\(\int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt {\sec (c+d x)}} \, dx\) [1288]

Optimal result

Integrand size = 43, antiderivative size = 307 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \]

[Out]

4/1155*a^3*(264*A+253*B+210*C)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/11*C*(a+a*cos(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)
^(3/2)+2/99*(11*B+6*C)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d/sec(d*x+c)^(3/2)+2/693*(99*A+143*B+105*C)*(a^3+a^
3*cos(d*x+c))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+4/231*a^3*(143*A+121*B+105*C)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+4/15*a
^3*(21*A+17*B+15*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*(143*A+121*B+105*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 = 0.91 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {4306, 3124, 3055, 3047, 3102, 2827, 2719, 2715, 2720} \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 (99 A+143 B+105 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{693 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (11 B+6 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{99 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \sec ^{\frac {3}{2}}(c+d x)} \]

[In]

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

[Out]

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

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 3124

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((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*((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*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*
B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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]

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 \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \left (\frac {1}{2} a (11 A+3 C)+\frac {1}{2} a (11 B+6 C) \cos (c+d x)\right ) \, dx}{11 a} \\ & = \frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^2 \left (\frac {3}{4} a^2 (33 A+11 B+15 C)+\frac {1}{4} a^2 (99 A+143 B+105 C) \cos (c+d x)\right ) \, dx}{99 a} \\ & = \frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x)) \left (\frac {15}{4} a^3 (33 A+22 B+21 C)+\frac {3}{4} a^3 (264 A+253 B+210 C) \cos (c+d x)\right ) \, dx}{693 a} \\ & = \frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \left (\frac {15}{4} a^4 (33 A+22 B+21 C)+\left (\frac {15}{4} a^4 (33 A+22 B+21 C)+\frac {3}{4} a^4 (264 A+253 B+210 C)\right ) \cos (c+d x)+\frac {3}{4} a^4 (264 A+253 B+210 C) \cos ^2(c+d x)\right ) \, dx}{693 a} \\ & = \frac {4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \left (\frac {231}{8} a^4 (21 A+17 B+15 C)+\frac {45}{8} a^4 (143 A+121 B+105 C) \cos (c+d x)\right ) \, dx}{3465 a} \\ & = \frac {4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{15} \left (2 a^3 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{77} \left (2 a^3 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {4 a^3 (21 A+17 B+15 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 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{231} \left (2 a^3 (143 A+121 B+105 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 (21 A+17 B+15 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 (143 A+121 B+105 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 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 B+6 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (99 A+143 B+105 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.57 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {a^3 \sqrt {\sec (c+d x)} \left (14784 (21 A+17 B+15 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+960 (143 A+121 B+105 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 (154 (108 A+151 B+165 C) \cos (c+d x)+5 (7260 A+6996 B+6741 C+36 (11 A+33 B+49 C) \cos (2 (c+d x))+154 (B+3 C) \cos (3 (c+d x))+63 C \cos (4 (c+d x)))) \sin (2 (c+d x))\right )}{55440 d} \]

[In]

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

[Out]

(a^3*Sqrt[Sec[c + d*x]]*(14784*(21*A + 17*B + 15*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 960*(143*A
+ 121*B + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(154*(108*A + 151*B + 165*C)*Cos[c + d*x] +
5*(7260*A + 6996*B + 6741*C + 36*(11*A + 33*B + 49*C)*Cos[2*(c + d*x)] + 154*(B + 3*C)*Cos[3*(c + d*x)] + 63*C
*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(55440*d)

Maple [A] (verified)

Time = 13.06 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.78

method	result	size
default	\(-\frac {4 \sqrt {\left (-1+2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{3} \left (10080 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 (-6160 B -43680 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 (3960 A +24200 B +77280 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 (-14256 A -37532 B -72240 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 (19866 A +29722 B +39270 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 (-6864 A -8118 B -8820 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 )+2145 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 )-4851 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 )+1815 B \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3927 B E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}+1575 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 )-3465 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 )}{3465 \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 {-1+2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(545\)
parts	\(\text {Expression too large to display}\)	\(1214\)

[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)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-4/3465*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(10080*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1
/2*c)^12+(-6160*B-43680*C)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(3960*A+24200*B+77280*C)*sin(1/2*d*x+1/2*c
)^8*cos(1/2*d*x+1/2*c)+(-14256*A-37532*B-72240*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(19866*A+29722*B+392
70*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-6864*A-8118*B-8820*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+
2145*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-485
1*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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+1815*B
*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3927*B*El
lipticE(cos(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)^(1/2)+1575*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3465*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)))/(-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 = 0.12 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.89 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (143 \, A + 121 \, B + 105 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (143 \, A + 121 \, B + 105 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (21 \, A + 17 \, B + 15 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (21 \, A + 17 \, B + 15 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (315 \, C a^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 45 \, {\left (11 \, A + 33 \, B + 42 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 77 \, {\left (27 \, A + 34 \, B + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 30 \, {\left (143 \, A + 121 \, B + 105 \, C\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{3465 \, d} \]

[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)^(1/2),x, algorithm="fricas")

[Out]

-2/3465*(15*I*sqrt(2)*(143*A + 121*B + 105*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) -
15*I*sqrt(2)*(143*A + 121*B + 105*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqr
t(2)*(21*A + 17*B + 15*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))
) + 231*I*sqrt(2)*(21*A + 17*B + 15*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*
sin(d*x + c))) - (315*C*a^3*cos(d*x + c)^5 + 385*(B + 3*C)*a^3*cos(d*x + c)^4 + 45*(11*A + 33*B + 42*C)*a^3*co
s(d*x + c)^3 + 77*(27*A + 34*B + 30*C)*a^3*cos(d*x + c)^2 + 30*(143*A + 121*B + 105*C)*a^3*cos(d*x + c))*sin(d
*x + c)/sqrt(cos(d*x + c)))/d

Sympy [F]

\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=a^{3} \left (\int \frac {A}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 A \cos {\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 A \cos ^{2}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {B \cos {\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 B \cos ^{2}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 B \cos ^{3}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {B \cos ^{4}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 C \cos ^{3}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 C \cos ^{4}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {C \cos ^{5}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx\right ) \]

[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)**(1/2),x)

[Out]

a**3*(Integral(A/sqrt(sec(c + d*x)), x) + Integral(3*A*cos(c + d*x)/sqrt(sec(c + d*x)), x) + Integral(3*A*cos(
c + d*x)**2/sqrt(sec(c + d*x)), x) + Integral(A*cos(c + d*x)**3/sqrt(sec(c + d*x)), x) + Integral(B*cos(c + d*
x)/sqrt(sec(c + d*x)), x) + Integral(3*B*cos(c + d*x)**2/sqrt(sec(c + d*x)), x) + Integral(3*B*cos(c + d*x)**3
/sqrt(sec(c + d*x)), x) + Integral(B*cos(c + d*x)**4/sqrt(sec(c + d*x)), x) + Integral(C*cos(c + d*x)**2/sqrt(
sec(c + d*x)), x) + Integral(3*C*cos(c + d*x)**3/sqrt(sec(c + d*x)), x) + Integral(3*C*cos(c + d*x)**4/sqrt(se
c(c + d*x)), x) + Integral(C*cos(c + d*x)**5/sqrt(sec(c + d*x)), x))

Maxima [F]

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

[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)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[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)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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