\(\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^5(c+d x) \, dx\) [1028]

Optimal result

Integrand size = 43, antiderivative size = 503 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{192 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (3 A b^3-128 a^3 B-136 a b^2 B-12 a^2 b (19 A+28 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{192 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{64 a^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a^2 d}+\frac {\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{96 a d}+\frac {(3 A b+8 a B) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d} \]

[Out]

1/192*(9*A*b^3-128*B*a^3-24*B*a*b^2-12*a^2*b*(13*A+20*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elli
pticE(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^2/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-
1/192*(3*A*b^3-128*B*a^3-136*B*a*b^2-12*a^2*b*(19*A+28*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ell
ipticF(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)/a/d/(a+b*cos(d*x+c))^(1/2)+1
/64*(3*A*b^4+96*B*a^3*b-8*B*a*b^3+24*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^2/d/(a+b*co
s(d*x+c))^(1/2)+1/4*A*(a+b*cos(d*x+c))^(3/2)*sec(d*x+c)^3*tan(d*x+c)/d-1/192*(9*A*b^3-128*B*a^3-24*B*a*b^2-12*
a^2*b*(13*A+20*C))*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a^2/d+1/96*(3*A*b^2+56*B*a*b+12*a^2*(3*A+4*C))*sec(d*x+c)
*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a/d+1/24*(3*A*b+8*B*a)*sec(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/d

Rubi [A] (verified)

Time = 2.35 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3126, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\tan (c+d x) \sec (c+d x) \left (12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{96 a d}-\frac {\tan (c+d x) \left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{192 a^2 d}-\frac {\left (-128 a^3 B-12 a^2 b (19 A+28 C)-136 a b^2 B+3 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{192 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{192 a^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{64 a^2 d \sqrt {a+b \cos (c+d x)}}+\frac {(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{24 d}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d} \]

[In]

Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]

[Out]

((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (
2*b)/(a + b)])/(192*a^2*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - ((3*A*b^3 - 128*a^3*B - 136*a*b^2*B - 12*a^2*b
*(19*A + 28*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(192*a*d*Sqrt[a + b*
Cos[c + d*x]]) + ((3*A*b^4 + 96*a^3*b*B - 8*a*b^3*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*Sqrt[(a + b*C
os[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(64*a^2*d*Sqrt[a + b*Cos[c + d*x]]) - ((9*A*b
^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(192*a^2*d) + ((3
*A*b^2 + 56*a*b*B + 12*a^2*(3*A + 4*C))*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(96*a*d) + ((3*A*b
+ 8*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(24*d) + (A*(a + b*Cos[c + d*x])^(3/2)*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 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))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int \sqrt {a+b \cos (c+d x)} \left (\frac {1}{2} (3 A b+8 a B)+(3 a A+4 b B+4 a C) \cos (c+d x)+\frac {1}{2} b (3 A+8 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {(3 A b+8 a B) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{12} \int \frac {\left (\frac {1}{4} \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right )+\frac {1}{2} \left (33 a A b+16 a^2 B+24 b^2 B+48 a b C\right ) \cos (c+d x)+\frac {3}{4} b (9 A b+8 a B+16 b C) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{96 a d}+\frac {(3 A b+8 a B) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {\int \frac {\left (\frac {1}{8} \left (-9 A b^3+128 a^3 B+24 a b^2 B+8 a^2 \left (\frac {39 A b}{2}+30 b C\right )\right )+\frac {1}{4} a \left (104 a b B+12 a^2 (3 A+4 C)+3 b^2 (19 A+32 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{24 a} \\ & = -\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a^2 d}+\frac {\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{96 a d}+\frac {(3 A b+8 a B) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {\int \frac {\left (\frac {3}{16} \left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right )+\frac {1}{8} a b \left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \cos (c+d x)+\frac {1}{16} b \left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{24 a^2} \\ & = -\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a^2 d}+\frac {\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{96 a d}+\frac {(3 A b+8 a B) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {\int \frac {\left (-\frac {3}{16} b \left (3 A b^4+96 a^3 b B-8 a b^3 B+24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)\right )+\frac {1}{16} a b \left (3 A b^3-128 a^3 B-136 a b^2 B-12 a^2 b (19 A+28 C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{24 a^2 b}+\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{384 a^2} \\ & = -\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a^2 d}+\frac {\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{96 a d}+\frac {(3 A b+8 a B) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {\left (3 A b^4+96 a^3 b B-8 a b^3 B+24 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^2}-\frac {\left (3 A b^3-128 a^3 B-136 a b^2 B-12 a^2 b (19 A+28 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{384 a}+\frac {\left (\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 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^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \\ & = \frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 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^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a^2 d}+\frac {\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{96 a d}+\frac {(3 A b+8 a B) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {\left (\left (3 A b^4+96 a^3 b B-8 a b^3 B+24 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^2 \sqrt {a+b \cos (c+d x)}}-\frac {\left (\left (3 A b^3-128 a^3 B-136 a b^2 B-12 a^2 b (19 A+28 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 a \sqrt {a+b \cos (c+d x)}} \\ & = \frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 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^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (3 A b^3-128 a^3 B-136 a b^2 B-12 a^2 b (19 A+28 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 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (3 A b^4+96 a^3 b B-8 a b^3 B+24 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^2 d \sqrt {a+b \cos (c+d x)}}-\frac {\left (9 A b^3-128 a^3 B-24 a b^2 B-12 a^2 b (13 A+20 C)\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{192 a^2 d}+\frac {\left (3 A b^2+56 a b B+12 a^2 (3 A+4 C)\right ) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{96 a d}+\frac {(3 A b+8 a B) \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{24 d}+\frac {A (a+b \cos (c+d x))^{3/2} \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 = 7.69 (sec) , antiderivative size = 783, normalized size of antiderivative = 1.56 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\frac {2 \left (144 a^3 A b+12 a A b^3+224 a^2 b^2 B+192 a^3 b C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (288 a^4 A-12 a^2 A b^2+27 A b^4+448 a^3 b B-72 a b^3 B+384 a^4 C+48 a^2 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-156 a^2 A b^2+9 A b^4-128 a^3 b B-24 a b^3 B-240 a^2 b^2 C\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{768 a^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{24} \sec ^3(c+d x) (9 A b \sin (c+d x)+8 a B \sin (c+d x))+\frac {\sec ^2(c+d x) \left (36 a^2 A \sin (c+d x)+3 A b^2 \sin (c+d x)+56 a b B \sin (c+d x)+48 a^2 C \sin (c+d x)\right )}{96 a}+\frac {\sec (c+d x) \left (156 a^2 A b \sin (c+d x)-9 A b^3 \sin (c+d x)+128 a^3 B \sin (c+d x)+24 a b^2 B \sin (c+d x)+240 a^2 b C \sin (c+d x)\right )}{192 a^2}+\frac {1}{4} a A \sec ^3(c+d x) \tan (c+d x)\right )}{d} \]

