\(\int (a+b \cos (c+d x))^{5/2} (A+C \cos ^2(c+d x)) \sec ^5(c+d x) \, dx\) [645]

Optimal result

Integrand size = 35, antiderivative size = 437 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=-\frac {b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{192 a d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {b \left (4 a^2 (89 A+132 C)+b^2 (133 A+384 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{192 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (5 A b^4-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{64 a d \sqrt {a+b \cos (c+d x)}}+\frac {b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a d}+\frac {\left (5 A b^2+4 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{32 d}+\frac {5 A b (a+b \cos (c+d x))^{3/2} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \tan (c+d x)}{4 d} \]

[Out]

-1/192*b*(15*A*b^2+4*a^2*(71*A+108*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)*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/a/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+1/192*b*(4*a^2*(89*A+
132*C)+b^2*(133*A+384*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)
*(b/(a+b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/d/(a+b*cos(d*x+c))^(1/2)-1/64*(5*A*b^4-120*a^2*b^2*(A+2*C)-16
*a^4*(3*A+4*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(b/(a+
b))^(1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/a/d/(a+b*cos(d*x+c))^(1/2)+5/24*A*b*(a+b*cos(d*x+c))^(3/2)*sec(d*x+c
)^2*tan(d*x+c)/d+1/4*A*(a+b*cos(d*x+c))^(5/2)*sec(d*x+c)^3*tan(d*x+c)/d+1/192*b*(15*A*b^2+4*a^2*(71*A+108*C))*
(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a/d+1/32*(5*A*b^2+4*a^2*(3*A+4*C))*sec(d*x+c)*(a+b*cos(d*x+c))^(1/2)*tan(d*x
+c)/d

Rubi [A] (verified)

Time = 2.07 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {3127, 3126, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{192 a d}+\frac {b \left (4 a^2 (89 A+132 C)+b^2 (133 A+384 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{192 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (4 a^2 (71 A+108 C)+15 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{192 a d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (4 a^2 (3 A+4 C)+5 A b^2\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{32 d}-\frac {\left (-16 a^4 (3 A+4 C)-120 a^2 b^2 (A+2 C)+5 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{64 a d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{5/2}}{4 d}+\frac {5 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{24 d} \]

[In]

Int[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]

[Out]

-1/192*(b*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(a
*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (b*(4*a^2*(89*A + 132*C) + b^2*(133*A + 384*C))*Sqrt[(a + b*Cos[c + d
*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(192*d*Sqrt[a + b*Cos[c + d*x]]) - ((5*A*b^4 - 120*a^2*b^
2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)]
)/(64*a*d*Sqrt[a + b*Cos[c + d*x]]) + (b*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*
x])/(192*a*d) + ((5*A*b^2 + 4*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d) + (5
*A*b*(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*
x]^3*Tan[c + d*x])/(4*d)

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3081

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a +
b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3126

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)
*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) +
(c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*
c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n +
1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2
, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3127

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Si
n[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^
(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n
+ 2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2
*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3138

