\(\int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx\) [396]

Optimal result

Integrand size = 23, antiderivative size = 181 \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=-\frac {49 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {13 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{6 a^3 d}+\frac {49 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )} \]

[Out]

-49/10*(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))/a^3/d-13/6*(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))/a^3/d-1/5*sin(d*x+c)/d/cos(d*x
+c)^(7/2)/(a+a*sec(d*x+c))^3-8/15*sin(d*x+c)/a/d/cos(d*x+c)^(5/2)/(a+a*sec(d*x+c))^2-13/6*sin(d*x+c)/d/cos(d*x
+c)^(3/2)/(a^3+a^3*sec(d*x+c))+49/10*sin(d*x+c)/a^3/d/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4349, 3901, 4104, 3872, 3856, 2720, 3853, 2719} \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=-\frac {13 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{6 a^3 d}-\frac {49 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {49 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^3} \]

[In]

Int[1/(Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3),x]

[Out]

(-49*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - (13*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + (49*Sin[c + d*x])/(10*
a^3*d*Sqrt[Cos[c + d*x]]) - Sin[c + d*x]/(5*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3) - (8*Sin[c + d*x])/(1
5*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2) - (13*Sin[c + d*x])/(6*d*Cos[c + d*x]^(3/2)*(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 3901

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

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]

Rule 4349

Int[(u_)*((c_.)*sin[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Csc[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {9}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx \\ & = -\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (\frac {5 a}{2}-\frac {11}{2} a \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (12 a^2-\frac {41}{2} a^2 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = -\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \left (\frac {65 a^3}{4}-\frac {147}{4} a^3 \sec (c+d x)\right ) \, dx}{15 a^6} \\ & = -\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (13 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{12 a^3}+\frac {\left (49 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx}{20 a^3} \\ & = \frac {49 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {13 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3}-\frac {\left (49 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3} \\ & = -\frac {13 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{6 a^3 d}+\frac {49 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {49 \int \sqrt {\cos (c+d x)} \, dx}{20 a^3} \\ & = -\frac {49 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {13 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{6 a^3 d}+\frac {49 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {8 \sin (c+d x)}{15 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {13 \sin (c+d x)}{6 d \cos ^{\frac {3}{2}}(c+d x) \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 = 2.50 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.06 \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\left (1284 \cos \left (\frac {1}{2} (c-d x)\right )+921 \cos \left (\frac {1}{2} (3 c+d x)\right )+1243 \cos \left (\frac {1}{2} (c+3 d x)\right )+374 \cos \left (\frac {1}{2} (5 c+3 d x)\right )+670 \cos \left (\frac {1}{2} (3 c+5 d x)\right )+65 \cos \left (\frac {1}{2} (7 c+5 d x)\right )+147 \cos \left (\frac {1}{2} (5 c+7 d x)\right )\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )}{16 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {4 i \sqrt {2} e^{-i (c+d x)} \left (147 \left (1+e^{2 i (c+d x)}\right )+147 \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 i (c+d x)}\right )-65 e^{i (c+d x)} \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-e^{2 i (c+d x)}\right )\right ) \sec ^3(c+d x)}{d \left (-1+e^{2 i c}\right ) \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{15 a^3 (1+\sec (c+d x))^3} \]

[In]

Integrate[1/(Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3),x]

[Out]

(Cos[(c + d*x)/2]^6*(((1284*Cos[(c - d*x)/2] + 921*Cos[(3*c + d*x)/2] + 1243*Cos[(c + 3*d*x)/2] + 374*Cos[(5*c
+ 3*d*x)/2] + 670*Cos[(3*c + 5*d*x)/2] + 65*Cos[(7*c + 5*d*x)/2] + 147*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2
]*Sec[(c + d*x)/2]^5)/(16*d*Cos[c + d*x]^(7/2)) - ((4*I)*Sqrt[2]*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^
((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 65*E^(I*(c
+ d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))
])*Sec[c + d*x]^3)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(1
5*a^3*(1 + Sec[c + d*x])^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(554\) vs. \(2(213)=426\).

Time = 9.62 (sec) , antiderivative size = 555, normalized size of antiderivative = 3.07

method	result	size
default	\(-\frac {-2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (65 \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+4 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (65 \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (65 \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+588 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-1634 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+1488 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-439 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{60 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(555\)

[In]

int(1/cos(d*x+c)^(9/2)/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/60*(-2*(sin(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)*(65*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*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)^4+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)*(-2*sin(1/2*d*x+
1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(65*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*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*(sin(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)*(65*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*Ellip
ticE(cos(1/2*d*x+1/2*c),2^(1/2)))*cos(1/2*d*x+1/2*c)+588*(-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)^8-1634*(-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)^6+1488*(-2*s
in(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)^4-439*(-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)/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)/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 = 394, normalized size of antiderivative = 2.18 \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\frac {2 \, {\left (147 \, \cos \left (d x + c\right )^{3} + 376 \, \cos \left (d x + c\right )^{2} + 295 \, \cos \left (d x + c\right ) + 60\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 65 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{4} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} - i \, \sqrt {2} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 65 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{4} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + i \, \sqrt {2} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 147 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{4} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + i \, \sqrt {2} \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 147 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{4} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} - i \, \sqrt {2} \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right )\right )}} \]

[In]

integrate(1/cos(d*x+c)^(9/2)/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/60*(2*(147*cos(d*x + c)^3 + 376*cos(d*x + c)^2 + 295*cos(d*x + c) + 60)*sqrt(cos(d*x + c))*sin(d*x + c) - 65
*(-I*sqrt(2)*cos(d*x + c)^4 - 3*I*sqrt(2)*cos(d*x + c)^3 - 3*I*sqrt(2)*cos(d*x + c)^2 - I*sqrt(2)*cos(d*x + c)
)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 65*(I*sqrt(2)*cos(d*x + c)^4 + 3*I*sqrt(2)*cos(d
*x + c)^3 + 3*I*sqrt(2)*cos(d*x + c)^2 + I*sqrt(2)*cos(d*x + c))*weierstrassPInverse(-4, 0, cos(d*x + c) - I*s
in(d*x + c)) - 147*(I*sqrt(2)*cos(d*x + c)^4 + 3*I*sqrt(2)*cos(d*x + c)^3 + 3*I*sqrt(2)*cos(d*x + c)^2 + I*sqr
t(2)*cos(d*x + c))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 147*(-I
*sqrt(2)*cos(d*x + c)^4 - 3*I*sqrt(2)*cos(d*x + c)^3 - 3*I*sqrt(2)*cos(d*x + c)^2 - I*sqrt(2)*cos(d*x + c))*we
ierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(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. \[ \int \frac {1}{\cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(1/cos(d*x+c)**(9/2)/(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

Maxima [F(-1)]

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

[In]

integrate(1/cos(d*x+c)^(9/2)/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

Timed out

Giac [F]

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

[In]

integrate(1/cos(d*x+c)^(9/2)/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(1/((a*sec(d*x + c) + a)^3*cos(d*x + c)^(9/2)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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