\(\int (a+a \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \sec ^{\frac {3}{2}}(c+d x) \, dx\) [1180]

Optimal result

Integrand size = 35, antiderivative size = 257 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (35 A+13 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d} \]

[Out]

-4/105*a^3*(35*A-41*C)*sin(d*x+c)/d/sec(d*x+c)^(1/2)-2/7*(7*A-C)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d/sec(d*x
+c)^(1/2)-2/35*(35*A-11*C)*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2*A*(a+a*cos(d*x+c))^3*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+4/21*a^3*(35*A+13*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 = 0.78 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4306, 3123, 3055, 3047, 3102, 2827, 2720, 2719} \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=-\frac {4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{35 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (35 A+13 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^3 (5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}-\frac {2 (7 A-C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 a d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sqrt {\sec (c+d x)} (a \cos (c+d x)+a)^3}{d} \]

[In]

Int[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),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*(35*A + 13*
C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(35*A - 41*C)*Sin[c + d*x]
)/(105*d*Sqrt[Sec[c + d*x]]) - (2*(7*A - C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d*Sqrt[Sec[c + d*x]]
) - (2*(35*A - 11*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Cos[c + d*
x])^3*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d

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 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 {3}{2}}(c+d x)} \, dx \\ & = \frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{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 (7 A-C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{a} \\ & = -\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{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 (35 A+C)-\frac {1}{4} a^2 (35 A-11 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{7 a} \\ & = -\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{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 {1}{2} a^3 (35 A+4 C)-\frac {1}{4} a^3 (35 A-41 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{35 a} \\ & = -\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} a^4 (35 A+4 C)+\left (-\frac {1}{4} a^4 (35 A-41 C)+\frac {1}{2} a^4 (35 A+4 C)\right ) \cos (c+d x)-\frac {1}{4} a^4 (35 A-41 C) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{35 a} \\ & = -\frac {4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {5}{8} a^4 (35 A+13 C)+\frac {21}{8} a^4 (5 A+7 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{105 a} \\ & = -\frac {4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{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}{21} \left (2 a^3 (35 A+13 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 (35 A+13 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 (7 A-C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 a d \sqrt {\sec (c+d x)}}-\frac {2 (35 A-11 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{35 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sqrt {\sec (c+d x)} \sin (c+d x)}{d} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 4.31 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.85 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {a^3 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (1680 i A \cos (c+d x)+2352 i C \cos (c+d x)+80 (35 A+13 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-112 i (5 A+7 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+840 A \sin (c+d x)+126 C \sin (c+d x)+140 A \sin (2 (c+d x))+550 C \sin (2 (c+d x))+126 C \sin (3 (c+d x))+15 C \sin (4 (c+d x))\right )}{420 d} \]

[In]

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

[Out]

(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((1680*I)*A*Cos[c + d*x] + (2352*I)*C*Cos[c + d*x] + 80*(35*A
+ 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(5*A + 7*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(
c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 840*A*Sin[c + d*x] + 126*C*Sin[c + d*x] +
140*A*Sin[2*(c + d*x)] + 550*C*Sin[2*(c + d*x)] + 126*C*Sin[3*(c + d*x)] + 15*C*Sin[4*(c + d*x)]))/(420*d*E^(I
*d*x))

Maple [A] (verified)

Time = 5.65 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.40

method	result	size
default	\(-\frac {4 a^{3} \left (120 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-432 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+70 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+602 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-140 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) A +175 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-105 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-208 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) C +65 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-1+2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(360\)
parts	\(\text {Expression too large to display}\)	\(1064\)

[In]

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

[Out]

-4/105*a^3*(120*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-432*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+70*A*c
os(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+602*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4-140*sin(1/2*d*x+1/2*c)^2*
cos(1/2*d*x+1/2*c)*A+175*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))-105*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))-208*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)*C+65*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))-147*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)))/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 = 213, normalized size of antiderivative = 0.83 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (35 \, A + 13 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (35 \, A + 13 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (5 \, A + 7 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (15 \, C a^{3} \cos \left (d x + c\right )^{3} + 63 \, C a^{3} \cos \left (d x + c\right )^{2} + 5 \, {\left (7 \, A + 26 \, C\right )} a^{3} \cos \left (d x + c\right ) + 105 \, A a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d} \]

[In]

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

[Out]

-2/105*(5*I*sqrt(2)*(35*A + 13*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 5*I*sqrt(2)*
(35*A + 13*C)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*I*sqrt(2)*(5*A + 7*C)*a^3*wei
erstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*I*sqrt(2)*(5*A + 7*C)*a^3*
weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (15*C*a^3*cos(d*x + c)^3 +
63*C*a^3*cos(d*x + c)^2 + 5*(7*A + 26*C)*a^3*cos(d*x + c) + 105*A*a^3)*sin(d*x + c)/sqrt(cos(d*x + c)))/d

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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