\(\int \frac {\cos ^{\frac {3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [390]

Optimal result

Integrand size = 23, antiderivative size = 181 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}+\frac {11 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )} \]

[Out]

-119/10*(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+11/2*(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+11/2*sin(d*x+c)*cos(d*x
+c)^(1/2)/a^3/d-1/5*sin(d*x+c)*cos(d*x+c)^(1/2)/d/(a+a*sec(d*x+c))^3-2/3*sin(d*x+c)*cos(d*x+c)^(1/2)/a/d/(a+a*
sec(d*x+c))^2-119/30*sin(d*x+c)*cos(d*x+c)^(1/2)/d/(a^3+a^3*sec(d*x+c))

Rubi [A] (verified)

Time = 0.46 (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, 3902, 4105, 3872, 3854, 3856, 2720, 2719} \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {11 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}-\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 a^3 d}-\frac {119 \sin (c+d x) \sqrt {\cos (c+d x)}}{30 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 a d (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \sec (c+d x)+a)^3} \]

[In]

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

[Out]

(-119*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + (11*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + (11*Sqrt[Cos[c + d*x]
]*Sin[c + d*x])/(2*a^3*d) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*Sqrt[Cos[c + d
*x]]*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - (119*Sqrt[Cos[c + d*x]]*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 3854

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Csc[c + d*x])^(n + 1)/(b*d*n)), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 3902

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

Rule 4105

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(
2*m + 1))), x] - Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*B*n - a*A*
(2*m + n + 1) + (A*b - a*B)*(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] && 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 {1}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx \\ & = -\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {13 a}{2}+\frac {7}{2} a \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {69 a^2}{2}+25 a^2 \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4} \\ & = -\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {495 a^3}{4}+\frac {357}{4} a^3 \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{15 a^6} \\ & = -\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (119 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3}+\frac {\left (33 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{4 a^3} \\ & = \frac {11 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {119 \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}+\frac {\left (11 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{4 a^3} \\ & = -\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (c+d x)} \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {11 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{4 a^3} \\ & = -\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}+\frac {11 \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {119 \sqrt {\cos (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 = 1.44 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\sqrt {\cos (c+d x)} \csc (c+d x) \left ((511+2260 \cos (c+d x)-559 \cos (2 (c+d x))-910 \cos (3 (c+d x))+245 \cos (4 (c+d x))+90 \cos (5 (c+d x))-5 \cos (6 (c+d x))) \csc ^4(c+d x)-1320 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}-5440 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{2},\frac {7}{4},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{240 a^3 d} \]

[In]

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

[Out]

(Sqrt[Cos[c + d*x]]*Csc[c + d*x]*((511 + 2260*Cos[c + d*x] - 559*Cos[2*(c + d*x)] - 910*Cos[3*(c + d*x)] + 245
*Cos[4*(c + d*x)] + 90*Cos[5*(c + d*x)] - 5*Cos[6*(c + d*x)])*Csc[c + d*x]^4 - 1320*Hypergeometric2F1[1/4, 1/2
, 5/4, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2] - 5440*Cos[c + d*x]*Hypergeometric2F1[3/4, 7/2, 7/4, Cos[c + d*x]^
2]*Sqrt[Sin[c + d*x]^2]))/(240*a^3*d)

Maple [A] (verified)

Time = 8.74 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.56

method	result	size
default	\(-\frac {\sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+468 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+330 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+714 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1058 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+474 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-47 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3\right )}{60 a^{3} \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(283\)

[In]

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

[Out]

-1/60*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(160*cos(1/2*d*x+1/2*c)^10+468*cos(1/2*d*x+1/2*c
)^8+330*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*c
os(1/2*d*x+1/2*c)^5+714*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*cos(1/2*d*x+1/2*c)^5*El
lipticE(cos(1/2*d*x+1/2*c),2^(1/2))-1058*cos(1/2*d*x+1/2*c)^6+474*cos(1/2*d*x+1/2*c)^4-47*cos(1/2*d*x+1/2*c)^2
+3)/a^3/(-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)^5/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 = 354, normalized size of antiderivative = 1.96 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {2 \, {\left (20 \, \cos \left (d x + c\right )^{3} + 237 \, \cos \left (d x + c\right )^{2} + 376 \, \cos \left (d x + c\right ) + 165\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 165 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 3 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 165 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 3 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 357 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 3 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 357 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 3 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]

[In]

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

[Out]

1/60*(2*(20*cos(d*x + c)^3 + 237*cos(d*x + c)^2 + 376*cos(d*x + c) + 165)*sqrt(cos(d*x + c))*sin(d*x + c) - 16
5*(I*sqrt(2)*cos(d*x + c)^3 + 3*I*sqrt(2)*cos(d*x + c)^2 + 3*I*sqrt(2)*cos(d*x + c) + I*sqrt(2))*weierstrassPI
nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 165*(-I*sqrt(2)*cos(d*x + c)^3 - 3*I*sqrt(2)*cos(d*x + c)^2 - 3
*I*sqrt(2)*cos(d*x + c) - I*sqrt(2))*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 357*(I*sqrt(2
)*cos(d*x + c)^3 + 3*I*sqrt(2)*cos(d*x + c)^2 + 3*I*sqrt(2)*cos(d*x + c) + I*sqrt(2))*weierstrassZeta(-4, 0, w
eierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 357*(-I*sqrt(2)*cos(d*x + c)^3 - 3*I*sqrt(2)*cos(d
*x + c)^2 - 3*I*sqrt(2)*cos(d*x + c) - I*sqrt(2))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x +
c) - I*sin(d*x + c))))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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