3.1281 \(\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^{\frac{13}{2}}(c+d x) \, dx\)

Optimal. Leaf size=343 \[ \frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{693 d}+\frac{4 a^3 (105 A+121 B+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}-\frac{4 a^3 (15 A+17 B+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (6 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d} \]

[Out]

(-4*a^3*(15*A + 17*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*
(105*A + 121*B + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(15*
A + 17*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*Sec[c + d*x]^(3/2)*S
in[c + d*x])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d) + (2*(105*A +
143*B + 99*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d) + (2*(6*A + 11*B)*(a^2 + a^2*C
os[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*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 = 0.843238, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {4221, 3043, 2975, 2968, 3021, 2748, 2636, 2641, 2639} \[ \frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{693 d}+\frac{4 a^3 (105 A+121 B+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}-\frac{4 a^3 (15 A+17 B+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 (6 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^3*(15*A + 17*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*
(105*A + 121*B + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(15*
A + 17*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*Sec[c + d*x]^(3/2)*S
in[c + d*x])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d) + (2*(105*A +
143*B + 99*C)*(a^3 + a^3*Cos[c + d*x])*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(693*d) + (2*(6*A + 11*B)*(a^2 + a^2*C
os[c + d*x])^2*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(99*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 3043

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + 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 -
B*d)*(a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(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, B, 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+B \cos (c+d x)+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+B \cos (c+d x)+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 (\frac{1}{2} a (6 A+11 B)+\frac{1}{2} a (3 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 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{1}{4} a^2 (105 A+143 B+99 C)+\frac{3}{4} a^2 (15 A+11 B+33 C) \cos (c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 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{3}{4} a^3 (210 A+253 B+264 C)+\frac{15}{4} a^3 (21 A+22 B+33 C) \cos (c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 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{3}{4} a^4 (210 A+253 B+264 C)+\left (\frac{15}{4} a^4 (21 A+22 B+33 C)+\frac{3}{4} a^4 (210 A+253 B+264 C)\right ) \cos (c+d x)+\frac{15}{4} a^4 (21 A+22 B+33 C) \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac{4 a^3 (210 A+253 B+264 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 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+121 B+143 C)+\frac{231}{8} a^4 (15 A+17 B+21 C) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{3465 a}\\ &=\frac{4 a^3 (210 A+253 B+264 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 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}{15} \left (2 a^3 (15 A+17 B+21 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+121 B+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 (15 A+17 B+21 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^3 (105 A+121 B+143 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{4 a^3 (210 A+253 B+264 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 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}{15} \left (2 a^3 (15 A+17 B+21 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+121 B+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 (15 A+17 B+21 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (105 A+121 B+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 (15 A+17 B+21 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{15 d}+\frac{4 a^3 (105 A+121 B+143 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac{4 a^3 (210 A+253 B+264 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{99 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 [A]  time = 3.14525, size = 242, normalized size = 0.71 \[ \frac{a^3 \sec ^{\frac{11}{2}}(c+d x) \left (480 (105 A+121 B+143 C) \cos ^{\frac{11}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-7392 (15 A+17 B+21 C) \cos ^{\frac{11}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) (154 (375 A+377 B+396 C) \cos (c+d x)+60 (336 A+341 B+319 C) \cos (2 (c+d x))+21945 A \cos (3 (c+d x))+3150 A \cos (4 (c+d x))+3465 A \cos (5 (c+d x))+19530 A+24871 B \cos (3 (c+d x))+3630 B \cos (4 (c+d x))+3927 B \cos (5 (c+d x))+16830 B+28413 C \cos (3 (c+d x))+4290 C \cos (4 (c+d x))+4851 C \cos (5 (c+d x))+14850 C)\right )}{27720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Sec[c + d*x]^(11/2)*(-7392*(15*A + 17*B + 21*C)*Cos[c + d*x]^(11/2)*EllipticE[(c + d*x)/2, 2] + 480*(105*
A + 121*B + 143*C)*Cos[c + d*x]^(11/2)*EllipticF[(c + d*x)/2, 2] + 2*(19530*A + 16830*B + 14850*C + 154*(375*A
+ 377*B + 396*C)*Cos[c + d*x] + 60*(336*A + 341*B + 319*C)*Cos[2*(c + d*x)] + 21945*A*Cos[3*(c + d*x)] + 2487
1*B*Cos[3*(c + d*x)] + 28413*C*Cos[3*(c + d*x)] + 3150*A*Cos[4*(c + d*x)] + 3630*B*Cos[4*(c + d*x)] + 4290*C*C
os[4*(c + d*x)] + 3465*A*Cos[5*(c + d*x)] + 3927*B*Cos[5*(c + d*x)] + 4851*C*Cos[5*(c + d*x)])*Sin[c + d*x]))/
(27720*d)

________________________________________________________________________________________

Maple [B]  time = 5.989, size = 1424, normalized size = 4.2 \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+B*cos(d*x+c)+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*((3/8*A+3/8*B+1/8*C)*(-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)))+(1/8*B+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)))+(3/8*A+1/8*B)*(-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
(cos(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))))+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)-1/5*(1/8*A+3/8*B
+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*Ell
ipticE(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*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} + B \cos \left (d x + c\right ) + 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+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + 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} +{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} +{\left (A + 3 \, B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} +{\left (3 \, A + 3 \, B + C\right )} a^{3} \cos \left (d x + c\right )^{2} +{\left (3 \, A + B\right )} 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+B*cos(d*x+c)+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 + (B + 3*C)*a^3*cos(d*x + c)^4 + (A + 3*B + 3*C)*a^3*cos(d*x + c)^3 + (3*A + 3*
B + C)*a^3*cos(d*x + c)^2 + (3*A + B)*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+B*cos(d*x+c)+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} + B \cos \left (d x + c\right ) + 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+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(13/2),x, algorithm="giac")

[Out]

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