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) +
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]]
, x], x] - Dist[1/(b*d), Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e +
f*x]]*(c + d*Sin[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \cos (c+d x))^{3/2} \left (\frac {5 A b}{2}+a (3 A+4 C) \cos (c+d x)+\frac {1}{2} b (A+8 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {5 A b (a+b \cos (c+d x))^{3/2} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{12} \int \sqrt {a+b \cos (c+d x)} \left (\frac {3}{4} \left (5 A b^2+4 a^2 (3 A+4 C)\right )+\frac {1}{2} a b (31 A+48 C) \cos (c+d x)+\frac {1}{4} b^2 (11 A+48 C) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {\left (5 A b^2+4 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{32 d}+\frac {5 A b (a+b \cos (c+d x))^{3/2} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{24} \int \frac {\left (\frac {1}{8} \left (15 A b^3+8 a^2 \left (\frac {71 A b}{2}+54 b C\right )\right )+\frac {1}{4} a \left (12 a^2 (3 A+4 C)+b^2 (161 A+288 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (12 a^2 (3 A+4 C)+b^2 (59 A+192 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a d}+\frac {\left (5 A b^2+4 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{32 d}+\frac {5 A b (a+b \cos (c+d x))^{3/2} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {\int \frac {\left (-\frac {3}{16} \left (5 A b^4-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right )+\frac {1}{8} a b \left (12 a^2 (3 A+4 C)+b^2 (59 A+192 C)\right ) \cos (c+d x)-\frac {1}{16} b^2 \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{24 a} \\ & = \frac {b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a d}+\frac {\left (5 A b^2+4 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{32 d}+\frac {5 A b (a+b \cos (c+d x))^{3/2} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {\int \frac {\left (\frac {3}{16} b \left (5 A b^4-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right )-\frac {1}{16} a b^2 \left (4 a^2 (89 A+132 C)+b^2 (133 A+384 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{24 a b}-\frac {\left (b \left (15 A b^2+4 a^2 (71 A+108 C)\right )\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{384 a} \\ & = \frac {b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a d}+\frac {\left (5 A b^2+4 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{32 d}+\frac {5 A b (a+b \cos (c+d x))^{3/2} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {\left (5 A b^4-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{128 a}+\frac {1}{384} \left (b \left (4 a^2 (89 A+132 C)+b^2 (133 A+384 C)\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx-\frac {\left (b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{384 a \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \\ & = -\frac {b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{192 a d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a d}+\frac {\left (5 A b^2+4 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{32 d}+\frac {5 A b (a+b \cos (c+d x))^{3/2} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {\left (\left (5 A b^4-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{128 a \sqrt {a+b \cos (c+d x)}}+\frac {\left (b \left (4 a^2 (89 A+132 C)+b^2 (133 A+384 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{384 \sqrt {a+b \cos (c+d x)}} \\ & = -\frac {b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{192 a d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {b \left (4 a^2 (89 A+132 C)+b^2 (133 A+384 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{192 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (5 A b^4-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{64 a d \sqrt {a+b \cos (c+d x)}}+\frac {b \left (15 A b^2+4 a^2 (71 A+108 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a d}+\frac {\left (5 A b^2+4 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{32 d}+\frac {5 A b (a+b \cos (c+d x))^{3/2} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \tan (c+d x)}{4 d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.30 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.61 \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\frac {2 \left (144 a^3 A b+236 a A b^3+192 a^3 b C+768 a b^3 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (288 a^4 A+436 a^2 A b^2-45 A b^4+384 a^4 C+1008 a^2 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-284 a^2 A b^2-15 A b^4-432 a^2 b^2 C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{768 a d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{96} \sec ^2(c+d x) \left (36 a^2 A \sin (c+d x)+59 A b^2 \sin (c+d x)+48 a^2 C \sin (c+d x)\right )+\frac {\sec (c+d x) \left (284 a^2 A b \sin (c+d x)+15 A b^3 \sin (c+d x)+432 a^2 b C \sin (c+d x)\right )}{192 a}+\frac {17}{24} a A b \sec ^2(c+d x) \tan (c+d x)+\frac {1}{4} a^2 A \sec ^3(c+d x) \tan (c+d x)\right )}{d} \]

[In]

Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]

[Out]

((2*(144*a^3*A*b + 236*a*A*b^3 + 192*a^3*b*C + 768*a*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c +
d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(288*a^4*A + 436*a^2*A*b^2 - 45*A*b^4 + 384*a^4*C + 1008
*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d
*x]] - ((2*I)*(-284*a^2*A*b^2 - 15*A*b^4 - 432*a^2*b^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos
[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c +
d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(
a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*
Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (
a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(768*a*d) +
(Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^2*(36*a^2*A*Sin[c + d*x] + 59*A*b^2*Sin[c + d*x] + 48*a^2*C*Sin[c +
d*x]))/96 + (Sec[c + d*x]*(284*a^2*A*b*Sin[c + d*x] + 15*A*b^3*Sin[c + d*x] + 432*a^2*b*C*Sin[c + d*x]))/(192*
a) + (17*a*A*b*Sec[c + d*x]^2*Tan[c + d*x])/24 + (a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/4))/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3455\) vs. \(2(490)=980\).

