\(\int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x)) \, dx\) [731]

Optimal result

Integrand size = 35, antiderivative size = 507 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 b d}+\frac {\sqrt {a+b} \left (16 a^2 A+26 a A b+33 A b^2+24 a^2 C+144 a b C-48 b^2 C\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 d}-\frac {5 b \sqrt {a+b} \left (A b^2+4 a^2 (A+2 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a d}+\frac {\left (15 A b^2+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {5 A b \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d} \]

[Out]

5/12*A*b*cos(d*x+c)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/3*A*cos(d*x+c)^2*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d
+1/24*(a-b)*(3*b^2*(11*A-16*C)+8*a^2*(2*A+3*C))*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)
/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b/d+1/24*(16*A*a^2+2
6*A*a*b+33*A*b^2+24*C*a^2+144*C*a*b-48*C*b^2)*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(
a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/d-5/8*b*(A*b^2+4*a^2*(
A+2*C))*cot(d*x+c)*EllipticPi((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(
1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a/d+1/24*(15*A*b^2+8*a^2*(2*A+3*C))*sin(d*x+c)*(a+b
*sec(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4180, 4179, 4143, 4006, 3869, 3917, 4089} \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\sqrt {a+b} \left (16 a^2 A+24 a^2 C+26 a A b+144 a b C+33 A b^2-48 b^2 C\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{24 d}+\frac {(a-b) \sqrt {a+b} \left (8 a^2 (2 A+3 C)+3 b^2 (11 A-16 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{24 b d}-\frac {5 b \sqrt {a+b} \left (4 a^2 (A+2 C)+A b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{8 a d}+\frac {\left (8 a^2 (2 A+3 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{24 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}+\frac {5 A b \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{12 d} \]

[In]

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

[Out]

((a - b)*Sqrt[a + b]*(3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c
+ d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a -
b))])/(24*b*d) + (Sqrt[a + b]*(16*a^2*A + 26*a*A*b + 33*A*b^2 + 24*a^2*C + 144*a*b*C - 48*b^2*C)*Cot[c + d*x]
*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]
*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (5*b*Sqrt[a + b]*(A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*Ell
ipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/
(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c +
d*x]]*Sin[c + d*x])/(24*d) + (5*A*b*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c +
d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)

Rule 3869

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b
*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b)*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b
*Csc[c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

Rule 3917

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(Rt[a + b, 2]/(b*
f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin
[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4006

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c, In
t[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[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]

Rule 4089

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rule 4143

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

Rule 4179

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

Rule 4180

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b
_.) + (a_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Dis
t[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*(C*n + A*(n + 1))*Csc[e +
f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2,
0] && GtQ[m, 0] && LeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {5 A b}{2}+a (2 A+3 C) \sec (c+d x)-\frac {1}{2} b (A-6 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {5 A b \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {1}{4} \left (15 A b^2+4 a^2 (4 A+6 C)\right )+\frac {1}{2} a b (11 A+24 C) \sec (c+d x)-\frac {3}{4} b^2 (3 A-8 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {\left (15 A b^2+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {5 A b \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{6} \int \frac {\frac {15}{8} b \left (A b^2+4 a^2 (A+2 C)\right )+\frac {1}{4} a b^2 (13 A+72 C) \sec (c+d x)-\frac {1}{8} b \left (3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (15 A b^2+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {5 A b \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{6} \int \frac {\frac {15}{8} b \left (A b^2+4 a^2 (A+2 C)\right )+\left (\frac {1}{4} a b^2 (13 A+72 C)+\frac {1}{8} b \left (3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx-\frac {1}{48} \left (b \left (3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {(a-b) \sqrt {a+b} \left (3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 b d}+\frac {\left (15 A b^2+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {5 A b \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{48} \left (b \left (16 a^2 A+26 a A b+33 A b^2+24 a^2 C+144 a b C-48 b^2 C\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx+\frac {1}{16} \left (5 b \left (A b^2+4 a^2 (A+2 C)\right )\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {(a-b) \sqrt {a+b} \left (3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 b d}+\frac {\sqrt {a+b} \left (16 a^2 A+26 a A b+33 A b^2+24 a^2 C+144 a b C-48 b^2 C\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 d}-\frac {5 b \sqrt {a+b} \left (A b^2+4 a^2 (A+2 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a d}+\frac {\left (15 A b^2+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {5 A b \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1501\) vs. \(2(507)=1014\).

