\(\int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 (A+C \sec ^2(c+d x)) \, dx\) [215]

Optimal result

Integrand size = 35, antiderivative size = 237 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 a^2 (5 A+3 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {8 a^2 (7 A+3 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^2 (5 A+3 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a^2 (35 A+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {8 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d} \]

[Out]

2/105*a^2*(35*A+33*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/7*C*sec(d*x+c)^(3/2)*(a+a*sec(d*x+c))^2*sin(d*x+c)/d+8/3
5*C*sec(d*x+c)^(3/2)*(a^2+a^2*sec(d*x+c))*sin(d*x+c)/d+4/5*a^2*(5*A+3*C)*sin(d*x+c)*sec(d*x+c)^(1/2)/d-4/5*a^2
*(5*A+3*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+8/21*a^2*(7*A+3*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 = 0.48 (sec) , antiderivative size = 237, 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 = {4174, 4103, 4082, 3872, 3856, 2720, 3853, 2719} \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a^2 (35 A+33 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {4 a^2 (5 A+3 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {8 a^2 (7 A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {4 a^2 (5 A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {8 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{35 d}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \]

[In]

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

[Out]

(-4*a^2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(7*A + 3*C
)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(5*A + 3*C)*Sqrt[Sec[c + d*
x]]*Sin[c + d*x])/(5*d) + (2*a^2*(35*A + 33*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(3
/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + (8*C*Sec[c + d*x]^(3/2)*(a^2 + a^2*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 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]

Rule 4174

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (\frac {1}{2} a (7 A+C)+2 a C \sec (c+d x)\right ) \, dx}{7 a} \\ & = \frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {8 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {4 \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x)) \left (\frac {1}{4} a^2 (35 A+9 C)+\frac {1}{4} a^2 (35 A+33 C) \sec (c+d x)\right ) \, dx}{35 a} \\ & = \frac {2 a^2 (35 A+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {8 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {8 \int \sqrt {\sec (c+d x)} \left (\frac {5}{2} a^3 (7 A+3 C)+\frac {21}{4} a^3 (5 A+3 C) \sec (c+d x)\right ) \, dx}{105 a} \\ & = \frac {2 a^2 (35 A+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {8 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {1}{5} \left (2 a^2 (5 A+3 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{21} \left (4 a^2 (7 A+3 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {4 a^2 (5 A+3 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a^2 (35 A+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {8 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac {1}{5} \left (2 a^2 (5 A+3 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (4 a^2 (7 A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {8 a^2 (7 A+3 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^2 (5 A+3 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a^2 (35 A+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {8 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac {1}{5} \left (2 a^2 (5 A+3 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {4 a^2 (5 A+3 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 {8 a^2 (7 A+3 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^2 (5 A+3 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a^2 (35 A+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {8 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{35 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 = 8.99 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.84 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 e^{-i d x} \cos ^4(c+d x) \csc (c) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (1+\sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \left (7 \sqrt {2} (5 A+3 C) e^{2 i d x} \left (-1+e^{2 i c}\right ) \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )-\frac {e^{-i (c-d x)} \left (-1+e^{2 i c}\right ) \left (35 A \left (1+e^{2 i (c+d x)}\right )^2 \left (-1+6 e^{i (c+d x)}+e^{2 i (c+d x)}+6 e^{3 i (c+d x)}\right )+6 C \left (-10+7 e^{i (c+d x)}-20 e^{2 i (c+d x)}+63 e^{3 i (c+d x)}+20 e^{4 i (c+d x)}+77 e^{5 i (c+d x)}+10 e^{6 i (c+d x)}+21 e^{7 i (c+d x)}\right )\right ) \sqrt {\sec (c+d x)}}{2 \left (1+e^{2 i (c+d x)}\right )^3}+20 (7 A+3 C) e^{i d x} \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)} \sin (c)\right )}{105 d (A+2 C+A \cos (2 (c+d x)))} \]

[In]

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

[Out]

(a^2*Cos[c + d*x]^4*Csc[c]*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(7*Sqrt[2]*(5*A + 3*
C)*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sqrt[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))*(35*A*(1 + E^((2*I)*(c + d*x)
))^2*(-1 + 6*E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) + 6*E^((3*I)*(c + d*x))) + 6*C*(-10 + 7*E^(I*(c + d*x)) - 2
0*E^((2*I)*(c + d*x)) + 63*E^((3*I)*(c + d*x)) + 20*E^((4*I)*(c + d*x)) + 77*E^((5*I)*(c + d*x)) + 10*E^((6*I)
*(c + d*x)) + 21*E^((7*I)*(c + d*x))))*Sqrt[Sec[c + d*x]])/(2*E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^3) + 2
0*(7*A + 3*C)*E^(I*d*x)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*Sin[c]))/(105*d*E^(I*d
*x)*(A + 2*C + A*Cos[2*(c + d*x)]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(891\) vs. \(2(261)=522\).

Time = 3.76 (sec) , antiderivative size = 892, normalized size of antiderivative = 3.76

method	result	size
default	\(\text {Expression too large to display}\)	\(892\)
parts	\(\text {Expression too large to display}\)	\(1153\)

[In]

int((a+a*sec(d*x+c))^2*(A+C*sec(d*x+c)^2)*sec(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-a^2*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A*(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)
)+2*C*(-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)))+4*A/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*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2-(sin(1/2*d*x+1/2*c)^2)
^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2))+4/5*C/sin(1/2*d*x+1/2*c)^2/(8*s
in(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)*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2
*c)-12*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(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)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+12*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(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+8*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/
2*c)^2-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(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)+8*(1/4*A+1/4*C)*(-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))))/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 = 0.12 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.09 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (10 i \, \sqrt {2} {\left (7 \, A + 3 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 10 i \, \sqrt {2} {\left (7 \, A + 3 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (5 \, A + 3 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (5 \, A + 3 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (42 \, {\left (5 \, A + 3 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 5 \, {\left (7 \, A + 12 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 42 \, C a^{2} \cos \left (d x + c\right ) + 15 \, C a^{2}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d \cos \left (d x + c\right )^{3}} \]

[In]

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

[Out]

-2/105*(10*I*sqrt(2)*(7*A + 3*C)*a^2*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))
- 10*I*sqrt(2)*(7*A + 3*C)*a^2*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*I
*sqrt(2)*(5*A + 3*C)*a^2*cos(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin
(d*x + c))) - 21*I*sqrt(2)*(5*A + 3*C)*a^2*cos(d*x + c)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, co
s(d*x + c) - I*sin(d*x + c))) - (42*(5*A + 3*C)*a^2*cos(d*x + c)^3 + 5*(7*A + 12*C)*a^2*cos(d*x + c)^2 + 42*C*
a^2*cos(d*x + c) + 15*C*a^2)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^3)

Sympy [F(-1)]

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

[In]

integrate((a+a*sec(d*x+c))**2*(A+C*sec(d*x+c)**2)*sec(d*x+c)**(1/2),x)

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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