\(\int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \, dx\) [142]

Optimal result

Integrand size = 35, antiderivative size = 279 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {4 a^3 (221 A+175 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {4 a^3 (121 A+95 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (121 A+95 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^3 (221 A+175 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{585 d}+\frac {40 a^3 (143 A+118 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {12 C \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac {2 (143 A+145 C) \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d} \]

[Out]

4/195*a^3*(221*A+175*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))/
d+4/231*a^3*(121*A+95*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))
/d+4/585*a^3*(221*A+175*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+40/9009*a^3*(143*A+118*C)*cos(d*x+c)^(5/2)*sin(d*x+c)
/d+2/13*C*cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+12/143*C*cos(d*x+c)^(5/2)*(a^2+a^2*cos(d*x+c))^2*si
n(d*x+c)/a/d+2/1287*(143*A+145*C)*cos(d*x+c)^(5/2)*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d+4/231*a^3*(121*A+95*C)*si
n(d*x+c)*cos(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3125, 3055, 3047, 3102, 2827, 2715, 2720, 2719} \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {4 a^3 (121 A+95 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (221 A+175 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {40 a^3 (143 A+118 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{9009 d}+\frac {4 a^3 (221 A+175 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{585 d}+\frac {2 (143 A+145 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{1287 d}+\frac {4 a^3 (121 A+95 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {12 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{143 a d}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d} \]

[In]

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

[Out]

(4*a^3*(221*A + 175*C)*EllipticE[(c + d*x)/2, 2])/(195*d) + (4*a^3*(121*A + 95*C)*EllipticF[(c + d*x)/2, 2])/(
231*d) + (4*a^3*(121*A + 95*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(221*A + 175*C)*Cos[c + d*x]^
(3/2)*Sin[c + d*x])/(585*d) + (40*a^3*(143*A + 118*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*C*Cos[c +
d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d) + (12*C*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])^2*
Sin[c + d*x])/(143*a*d) + (2*(143*A + 145*C)*Cos[c + d*x]^(5/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d
)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[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 3055

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)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x
])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*
x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))
*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, 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 3102

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 3125

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Si
mp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^
(-1)] && NeQ[m + n + 2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {2 \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (\frac {1}{2} a (13 A+5 C)+3 a C \cos (c+d x)\right ) \, dx}{13 a} \\ & = \frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {12 C \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac {4 \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left (\frac {1}{4} a^2 (143 A+85 C)+\frac {1}{4} a^2 (143 A+145 C) \cos (c+d x)\right ) \, dx}{143 a} \\ & = \frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {12 C \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac {2 (143 A+145 C) \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}+\frac {8 \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x)) \left (\frac {1}{4} a^3 (1001 A+745 C)+\frac {5}{2} a^3 (143 A+118 C) \cos (c+d x)\right ) \, dx}{1287 a} \\ & = \frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {12 C \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac {2 (143 A+145 C) \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}+\frac {8 \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {1}{4} a^4 (1001 A+745 C)+\left (\frac {5}{2} a^4 (143 A+118 C)+\frac {1}{4} a^4 (1001 A+745 C)\right ) \cos (c+d x)+\frac {5}{2} a^4 (143 A+118 C) \cos ^2(c+d x)\right ) \, dx}{1287 a} \\ & = \frac {40 a^3 (143 A+118 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {12 C \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac {2 (143 A+145 C) \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}+\frac {16 \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {117}{8} a^4 (121 A+95 C)+\frac {77}{8} a^4 (221 A+175 C) \cos (c+d x)\right ) \, dx}{9009 a} \\ & = \frac {40 a^3 (143 A+118 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {12 C \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac {2 (143 A+145 C) \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}+\frac {1}{77} \left (2 a^3 (121 A+95 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{117} \left (2 a^3 (221 A+175 C)\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx \\ & = \frac {4 a^3 (121 A+95 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^3 (221 A+175 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{585 d}+\frac {40 a^3 (143 A+118 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {12 C \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac {2 (143 A+145 C) \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d}+\frac {1}{231} \left (2 a^3 (121 A+95 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{195} \left (2 a^3 (221 A+175 C)\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = \frac {4 a^3 (221 A+175 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {4 a^3 (121 A+95 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (121 A+95 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^3 (221 A+175 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{585 d}+\frac {40 a^3 (143 A+118 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {12 C \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac {2 (143 A+145 C) \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 7.16 (sec) , antiderivative size = 1028, normalized size of antiderivative = 3.68 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-\frac {(221 A+175 C) \cot (c)}{390 d}+\frac {(2134 A+1811 C) \cos (d x) \sin (c)}{7392 d}+\frac {(7592 A+7825 C) \cos (2 d x) \sin (2 c)}{74880 d}+\frac {(132 A+215 C) \cos (3 d x) \sin (3 c)}{4928 d}+\frac {(13 A+59 C) \cos (4 d x) \sin (4 c)}{3744 d}+\frac {3 C \cos (5 d x) \sin (5 c)}{704 d}+\frac {C \cos (6 d x) \sin (6 c)}{1664 d}+\frac {(2134 A+1811 C) \cos (c) \sin (d x)}{7392 d}+\frac {(7592 A+7825 C) \cos (2 c) \sin (2 d x)}{74880 d}+\frac {(132 A+215 C) \cos (3 c) \sin (3 d x)}{4928 d}+\frac {(13 A+59 C) \cos (4 c) \sin (4 d x)}{3744 d}+\frac {3 C \cos (5 c) \sin (5 d x)}{704 d}+\frac {C \cos (6 c) \sin (6 d x)}{1664 d}\right )-\frac {11 A (a+a \cos (c+d x))^3 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{42 d \sqrt {1+\cot ^2(c)}}-\frac {95 C (a+a \cos (c+d x))^3 \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{462 d \sqrt {1+\cot ^2(c)}}-\frac {17 A (a+a \cos (c+d x))^3 \csc (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \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 ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{60 d}-\frac {35 C (a+a \cos (c+d x))^3 \csc (c) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \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 ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{156 d} \]

