\(\int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx\) [216]

Optimal result

Integrand size = 33, antiderivative size = 292 \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=-\frac {7 (7 A-17 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}-\frac {(13 A-33 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}+\frac {7 (7 A-17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac {(13 A-33 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}+\frac {(A-B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )} \]

[Out]

-1/6*(13*A-33*B)*sec(d*x+c)^(3/2)*sin(d*x+c)/a^3/d+1/5*(A-B)*sec(d*x+c)^(9/2)*sin(d*x+c)/d/(a+a*sec(d*x+c))^3+
1/3*(A-2*B)*sec(d*x+c)^(7/2)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^2+7/30*(7*A-17*B)*sec(d*x+c)^(5/2)*sin(d*x+c)/d/(
a^3+a^3*sec(d*x+c))+7/10*(7*A-17*B)*sin(d*x+c)*sec(d*x+c)^(1/2)/a^3/d-7/10*(7*A-17*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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a^3/d-1/6*(13*
A-33*B)*(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)/a^3/d

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4104, 3872, 3853, 3856, 2719, 2720} \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {7 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{30 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {(13 A-33 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{6 a^3 d}+\frac {7 (7 A-17 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{10 a^3 d}-\frac {(13 A-33 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{6 a^3 d}-\frac {7 (7 A-17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac {(A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2} \]

[In]

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

[Out]

(-7*(7*A - 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 33*B)*
Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (7*(7*A - 17*B)*Sqrt[Sec[c + d*x]
]*Sin[c + d*x])/(10*a^3*d) - ((13*A - 33*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a^3*d) + ((A - B)*Sec[c + d*x]
^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A - 2*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*a*d*(a + a*
Sec[c + d*x])^2) + (7*(7*A - 17*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))

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 4104

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

Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (\frac {7}{2} a (A-B)-\frac {1}{2} a (3 A-13 B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = \frac {(A-B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (\frac {25}{2} a^2 (A-2 B)-\frac {3}{2} a^2 (8 A-23 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = \frac {(A-B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {21}{4} a^3 (7 A-17 B)-\frac {15}{4} a^3 (13 A-33 B) \sec (c+d x)\right ) \, dx}{15 a^6} \\ & = \frac {(A-B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(13 A-33 B) \int \sec ^{\frac {5}{2}}(c+d x) \, dx}{4 a^3}+\frac {(7 (7 A-17 B)) \int \sec ^{\frac {3}{2}}(c+d x) \, dx}{20 a^3} \\ & = \frac {7 (7 A-17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac {(13 A-33 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}+\frac {(A-B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(13 A-33 B) \int \sqrt {\sec (c+d x)} \, dx}{12 a^3}-\frac {(7 (7 A-17 B)) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3} \\ & = \frac {7 (7 A-17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac {(13 A-33 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}+\frac {(A-B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left ((13 A-33 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3}-\frac {\left (7 (7 A-17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3} \\ & = -\frac {7 (7 A-17 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}-\frac {(13 A-33 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}+\frac {7 (7 A-17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac {(13 A-33 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}+\frac {(A-B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 8.32 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.71 \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {e^{-i d x} (A+B \sec (c+d x)) \left (\frac {7 i (7 A-17 B) e^{i d x} \left (1+e^{i (c+d x)}\right )^6 \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )}{2 \sqrt {2} \left (1+e^{2 i (c+d x)}\right )^{3/2}}+\frac {e^{-\frac {1}{2} i (5 c+3 d x)} \cos \left (\frac {1}{2} (c+d x)\right ) \left (-i A \left (65+374 e^{i (c+d x)}+986 e^{2 i (c+d x)}+1658 e^{3 i (c+d x)}+2164 e^{4 i (c+d x)}+1954 e^{5 i (c+d x)}+1390 e^{6 i (c+d x)}+670 e^{7 i (c+d x)}+147 e^{8 i (c+d x)}\right )+i B \left (165+944 e^{i (c+d x)}+2476 e^{2 i (c+d x)}+4148 e^{3 i (c+d x)}+5134 e^{4 i (c+d x)}+4664 e^{5 i (c+d x)}+3340 e^{6 i (c+d x)}+1620 e^{7 i (c+d x)}+357 e^{8 i (c+d x)}\right )-5 (13 A-33 B) \left (1+e^{i (c+d x)}\right )^5 \left (1+e^{2 i (c+d x)}\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )\right ) \sec ^{\frac {5}{2}}(c+d x)}{8 \left (1+e^{2 i (c+d x)}\right )}\right )}{15 a^3 d (B+A \cos (c+d x)) (1+\sec (c+d x))^3} \]

[In]

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

[Out]

