\(\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx\) [470]

Optimal result

Integrand size = 33, antiderivative size = 244 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {4 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d} \]

[Out]

4/105*a^3*(41*A+42*B)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/7*a*A*sec(d*x+c)^(3/2)*(a+a*sec(d*x+c))^2*sin(d*x+c)/d+2
/35*(11*A+7*B)*sec(d*x+c)^(3/2)*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d+4/5*a^3*(7*A+9*B)*sin(d*x+c)*sec(d*x+c)^(1/2
)/d-4/5*a^3*(7*A+9*B)*(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))*co
s(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+4/21*a^3*(13*A+21*B)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellipti
cF(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 = 0.54 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3039, 4103, 4082, 3872, 3856, 2720, 3853, 2719} \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {4 a^3 (41 A+42 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {2 (11 A+7 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{35 d}+\frac {4 a^3 (7 A+9 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {4 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \]

[In]

Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]

[Out]

(-4*a^3*(7*A + 9*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 21
*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(7*A + 9*B)*Sqrt[Sec[c +
d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(41*A + 42*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*A*Sec[c + d*x
]^(3/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(11*A + 7*B)*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x
])*Sin[c + d*x])/(35*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 3039

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[g^(m + n), Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(
d + c*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[p] && I
ntegerQ[m] && IntegerQ[n]

Rule 3853

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

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 4082

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.
) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*(n + 1))), x] + Dist[1/(n + 1), Int[(d
*Csc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e,
f, A, B}, x] && NeQ[A*b - a*B, 0] &&  !LeQ[n, -1]

Rule 4103

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m +
n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*d*(m + n) + B*(b*d
*n) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] &&
NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 (B+A \sec (c+d x)) \, dx \\ & = \frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2}{7} \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (\frac {1}{2} a (A+7 B)+\frac {1}{2} a (11 A+7 B) \sec (c+d x)\right ) \, dx \\ & = \frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {4}{35} \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x)) \left (\frac {1}{2} a^2 (8 A+21 B)+\frac {1}{2} a^2 (41 A+42 B) \sec (c+d x)\right ) \, dx \\ & = \frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {8}{105} \int \sqrt {\sec (c+d x)} \left (\frac {5}{4} a^3 (13 A+21 B)+\frac {21}{4} a^3 (7 A+9 B) \sec (c+d x)\right ) \, dx \\ & = \frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {1}{5} \left (2 a^3 (7 A+9 B)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{21} \left (2 a^3 (13 A+21 B)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {4 a^3 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac {1}{5} \left (2 a^3 (7 A+9 B)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {4 a^3 (13 A+21 B) \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 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac {1}{5} \left (2 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {4 a^3 (7 A+9 B) \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 (13 A+21 B) \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 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 3.52 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.78 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {a^3 e^{-i d x} (1+\cos (c+d x))^3 \csc (c) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (7 \sqrt {2} (7 A+9 B) e^{2 i d x} \left (-1+e^{2 i c}\right ) \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )-\frac {e^{-i (c-d x)} \left (-1+e^{2 i c}\right ) \left (21 B \left (-5+16 e^{i (c+d x)}-5 e^{2 i (c+d x)}+54 e^{3 i (c+d x)}+5 e^{4 i (c+d x)}+56 e^{5 i (c+d x)}+5 e^{6 i (c+d x)}+18 e^{7 i (c+d x)}\right )+2 A \left (-65+84 e^{i (c+d x)}-95 e^{2 i (c+d x)}+441 e^{3 i (c+d x)}+95 e^{4 i (c+d x)}+504 e^{5 i (c+d x)}+65 e^{6 i (c+d x)}+147 e^{7 i (c+d x)}\right )+10 i (13 A+21 B) \left (1+e^{2 i (c+d x)}\right )^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )\right ) \sqrt {\sec (c+d x)}}{2 \left (1+e^{2 i (c+d x)}\right )^3}\right )}{420 d} \]

[In]

Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]

[Out]

