3.1181 \(\int (a+a \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \sqrt{\sec (c+d x)} \, dx\)

Optimal. Leaf size=253 \[ \frac{8 a^3 (21 A+16 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+73 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{315 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (27 A+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \sqrt{\sec (c+d x)}} \]

[Out]

(4*a^3*(27*A + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A +
11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(21*A + 16*C)*Sin[c + d
*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (4*C*(a
^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]]) + (2*(63*A + 73*C)*(a^3 + a^3*Cos[c + d*x])
*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.66961, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4221, 3046, 2976, 2968, 3023, 2748, 2641, 2639} \[ \frac{8 a^3 (21 A+16 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+73 C) \sin (c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{315 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (27 A+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 C \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \sqrt{\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^3*(27*A + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A +
11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(21*A + 16*C)*Sin[c + d
*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Sqrt[Sec[c + d*x]]) + (4*C*(a
^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]]) + (2*(63*A + 73*C)*(a^3 + a^3*Cos[c + d*x])
*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]])

Rule 4221

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]

Rule 3046

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*Simp
[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]

Rule 2976

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)))*S
in[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 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 3023

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 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 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 (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \, dx &=\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 )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^3 \left (\frac{1}{2} a (9 A+C)+3 a C \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{9 a}\\ &=\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{4 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\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 (63 A+13 C)+\frac{1}{4} a^2 (63 A+73 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{63 a}\\ &=\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{4 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+73 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x)) \left (\frac{3}{4} a^3 (63 A+23 C)+\frac{9}{2} a^3 (21 A+16 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{315 a}\\ &=\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{4 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+73 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{4} a^4 (63 A+23 C)+\left (\frac{9}{2} a^4 (21 A+16 C)+\frac{3}{4} a^4 (63 A+23 C)\right ) \cos (c+d x)+\frac{9}{2} a^4 (21 A+16 C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{315 a}\\ &=\frac{8 a^3 (21 A+16 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{4 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+73 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{45}{8} a^4 (21 A+11 C)+\frac{63}{8} a^4 (27 A+17 C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{945 a}\\ &=\frac{8 a^3 (21 A+16 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{4 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+73 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}+\frac{1}{21} \left (2 a^3 (21 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{15} \left (2 a^3 (27 A+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{4 a^3 (27 A+17 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 (21 A+11 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 d}+\frac{8 a^3 (21 A+16 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 C (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt{\sec (c+d x)}}+\frac{4 C \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+73 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 2.58173, size = 206, normalized size = 0.81 \[ \frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (-112 i (27 A+17 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right )+\cos (c+d x) (30 (84 A+97 C) \sin (c+d x)+14 (18 A+73 C) \sin (2 (c+d x))+9072 i A+270 C \sin (3 (c+d x))+35 C \sin (4 (c+d x))+5712 i C)+240 (21 A+11 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{1260 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(240*(21*A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2
] - (112*I)*(27*A + 17*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))] + Cos[c + d*x]*((9072*I)*A + (5712*I)*C + 30*(84*A + 97*C)*Sin[c + d*x] + 14*(18*A + 73*C)*Si
n[2*(c + d*x)] + 270*C*Sin[3*(c + d*x)] + 35*C*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))

________________________________________________________________________________________

Maple [A]  time = 1.328, size = 408, normalized size = 1.6 \begin{align*} -{\frac{4\,{a}^{3}}{315\,d}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( -560\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}\cos \left ( 1/2\,dx+c/2 \right ) +2200\,C \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}\cos \left ( 1/2\,dx+c/2 \right ) + \left ( -252\,A-3412\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( 882\,A+2702\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) + \left ( -378\,A-738\,C \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +315\,A\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -567\,A\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +165\,C\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -357\,C\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-560*C*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1
/2*c)+2200*C*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-252*A-3412*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(
882*A+2702*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-378*A-738*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+3
15*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))-567*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))+165*C*(si
n(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))-357*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)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C a^{3} \cos \left (d x + c\right )^{5} + 3 \, C a^{3} \cos \left (d x + c\right )^{4} +{\left (A + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} +{\left (3 \, A + C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \, A a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \sqrt{\sec \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

integral((C*a^3*cos(d*x + c)^5 + 3*C*a^3*cos(d*x + c)^4 + (A + 3*C)*a^3*cos(d*x + c)^3 + (3*A + C)*a^3*cos(d*x
+ c)^2 + 3*A*a^3*cos(d*x + c) + A*a^3)*sqrt(sec(d*x + c)), x)

________________________________________________________________________________________

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+C*cos(d*x+c)**2)*sec(d*x+c)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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