((A + B*Sec[c + d*x])*((((7*I)/2)*(7*A - 17*B)*E^(I*d*x)*(1 + E^(I*(c + d*x)))^6*Sqrt[E^(I*(c + d*x))/(1 + E^(
(2*I)*(c + d*x)))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/(Sqrt[2]*(1 + E^((2*I)*(c + d*x)))^
(3/2)) + (Cos[(c + d*x)/2]*((-I)*A*(65 + 374*E^(I*(c + d*x)) + 986*E^((2*I)*(c + d*x)) + 1658*E^((3*I)*(c + d*
x)) + 2164*E^((4*I)*(c + d*x)) + 1954*E^((5*I)*(c + d*x)) + 1390*E^((6*I)*(c + d*x)) + 670*E^((7*I)*(c + d*x))
+ 147*E^((8*I)*(c + d*x))) + I*B*(165 + 944*E^(I*(c + d*x)) + 2476*E^((2*I)*(c + d*x)) + 4148*E^((3*I)*(c + d
*x)) + 5134*E^((4*I)*(c + d*x)) + 4664*E^((5*I)*(c + d*x)) + 3340*E^((6*I)*(c + d*x)) + 1620*E^((7*I)*(c + d*x
)) + 357*E^((8*I)*(c + d*x))) - 5*(13*A - 33*B)*(1 + E^(I*(c + d*x)))^5*(1 + E^((2*I)*(c + d*x)))*Sqrt[Cos[c +
d*x]]*EllipticF[(c + d*x)/2, 2])*Sec[c + d*x]^(5/2))/(8*E^((I/2)*(5*c + 3*d*x))*(1 + E^((2*I)*(c + d*x))))))/
(15*a^3*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x])^3)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(875\) vs. \(2(312)=624\).

Time = 46.96 (sec) , antiderivative size = 876, normalized size of antiderivative = 3.00

method	result	size
default	\(\text {Expression too large to display}\)	\(876\)

[In]

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

[Out]

-1/60*(4*(-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)^2-1)^(1/2)*(sin(1/2*d*x+1/
2*c)^2)^(1/2)*(65*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-165*B*El
lipticF(cos(1/2*d*x+1/2*c),2^(1/2))+357*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*cos(1/2*d*x+1/2*c)*sin(1/2*d*
x+1/2*c)^6-10*(-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)^2-1)^(1/2)*(sin(1/2*d
*x+1/2*c)^2)^(1/2)*(65*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-165
*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+357*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*sin(1/2*d*x+1/2*c)^4*cos
(1/2*d*x+1/2*c)+8*(-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)^2-1)^(1/2)*(sin(1
/2*d*x+1/2*c)^2)^(1/2)*(65*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))
-165*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+357*B*EllipticE(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*(-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)^2-1)^(1/2)*(s
in(1/2*d*x+1/2*c)^2)^(1/2)*(65*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1
/2))-165*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+357*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*cos(1/2*d*x+1/2*
c)-168*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(7*A-17*B)*sin(1/2*d*x+1/2*c)^10+8*(-2*sin(1/2*d*x
+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(482*A-1167*B)*sin(1/2*d*x+1/2*c)^8-10*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*
d*x+1/2*c)^2)^(1/2)*(461*A-1111*B)*sin(1/2*d*x+1/2*c)^6+14*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2
)*(169*A-404*B)*sin(1/2*d*x+1/2*c)^4-(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(439*A-1029*B)*sin(1
/2*d*x+1/2*c)^2)/a^3/cos(1/2*d*x+1/2*c)^5/(-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)^2-1)^(3/2)/sin(1/2*d*x+1/2*c)/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 = 536, normalized size of antiderivative = 1.84 \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=-\frac {5 \, {\left (\sqrt {2} {\left (-13 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (-13 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-13 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-13 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (13 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (13 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (13 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (13 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (7 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (7 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (7 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (7 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )\right )} {\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 \, {\left (\sqrt {2} {\left (-7 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (-7 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-7 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-7 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )\right )} {\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 {2 \, {\left (21 \, {\left (7 \, A - 17 \, B\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (188 \, A - 453 \, B\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (59 \, A - 139 \, B\right )} \cos \left (d x + c\right )^{2} + 60 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right ) + 20 \, B\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right )\right )}} \]

[In]

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

[Out]

-1/60*(5*(sqrt(2)*(-13*I*A + 33*I*B)*cos(d*x + c)^4 + 3*sqrt(2)*(-13*I*A + 33*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*
(-13*I*A + 33*I*B)*cos(d*x + c)^2 + sqrt(2)*(-13*I*A + 33*I*B)*cos(d*x + c))*weierstrassPInverse(-4, 0, cos(d*
x + c) + I*sin(d*x + c)) + 5*(sqrt(2)*(13*I*A - 33*I*B)*cos(d*x + c)^4 + 3*sqrt(2)*(13*I*A - 33*I*B)*cos(d*x +
c)^3 + 3*sqrt(2)*(13*I*A - 33*I*B)*cos(d*x + c)^2 + sqrt(2)*(13*I*A - 33*I*B)*cos(d*x + c))*weierstrassPInver
se(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*(sqrt(2)*(7*I*A - 17*I*B)*cos(d*x + c)^4 + 3*sqrt(2)*(7*I*A - 17
*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(7*I*A - 17*I*B)*cos(d*x + c)^2 + sqrt(2)*(7*I*A - 17*I*B)*cos(d*x + c))*weie
rstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*(sqrt(2)*(-7*I*A + 17*I*B)*
cos(d*x + c)^4 + 3*sqrt(2)*(-7*I*A + 17*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(-7*I*A + 17*I*B)*cos(d*x + c)^2 + sqr
t(2)*(-7*I*A + 17*I*B)*cos(d*x + c))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*
x + c))) - 2*(21*(7*A - 17*B)*cos(d*x + c)^4 + 2*(188*A - 453*B)*cos(d*x + c)^3 + 5*(59*A - 139*B)*cos(d*x + c
)^2 + 60*(A - 2*B)*cos(d*x + c) + 20*B)*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^3*d*cos(d*x + c)^4 + 3*a^3*d*cos(d
*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + a^3*d*cos(d*x + c))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)