3.1175 \(\int (a+a \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \sec ^{\frac{13}{2}}(c+d x) \, dx\)

Optimal. Leaf size=319 \[ \frac{8 a^3 (35 A+44 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d}+\frac{4 a^3 (105 A+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{4 a^3 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 (35 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{231 d}+\frac{4 a^3 (105 A+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}-\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{33 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d} \]

[Out]

(-4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(105*A + 1
43*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(5*A + 7*C)*Sqrt[Sec[c
+ d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(105*A + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (8*a^3*(35*A
+ 44*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d) + (2*(35*A + 33*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(7/2
)*Sin[c + d*x])/(231*d) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(33*a*d) + (2*A*(a
+ a*Cos[c + d*x])^3*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)

________________________________________________________________________________________

Rubi [A]  time = 0.768622, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {4221, 3044, 2975, 2968, 3021, 2748, 2636, 2641, 2639} \[ \frac{8 a^3 (35 A+44 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d}+\frac{4 a^3 (105 A+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{4 a^3 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 (35 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{231 d}+\frac{4 a^3 (105 A+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}-\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{33 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]

[Out]

(-4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(105*A + 1
43*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(5*A + 7*C)*Sqrt[Sec[c
+ d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(105*A + 143*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (8*a^3*(35*A
+ 44*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d) + (2*(35*A + 33*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(7/2
)*Sin[c + d*x])/(231*d) + (4*A*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(33*a*d) + (2*A*(a
+ a*Cos[c + d*x])^3*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(11*d)

Rule 4221

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

Rule 3044

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^
m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n +
2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b
*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0
])

Rule 2975

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*S
in[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c + a*d)), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])
^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n +
1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d
, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n]
|| EqQ[c, 0])

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

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 2636

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

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 (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac{13}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{13}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^3 \left (3 a A+\frac{1}{2} a (3 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac{4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{33 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^2 \left (\frac{3}{4} a^2 (35 A+33 C)+\frac{9}{4} a^2 (5 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac{2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{33 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x)) \left (\frac{9}{2} a^3 (35 A+44 C)+\frac{45}{4} a^3 (7 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac{2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{33 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{9}{2} a^4 (35 A+44 C)+\left (\frac{45}{4} a^4 (7 A+11 C)+\frac{9}{2} a^4 (35 A+44 C)\right ) \cos (c+d x)+\frac{45}{4} a^4 (7 A+11 C) \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac{8 a^3 (35 A+44 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac{2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{33 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{45}{8} a^4 (105 A+143 C)+\frac{693}{8} a^4 (5 A+7 C) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{3465 a}\\ &=\frac{8 a^3 (35 A+44 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac{2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{33 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac{1}{5} \left (2 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{77} \left (2 a^3 (105 A+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{4 a^3 (5 A+7 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^3 (105 A+143 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{8 a^3 (35 A+44 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac{2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{33 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}-\frac{1}{5} \left (2 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (2 a^3 (105 A+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 (105 A+143 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{231 d}+\frac{4 a^3 (5 A+7 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{5 d}+\frac{4 a^3 (105 A+143 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{8 a^3 (35 A+44 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac{2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{33 a d}+\frac{2 A (a+a \cos (c+d x))^3 \sec ^{\frac{11}{2}}(c+d x) \sin (c+d x)}{11 d}\\ \end{align*}