(a^3*(1 + Cos[c + d*x])^3*Csc[c]*Sec[(c + d*x)/2]^6*(7*Sqrt[2]*(7*A + 9*B)*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sq
rt[E^(I*(c + d*x))/(1 + E^((2*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))] - ((-1 + E^((2*I)*c))*(21*B*(-5 + 16*E^(I*(c + d*x)) - 5*E^((2*I)*(c + d*x)) + 54*E^((3*I
)*(c + d*x)) + 5*E^((4*I)*(c + d*x)) + 56*E^((5*I)*(c + d*x)) + 5*E^((6*I)*(c + d*x)) + 18*E^((7*I)*(c + d*x))
) + 2*A*(-65 + 84*E^(I*(c + d*x)) - 95*E^((2*I)*(c + d*x)) + 441*E^((3*I)*(c + d*x)) + 95*E^((4*I)*(c + d*x))
+ 504*E^((5*I)*(c + d*x)) + 65*E^((6*I)*(c + d*x)) + 147*E^((7*I)*(c + d*x))) + (10*I)*(13*A + 21*B)*(1 + E^((
2*I)*(c + d*x)))^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])*Sqrt[Sec[c + d*x]])/(2*E^(I*(c - d*x))*(1 + E
^((2*I)*(c + d*x)))^3)))/(420*d*E^(I*d*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(901\) vs. \(2(268)=536\).

Time = 179.70 (sec) , antiderivative size = 902, normalized size of antiderivative = 3.70

method	result	size
default	\(\text {Expression too large to display}\)	\(902\)
parts	\(\text {Expression too large to display}\)	\(1171\)

[In]

int((a+cos(d*x+c)*a)^3*(A+B*cos(d*x+c))*sec(d*x+c)^(9/2),x,method=_RETURNVERBOSE)

[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*(1/8*B*(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*A*(-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)/(cos(1/2*d*x+1/2*c
)^2-1/2)^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)/(cos(1/2*d*x+1/2*c)^2-
1/2)^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)))+1/5*(1/8*B+3/8*A)/(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*(24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-12*(2*
sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1
/2*c)^4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^
(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*(
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)+(3/8*A+3/8*B)*(-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)/(cos(1/2*d*x+1/2*c)^2-1/2)^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)))+(1/8*A+
3/8*B)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2
*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*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)*Ellipti
cE(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

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.08 \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=-\frac {2 \, {\left (5 i \, \sqrt {2} {\left (13 \, A + 21 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (13 \, A + 21 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (7 \, A + 9 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (7 \, A + 9 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (42 \, {\left (7 \, A + 9 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 5 \, {\left (26 \, A + 21 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 21 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + 15 \, A a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d \cos \left (d x + c\right )^{3}} \]

[In]

integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c))*sec(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

-2/105*(5*I*sqrt(2)*(13*A + 21*B)*a^3*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))
- 5*I*sqrt(2)*(13*A + 21*B)*a^3*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21
*I*sqrt(2)*(7*A + 9*B)*a^3*cos(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*s
in(d*x + c))) - 21*I*sqrt(2)*(7*A + 9*B)*a^3*cos(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0,
cos(d*x + c) - I*sin(d*x + c))) - (42*(7*A + 9*B)*a^3*cos(d*x + c)^3 + 5*(26*A + 21*B)*a^3*cos(d*x + c)^2 + 21
*(3*A + B)*a^3*cos(d*x + c) + 15*A*a^3)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^3)

Sympy [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+a*cos(d*x+c))**3*(A+B*cos(d*x+c))*sec(d*x+c)**(9/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \]

[In]

integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c))*sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

integrate((B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^3*sec(d*x + c)^(9/2), x)

Giac [F]

\[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x } \]

[In]

integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c))*sec(d*x+c)^(9/2),x, algorithm="giac")

[Out]

integrate((B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^3*sec(d*x + c)^(9/2), x)

Mupad [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((A + B*cos(c + d*x))*(1/cos(c + d*x))^(9/2)*(a + a*cos(c + d*x))^3,x)

[Out]

int((A + B*cos(c + d*x))*(1/cos(c + d*x))^(9/2)*(a + a*cos(c + d*x))^3, x)