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

Optimal. Leaf size=303 \[ \frac{4 a^3 (105 A+121 B+143 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}-\frac{4 a^3 (15 A+17 B+21 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac{11}{2}}(c+d x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.748762, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {3043, 2975, 2968, 3021, 2748, 2636, 2641, 2639} \[ \frac{4 a^3 (105 A+121 B+143 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}-\frac{4 a^3 (15 A+17 B+21 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac{11}{2}}(c+d x)} \]

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))/Cos[c + d*x]^(13/2),x]

[Out]

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

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 \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 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 \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 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4 \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 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{8 \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 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{8 \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) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{16 \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) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{15} \left (2 a^3 (15 A+17 B+21 C)\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)\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{1}{15} \left (2 a^3 (15 A+17 B+21 C)\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{231} \left (2 a^3 (105 A+121 B+143 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{4 a^3 (15 A+17 B+21 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a^3 (105 A+121 B+143 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d}+\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x)}\\ \end{align*}

Mathematica [C]  time = 7.11775, size = 1418, normalized size = 4.68 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(((15*A + 17*B + 21*C)*Csc[c]*Sec[c])/(30*d) +
(A*Sec[c]*Sec[c + d*x]^6*Sin[d*x])/(44*d) + (Sec[c]*Sec[c + d*x]^5*(9*A*Sin[c] + 33*A*Sin[d*x] + 11*B*Sin[d*x]
))/(396*d) + (Sec[c]*Sec[c + d*x]^4*(231*A*Sin[c] + 77*B*Sin[c] + 378*A*Sin[d*x] + 297*B*Sin[d*x] + 99*C*Sin[d
*x]))/(2772*d) + (Sec[c]*Sec[c + d*x]*(525*A*Sin[c] + 605*B*Sin[c] + 715*C*Sin[c] + 1155*A*Sin[d*x] + 1309*B*S
in[d*x] + 1617*C*Sin[d*x]))/(2310*d) + (Sec[c]*Sec[c + d*x]^3*(1890*A*Sin[c] + 1485*B*Sin[c] + 495*C*Sin[c] +
2310*A*Sin[d*x] + 2618*B*Sin[d*x] + 2079*C*Sin[d*x]))/(13860*d) + (Sec[c]*Sec[c + d*x]^2*(2310*A*Sin[c] + 2618
*B*Sin[c] + 2079*C*Sin[c] + 3150*A*Sin[d*x] + 3630*B*Sin[d*x] + 4290*C*Sin[d*x]))/(13860*d)) - (5*A*(a + a*Cos
[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*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]]]])/(22*d*Sqrt[1 + Cot[c]^2]) - (11*B*(a + a*Cos[c + d*x])^3*Csc[c]
*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*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]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (13*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[
{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*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[Co
t[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) + (A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((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 + ArcTa
n[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]]))/(4*d
) + (17*B*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((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]]))/(60*d) + (7*C*(a + a*Cos[c + d*x])
^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((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]*Co
s[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)

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Maple [B]  time = 1.316, size = 1424, normalized size = 4.7 \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)/cos(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 \frac{{\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}}{\cos \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)/cos(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/cos(d*x + c)^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{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}}{\cos \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)/cos(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)/cos(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)/cos(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 \frac{{\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}}{\cos \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)/cos(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/cos(d*x + c)^(13/2), x)