[In]

Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]

[Out]

((2*(144*a^3*A*b + 12*a*A*b^3 + 224*a^2*b^2*B + 192*a^3*b*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 - 12*a^2*A*b^2 + 27*A*b^4 + 448*a^3*b*B - 72
*a*b^3*B + 384*a^4*C + 48*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)*(-156*a^2*A*b^2 + 9*A*b^4 - 128*a^3*b*B - 24*a*b^3*B - 240*a^2*b^2*C)*S
qrt[(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)*Elliptic
E[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^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^3*(9*A*b*Sin[c
+ d*x] + 8*a*B*Sin[c + d*x]))/24 + (Sec[c + d*x]^2*(36*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 56*a*b*B*Si
n[c + d*x] + 48*a^2*C*Sin[c + d*x]))/(96*a) + (Sec[c + d*x]*(156*a^2*A*b*Sin[c + d*x] - 9*A*b^3*Sin[c + d*x] +
128*a^3*B*Sin[c + d*x] + 24*a*b^2*B*Sin[c + d*x] + 240*a^2*b*C*Sin[c + d*x]))/(192*a^2) + (a*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. \(3550\) vs. \(2(556)=1112\).

Time = 16.65 (sec) , antiderivative size = 3551, normalized size of antiderivative = 7.06

method	result	size
default	\(\text {Expression too large to display}\)	\(3551\)
parts	\(\text {Expression too large to display}\)	\(5047\)

[In]

int((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x,method=_RETURNVERBOSE)

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2*b-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*C*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(
1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(c
os(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))+2*A*a^2*(-1/4*cos(1/2*d*x+1/2*c)/a*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*si
n(1/2*d*x+1/2*c)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)^4+7/24*b/a^2*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)
^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)^3-1/96*(36*a^2+35*b^2)/a^3*cos(1/2*d*x+1/2*c)
*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)^2+5/192*b*(20*a^2+21
*b^2)/a^4*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/
2*c)^2)-7/96*b/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1
/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-35/384*b^3/a^3*(sin
(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2
*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+25/96/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2
*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*b*Ellip
ticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-25/96*b^2/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^
2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c
),(-2*b/(a-b))^(1/2))+35/128/a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2
*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3
-35/128*b^4/a^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/
2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-3/8*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^
2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))-3/16/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2
*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(
1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b^2-35/128/a^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-
b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(
-2*b/(a-b))^(1/2))*b^4)+2*b*(B*b+2*C*a)*(-cos(1/2*d*x+1/2*c)/a*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/
2*c)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)+1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-
b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b
))^(1/2))-1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*
c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+1/2/a*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^
2)^(1/2)*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+1/2/a*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d
*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/
2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2)))+2*a*(2*A*b+B*a)*(-1/3*cos(1/2*d*x+1/2*c)/a*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b
)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)^3+5/12*b/a^2*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/
2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)^2-1/24*(16*a^2+15*b^2)/a^3*cos(1/2*d*x+1/
2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)+5/48*b^2/a^2*(si
n(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/
2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*co
s(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(
cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))
^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^
(1/2))+1/3/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c
)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-5/16*b^2/a^2*(sin(1/2
*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x
+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+5/16/a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*co
s(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(
cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3+1/4/a*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b
)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-
2*b/(a-b))^(1/2))+5/16*b^3/a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b
*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2)))+2
*(A*b^2+2*B*a*b+C*a^2)*(-1/2*cos(1/2*d*x+1/2*c)/a*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)
/(-1+2*cos(1/2*d*x+1/2*c)^2)^2+3/4*b/a^2*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c
)^2)^(1/2)/(-1+2*cos(1/2*d*x+1/2*c)^2)-1/8*b/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a
-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-
b))^(1/2))+3/8/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1
/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-3/8*b^2/a^2*(sin(
1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*
d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-1/2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(
1/2*d*x+1/2*c)^2*b+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(c
os(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))-3/8/a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)/(
a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b
/(a-b))^(1/2))*b^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. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+b*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**5,x)

[Out]

Timed out

Maxima [F]

\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{5} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/2)*sec(d*x + c)^5, x)

Giac [F]

\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{5} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/2)*sec(d*x + c)^5, x)

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\cos \left (c+d\,x\right )}^5} \,d x \]

[In]

int(((a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^5,x)

[Out]

int(((a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^5, x)