Time = 504.65 (sec) , antiderivative size = 3456, normalized size of antiderivative = 7.91

method	result	size
parts	\(\text {Expression too large to display}\)	\(3456\)
default	\(\text {Expression too large to display}\)	\(3651\)

[In]

int((a+cos(d*x+c)*b)^(5/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x,method=_RETURNVERBOSE)

[Out]

-1/192*A*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*((-9088*a^2*b^2-480*b^4)*cos(1/2*d*x+1/2*
c)*sin(1/2*d*x+1/2*c)^10+(5696*a^3*b+18176*a^2*b^2+2128*a*b^3+960*b^4)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)
+(-576*a^4-8544*a^3*b-15664*a^2*b^2-3192*a*b^3-720*b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(576*a^4+5008*
a^3*b+6576*a^2*b^2+1596*a*b^3+240*b^4)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-240*a^4-1080*a^3*b-1076*a^2*b
^2-266*a*b^3-30*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+16*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^
(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(356*b*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+133*b^3*Ellipti
cF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a-284*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+284*b^2
*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2-15*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3*
a+15*b^4*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-144*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/
2))*a^4-360*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b^2*a^2+15*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2
*b/(a-b))^(1/2))*b^4)*sin(1/2*d*x+1/2*c)^8-32*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x
+1/2*c)^2)^(1/2)*(356*b*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+133*b^3*EllipticF(cos(1/2*d*x+1/2
*c),(-2*b/(a-b))^(1/2))*a-284*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+284*b^2*EllipticE(cos(1/2
*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2-15*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3*a+15*b^4*EllipticE
(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-144*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a^4-360*Ellipt
icPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b^2*a^2+15*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b
^4)*sin(1/2*d*x+1/2*c)^6+24*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(
356*b*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+133*b^3*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(
1/2))*a-284*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+284*b^2*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/
(a-b))^(1/2))*a^2-15*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3*a+15*b^4*EllipticE(cos(1/2*d*x+1/2*c
),(-2*b/(a-b))^(1/2))-144*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a^4-360*EllipticPi(cos(1/2*d*x+1
/2*c),2,(-2*b/(a-b))^(1/2))*b^2*a^2+15*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b^4)*sin(1/2*d*x+1/
2*c)^4-8*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(356*b*EllipticF(cos
(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+133*b^3*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a-284*b*Ellip
ticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+284*b^2*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2-1
5*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3*a+15*b^4*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2
))-144*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a^4-360*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b)
)^(1/2))*b^2*a^2+15*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b^4)*sin(1/2*d*x+1/2*c)^2+356*b*(sin(1
/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/
(a-b))^(1/2))*a^3+133*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Ell
ipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a-284*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c
)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b+284*(sin(1/2*d*x+1/2*c)^2)^(1/2)
*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2-
15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*
c),(-2*b/(a-b))^(1/2))*a*b^3+15*b^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))
^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-144*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d
*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a^4-360*(sin(1/2*d*x+1/2*c)
^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/
2))*b^2*a^2+15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticPi(cos
(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b^4)/a/(2*cos(1/2*d*x+1/2*c)^2-1)^4/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin
(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(-2*b*sin(1/2*d*x+1/2*c)^2+a+b)^(1/2)/d-1/4*C*((2*b*cos(1/2*d*x+1/
2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-72*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6*a*b^2+(44*a^2*b+72*a*b^2)
*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-4*a^3-22*a^2*b-18*a*b^2)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+4*
(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(11*EllipticF(cos(1/2*d*x+1/2
*c),(-2*b/(a-b))^(1/2))*a^2*b+8*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3-9*EllipticE(cos(1/2*d*x+1
/2*c),(-2*b/(a-b))^(1/2))*a^2*b+9*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^2-4*EllipticPi(cos(1/2*
d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a^3-15*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a*b^2)*sin(1/2*d*x
+1/2*c)^4-4*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(11*EllipticF(cos
(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b+8*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3-9*EllipticE(c
os(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b+9*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^2-4*Ellipti
cPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a^3-15*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a*b^2)
*sin(1/2*d*x+1/2*c)^2+11*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Elli
pticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b+8*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+
1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-9*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-
2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b+9*(si
n(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2
*b/(a-b))^(1/2))*a*b^2-4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Elli
pticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*a^3-15*a*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*
d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2)))/(2*cos(1/2*d*x+1/2*c)^2-1
)^2/(-2*sin(1/2*d*x+1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(-2*b*sin(1/2*d*x+1/2*c)^2
+a+b)^(1/2)/d

Fricas [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c))**(5/2)*(A+C*cos(d*x+c)**2)*sec(d*x+c)**5,x)

[Out]

Timed out

Maxima [F]

\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{5} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^(5/2)*sec(d*x + c)^5, x)

Giac [F]

\[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{5} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^(5/2)*sec(d*x + c)^5, x)

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^5} \,d x \]

[In]

int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2))/cos(c + d*x)^5,x)

[Out]

int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2))/cos(c + d*x)^5, x)