Time = 22.17 (sec) , antiderivative size = 1501, normalized size of antiderivative = 2.96 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {1}{6} \left (a^2 A+24 b^2 C\right ) \sin (c+d x)+\frac {13}{12} a A b \sin (2 (c+d x))+\frac {1}{6} a^2 A \sin (3 (c+d x))\right )}{d (b+a \cos (c+d x))^2 (A+2 C+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \left (16 a^3 A \tan \left (\frac {1}{2} (c+d x)\right )+16 a^2 A b \tan \left (\frac {1}{2} (c+d x)\right )+33 a A b^2 \tan \left (\frac {1}{2} (c+d x)\right )+33 A b^3 \tan \left (\frac {1}{2} (c+d x)\right )+24 a^3 C \tan \left (\frac {1}{2} (c+d x)\right )+24 a^2 b C \tan \left (\frac {1}{2} (c+d x)\right )-48 a b^2 C \tan \left (\frac {1}{2} (c+d x)\right )-48 b^3 C \tan \left (\frac {1}{2} (c+d x)\right )-32 a^3 A \tan ^3\left (\frac {1}{2} (c+d x)\right )-66 a A b^2 \tan ^3\left (\frac {1}{2} (c+d x)\right )-48 a^3 C \tan ^3\left (\frac {1}{2} (c+d x)\right )+96 a b^2 C \tan ^3\left (\frac {1}{2} (c+d x)\right )+16 a^3 A \tan ^5\left (\frac {1}{2} (c+d x)\right )-16 a^2 A b \tan ^5\left (\frac {1}{2} (c+d x)\right )+33 a A b^2 \tan ^5\left (\frac {1}{2} (c+d x)\right )-33 A b^3 \tan ^5\left (\frac {1}{2} (c+d x)\right )+24 a^3 C \tan ^5\left (\frac {1}{2} (c+d x)\right )-24 a^2 b C \tan ^5\left (\frac {1}{2} (c+d x)\right )-48 a b^2 C \tan ^5\left (\frac {1}{2} (c+d x)\right )+48 b^3 C \tan ^5\left (\frac {1}{2} (c+d x)\right )+120 a^2 A b \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+30 A b^3 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+240 a^2 b C \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+120 a^2 A b \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+30 A b^3 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+240 a^2 b C \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+(a+b) \left (3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-2 b \left (24 b^2 (A-C)-a b (13 A+72 C)+a^2 (38 A+72 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}\right )}{12 d (b+a \cos (c+d x))^{5/2} (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x) \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}}} \]

[In]

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

[Out]