[In]

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

[Out]

Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/390*((221*A + 175*C)*Cot[c])/d + ((2134*A +
1811*C)*Cos[d*x]*Sin[c])/(7392*d) + ((7592*A + 7825*C)*Cos[2*d*x]*Sin[2*c])/(74880*d) + ((132*A + 215*C)*Cos[
3*d*x]*Sin[3*c])/(4928*d) + ((13*A + 59*C)*Cos[4*d*x]*Sin[4*c])/(3744*d) + (3*C*Cos[5*d*x]*Sin[5*c])/(704*d) +
(C*Cos[6*d*x]*Sin[6*c])/(1664*d) + ((2134*A + 1811*C)*Cos[c]*Sin[d*x])/(7392*d) + ((7592*A + 7825*C)*Cos[2*c]
*Sin[2*d*x])/(74880*d) + ((132*A + 215*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((13*A + 59*C)*Cos[4*c]*Sin[4*d*x])/
(3744*d) + (3*C*Cos[5*c]*Sin[5*d*x])/(704*d) + (C*Cos[6*c]*Sin[6*d*x])/(1664*d)) - (11*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 - ArcT
an[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*S
qrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (95*C*(a + a*Cos[c + d*x])^3*Csc[c]*Hypergeome
tricPFQ[{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 - A
rcTan[Cot[c]]]])/(462*d*Sqrt[1 + Cot[c]^2]) - (17*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((Hyper
geometricPFQ[{-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 + A
rcTan[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) - (35*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]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(156*d)

Maple [A] (verified)

Time = 27.02 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.66

method	result	size
default	\(-\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{3} \left (-221760 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1058400 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-80080 A -2122400 C \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (314600 A +2331040 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-487916 A -1535860 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (386386 A +633710 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-105534 A -121230 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+23595 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-51051 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+18525 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-40425 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45045 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(464\)
parts	\(\text {Expression too large to display}\)	\(1291\)

[In]

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

[Out]

-4/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-221760*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x
+1/2*c)^14+1058400*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-80080*A-2122400*C)*sin(1/2*d*x+1/2*c)^10*cos(1
/2*d*x+1/2*c)+(314600*A+2331040*C)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-487916*A-1535860*C)*sin(1/2*d*x+1
/2*c)^6*cos(1/2*d*x+1/2*c)+(386386*A+633710*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-105534*A-121230*C)*si
n(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+23595*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)*El
lipticF(cos(1/2*d*x+1/2*c),2^(1/2))-51051*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)*Elli
pticE(cos(1/2*d*x+1/2*c),2^(1/2))+18525*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)*Ellipt
icF(cos(1/2*d*x+1/2*c),2^(1/2))-40425*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)*Elliptic
E(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)/(2*cos(
1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [C] (verification not implemented)

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

Time = 0.13 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.93 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (195 i \, \sqrt {2} {\left (121 \, A + 95 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 195 i \, \sqrt {2} {\left (121 \, A + 95 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (221 \, A + 175 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (221 \, A + 175 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (3465 \, C a^{3} \cos \left (d x + c\right )^{5} + 12285 \, C a^{3} \cos \left (d x + c\right )^{4} + 385 \, {\left (13 \, A + 50 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 585 \, {\left (33 \, A + 38 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 154 \, {\left (221 \, A + 175 \, C\right )} a^{3} \cos \left (d x + c\right ) + 390 \, {\left (121 \, A + 95 \, C\right )} a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{45045 \, d} \]

[In]

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

[Out]

-2/45045*(195*I*sqrt(2)*(121*A + 95*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 195*I*s
qrt(2)*(121*A + 95*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqrt(2)*(221*A + 1
75*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*I*sqrt(2)*(2
21*A + 175*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (3465*C*
a^3*cos(d*x + c)^5 + 12285*C*a^3*cos(d*x + c)^4 + 385*(13*A + 50*C)*a^3*cos(d*x + c)^3 + 585*(33*A + 38*C)*a^3
*cos(d*x + c)^2 + 154*(221*A + 175*C)*a^3*cos(d*x + c) + 390*(121*A + 95*C)*a^3)*sqrt(cos(d*x + c))*sin(d*x +
c))/d

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.16 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.29 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {A\,a^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {6\,A\,a^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,C\,a^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,C\,a^3\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^3\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {15}{4};\ \frac {19}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{15\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]

[In]

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

[Out]

(A*a^3*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (6*A*a^3*cos(c + d*x)^
(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*A*a^3*cos(c
+ d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2)) - (2*A*a^3
*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2))
- (2*C*a^3*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^
(1/2)) - (6*C*a^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c
+ d*x)^2)^(1/2)) - (6*C*a^3*cos(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13
*d*(sin(c + d*x)^2)^(1/2)) - (2*C*a^3*cos(c + d*x)^(15/2)*sin(c + d*x)*hypergeom([1/2, 15/4], 19/4, cos(c + d*
x)^2))/(15*d*(sin(c + d*x)^2)^(1/2))