\(\int (a+a \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \sec ^{\frac {13}{2}}(c+d x) \, dx\) [1175]

Optimal result

Integrand size = 35, antiderivative size = 319 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=-\frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (105 A+143 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a^3 (5 A+7 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (105 A+143 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {8 a^3 (35 A+44 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 (35 A+33 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {4 A \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{33 a d}+\frac {2 A (a+a \cos (c+d x))^3 \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \]

[Out]

4/231*a^3*(105*A+143*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+8/385*a^3*(35*A+44*C)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/23
1*(35*A+33*C)*(a^3+a^3*cos(d*x+c))*sec(d*x+c)^(7/2)*sin(d*x+c)/d+4/33*A*(a^2+a^2*cos(d*x+c))^2*sec(d*x+c)^(9/2
)*sin(d*x+c)/a/d+2/11*A*(a+a*cos(d*x+c))^3*sec(d*x+c)^(11/2)*sin(d*x+c)/d+4/5*a^3*(5*A+7*C)*sin(d*x+c)*sec(d*x
+c)^(1/2)/d-4/5*a^3*(5*A+7*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(
1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+4/231*a^3*(105*A+143*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2
*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4306, 3123, 3054, 3047, 3100, 2827, 2716, 2720, 2719} \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {8 a^3 (35 A+44 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{385 d}+\frac {4 a^3 (105 A+143 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{231 d}+\frac {4 a^3 (5 A+7 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 (35 A+33 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{231 d}+\frac {4 a^3 (105 A+143 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}-\frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {4 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{33 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d} \]

[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 2716

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 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2827

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 3047

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 3054

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
*Sin[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 3100

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 3123

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*Si
n[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 4306

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]

Rubi steps \begin{align*} \text {integral}& = \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)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.43 (sec) , antiderivative size = 697, normalized size of antiderivative = 2.18 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\frac {A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} (a+a \cos (c+d x))^3 \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{6 \sqrt {2} d}+\frac {7 C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} (a+a \cos (c+d x))^3 \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{30 \sqrt {2} d}+\frac {5 A \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)}}{22 d}+\frac {13 C \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)}}{42 d}+(a+a \cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\sec (c+d x)} \left (\frac {(5 A+7 C) \cos (d x) \csc (c)}{10 d}+\frac {A \sec (c) \sec ^5(c+d x) \sin (d x)}{44 d}+\frac {\sec (c) \sec ^4(c+d x) (3 A \sin (c)+11 A \sin (d x))}{132 d}+\frac {\sec (c) \sec ^3(c+d x) (77 A \sin (c)+126 A \sin (d x)+33 C \sin (d x))}{924 d}+\frac {\sec (c) \sec ^2(c+d x) (630 A \sin (c)+165 C \sin (c)+770 A \sin (d x)+693 C \sin (d x))}{4620 d}+\frac {\sec (c) \sec (c+d x) (770 A \sin (c)+693 C \sin (c)+1050 A \sin (d x)+1430 C \sin (d x))}{4620 d}+\frac {(105 A+143 C) \tan (c)}{462 d}\right ) \]

[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))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1380\) vs. \(2(335)=670\).

Time = 507.77 (sec) , antiderivative size = 1381, normalized size of antiderivative = 4.33

method	result	size
default	\(\text {Expression too large to display}\)	\(1381\)
parts	\(\text {Expression too large to display}\)	\(1711\)

[In]

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

[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*(-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)/(cos(1/2*d*x+1/2*c)^2-1/2)^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)/(cos(1/2*d*x+1/2*c)^2-1/2)^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/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)
^2*cos(1/2*d*x+1/2*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)))+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)/(cos(1/2*d*
x+1/2*c)^2-1/2)^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)/(cos(1/2*d*x+1
/2*c)^2-1/2)^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)/(cos(1/2*d*x+1/2
*c)^2-1/2)^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)))+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)/(cos(1/2*d*x+1/2*c)^2-1/2)^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*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)/(cos(1/2*d*
x+1/2*c)^2-1/2)^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)/(cos(1/2*d*x+1/
2*c)^2-1/2)^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)))+1/5*(1/8*A+3/8*C)/(8*sin(1/2*d*x+1/2*c)^6-12*s
in(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*(24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6
-12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/
2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2
*c)-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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*(-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)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.94 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (462 \, {\left (5 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 10 \, {\left (105 \, A + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 77 \, {\left (10 \, A + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (42 \, A + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 385 \, A a^{3} \cos \left (d x + c\right ) + 105 \, A a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{1155 \, d \cos \left (d x + c\right )^{5}} \]

[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]

-2/1155*(5*I*sqrt(2)*(105*A + 143*C)*a^3*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x +
c)) - 5*I*sqrt(2)*(105*A + 143*C)*a^3*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))
+ 231*I*sqrt(2)*(5*A + 7*C)*a^3*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c)
+ I*sin(d*x + c))) - 231*I*sqrt(2)*(5*A + 7*C)*a^3*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(
-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (462*(5*A + 7*C)*a^3*cos(d*x + c)^5 + 10*(105*A + 143*C)*a^3*cos(d*x
+ c)^4 + 77*(10*A + 9*C)*a^3*cos(d*x + c)^3 + 15*(42*A + 11*C)*a^3*cos(d*x + c)^2 + 385*A*a^3*cos(d*x + c) + 1
05*A*a^3)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^5)

Sympy [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]

[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)

Giac [F]

\[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {13}{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {13}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(13/2)*(a + a*cos(c + d*x))^3,x)

[Out]

int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(13/2)*(a + a*cos(c + d*x))^3, x)