Mathematica [C]  time = 6.92472, size = 697, normalized size = 2.18 \[ \sqrt{\sec (c+d x)} \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^3 \left (\frac{(5 A+7 C) \csc (c) \cos (d x)}{10 d}+\frac{\sec (c) \sec ^3(c+d x) (77 A \sin (c)+126 A \sin (d x)+33 C \sin (d x))}{924 d}+\frac{\sec (c) \sec ^2(c+d x) (630 A \sin (c)+770 A \sin (d x)+165 C \sin (c)+693 C \sin (d x))}{4620 d}+\frac{\sec (c) \sec (c+d x) (770 A \sin (c)+1050 A \sin (d x)+693 C \sin (c)+1430 C \sin (d x))}{4620 d}+\frac{(105 A+143 C) \tan (c)}{462 d}+\frac{A \sec (c) \sin (d x) \sec ^5(c+d x)}{44 d}+\frac{\sec (c) \sec ^4(c+d x) (3 A \sin (c)+11 A \sin (d x))}{132 d}\right )+\frac{A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^3}{6 \sqrt{2} d}+\frac{5 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a \cos (c+d x)+a)^3}{22 d}+\frac{7 C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left (\left (-1+e^{2 i c}\right ) e^{2 i d x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^3}{30 \sqrt{2} d}+\frac{13 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) (a \cos (c+d x)+a)^3}{42 d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]

[Out]

(A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(a + a*Cos[c + d*x])^3*Csc[c]
*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*
I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6)/(6*Sqrt[2]*d*E^(I*d*x)) + (7*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d
*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(a + a*Cos[c + d*x])^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I
)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6)/(30*Sq
rt[2]*d*E^(I*d*x)) + (5*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/
2]^6*Sqrt[Sec[c + d*x]])/(22*d) + (13*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*EllipticF[(c + d*x)/2, 2]*Se
c[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]])/(42*d) + (a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]]
*(((5*A + 7*C)*Cos[d*x]*Csc[c])/(10*d) + (A*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(44*d) + (Sec[c]*Sec[c + d*x]^4*(3
*A*Sin[c] + 11*A*Sin[d*x]))/(132*d) + (Sec[c]*Sec[c + d*x]^3*(77*A*Sin[c] + 126*A*Sin[d*x] + 33*C*Sin[d*x]))/(
924*d) + (Sec[c]*Sec[c + d*x]^2*(630*A*Sin[c] + 165*C*Sin[c] + 770*A*Sin[d*x] + 693*C*Sin[d*x]))/(4620*d) + (S
ec[c]*Sec[c + d*x]*(770*A*Sin[c] + 693*C*Sin[c] + 1050*A*Sin[d*x] + 1430*C*Sin[d*x]))/(4620*d) + ((105*A + 143
*C)*Tan[c])/(462*d))

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Maple [B]  time = 5.112, size = 1408, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x)

[Out]

-16*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-1/5*(1/8*A+3/8*C)/(8*sin(1/2*d*x+1/2*c)^6-
12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2
))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^6*
cos(1/2*d*x+1/2*c)-12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)
^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2+24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+3*EllipticE(cos(1/2*d*x+1/2*c),2^(
1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)-8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c))
*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+1/8*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)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+2*(-2
*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2)/sin(1/2*d*x+1/2*c)^
2/(2*sin(1/2*d*x+1/2*c)^2-1)+(1/8*C+3/8*A)*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*
c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^
2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2
*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+3/8*A*(-1/144*cos(1/2
*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^5-7/180*cos(1/2*d
*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^3-14/15*sin(1/2*d*x
+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c
)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(co
s(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+
1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/
2))))+3/8*C*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/
2*c)^2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*
d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+1/8*A*(-1/352*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/
2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^6-9/616*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*
c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^4-15/154*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c
)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^2+15/77*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d
*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{13}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^3*sec(d*x + c)^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C a^{3} \cos \left (d x + c\right )^{5} + 3 \, C a^{3} \cos \left (d x + c\right )^{4} +{\left (A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} +{\left (3 \, A + C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, A a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sec \left (d x + c\right )^{\frac{13}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algorithm="fricas")

[Out]

integral((C*a^3*cos(d*x + c)^5 + 3*C*a^3*cos(d*x + c)^4 + (A + 3*C)*a^3*cos(d*x + c)^3 + (3*A + C)*a^3*cos(d*x
+ c)^2 + 3*A*a^3*cos(d*x + c) + A*a^3)*sec(d*x + c)^(13/2), 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((a+a*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)*sec(d*x+c)**(13/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac{13}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)^3*sec(d*x + c)^(13/2), x)