*cos(1
/2*d*x+1/2*c)+(-168*A*a^3-720*A*a^2*b-378*A*a*b^2-210*A*b^3-240*B*a^3-378*B*a^2*b-630*B*a*b^2-126*C*a^3-630*C*
a^2*b)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+225*A*a^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))+105*A*b^3*(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^3-567*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*b^2+75*B*a^3*(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))+315*B*a*b^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))-567*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^2*b-315*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))*b^3+315*a^2*b*C*(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))+315*C*b^3*(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))-189*C*(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^3-945*C*(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)/(-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
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Maxima [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*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
[Out]
Timed out
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{3} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{5} +{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{4} + A a^{3} \cos \left (d x + c\right )^{4} +{\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )\right )} \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*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
integral((C*b^3*cos(d*x + c)^4*sec(d*x + c)^5 + (3*C*a*b^2 + B*b^3)*cos(d*x + c)^4*sec(d*x + c)^4 + A*a^3*cos(
d*x + c)^4 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*cos(d*x + c)^4*sec(d*x + c)^3 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c
os(d*x + c)^4*sec(d*x + c)^2 + (B*a^3 + 3*A*a^2*b)*cos(d*x + c)^4*sec(d*x + c))*sqrt(cos(d*x + c)), x)
________________________________________________________________________________________
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*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [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 )}^{3} \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*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")
[Out]
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3*cos(d*x + c)^(9/2), x)