(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(((a^2*A + 24*b^2*C)*Sin[c + d*x])/6 + (13*a
*A*b*Sin[2*(c + d*x)])/12 + (a^2*A*Sin[3*(c + d*x)])/6))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*
x])) + ((a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(16*a^3*A*Tan[(c
+ d*x)/2] + 16*a^2*A*b*Tan[(c + d*x)/2] + 33*a*A*b^2*Tan[(c + d*x)/2] + 33*A*b^3*Tan[(c + d*x)/2] + 24*a^3*C*
Tan[(c + d*x)/2] + 24*a^2*b*C*Tan[(c + d*x)/2] - 48*a*b^2*C*Tan[(c + d*x)/2] - 48*b^3*C*Tan[(c + d*x)/2] - 32*
a^3*A*Tan[(c + d*x)/2]^3 - 66*a*A*b^2*Tan[(c + d*x)/2]^3 - 48*a^3*C*Tan[(c + d*x)/2]^3 + 96*a*b^2*C*Tan[(c + d
*x)/2]^3 + 16*a^3*A*Tan[(c + d*x)/2]^5 - 16*a^2*A*b*Tan[(c + d*x)/2]^5 + 33*a*A*b^2*Tan[(c + d*x)/2]^5 - 33*A*
b^3*Tan[(c + d*x)/2]^5 + 24*a^3*C*Tan[(c + d*x)/2]^5 - 24*a^2*b*C*Tan[(c + d*x)/2]^5 - 48*a*b^2*C*Tan[(c + d*x
)/2]^5 + 48*b^3*C*Tan[(c + d*x)/2]^5 + 120*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*S
qrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*A*b^3*Ell
ipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d
*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 240*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a +
b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 120*a^2
*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]
*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*A*b^3*EllipticPi[-1, ArcSin[Tan[(c +
d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2
+ b*Tan[(c + d*x)/2]^2)/(a + b)] + 240*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[
(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)
] + (a + b)*(3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqr
t[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/
(a + b)] - 2*b*(24*b^2*(A - C) - a*b*(13*A + 72*C) + a^2*(38*A + 72*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a
- b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Ta
n[(c + d*x)/2]^2)/(a + b)]))/(12*d*(b + a*Cos[c + d*x])^(5/2)*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2
)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)
/2]^2)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4748\) vs. \(2(462)=924\).

Time = 174.31 (sec) , antiderivative size = 4749, normalized size of antiderivative = 9.37

method	result	size
default	\(\text {Expression too large to display}\)	\(4749\)

[In]

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

[Out]

-1/24/d*(-48*C*sin(d*x+c)*b^3-288*C*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+
c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)+66*A*EllipticE(cot(d*x+c)-csc(d*
x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*
b^2*cos(d*x+c)+52*A*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/
(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^2*cos(d*x+c)+48*C*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b
))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)+3
3*A*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d
*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^2*cos(d*x+c)^2+26*A*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^2*cos(d*x+c)^2+24*C*Elliptic
E(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(
d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)^2-144*C*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(co
s(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)^2-24*C*a^2*b*cos(d*x+c)*si
n(d*x+c)-26*A*a*b^2*cos(d*x+c)*sin(d*x+c)+33*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(co
s(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^3+24*C*(cos(d*x+c)/(cos(d*x+c)+1))^(
1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^3-
48*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-cs
c(d*x+c),((a-b)/(a+b))^(1/2))*a*b^2-96*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+
c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^3*cos(d*x+c)+288*C*(1/(a+b)*(b+a*cos(d*x+c
))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2)
)*a*b^2*cos(d*x+c)-96*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*Elli
pticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a*b^2*cos(d*x+c)+16*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*Ellip
ticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b+32*A*Ell
ipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/
(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)+240*A*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*(cos(d*x
+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)+480*C*EllipticPi(co
t(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d
*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)-76*A*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)^2+16*A*EllipticE(cot(d*x+c)-cs
c(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2
)*a^2*b*cos(d*x+c)^2+120*A*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)^2+240*C*EllipticPi(cot(d*x+c)-csc(d*x
+c),-1,((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*
a^2*b*cos(d*x+c)^2-48*C*a*b^2*cos(d*x+c)*sin(d*x+c)-48*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^3-48*A*(1/(a+b)*(b+a*cos(d
*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(
1/2))*b^3+48*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot
(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^3+144*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/
(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a*b^2-96*A*(1/(a+b)*(b+a*cos(d*x+c)
)/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))
*b^3*cos(d*x+c)+96*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*Ellipti
cF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^3*cos(d*x+c)-48*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1
/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^3*cos(d*x+c)^2+48
*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(
d*x+c),((a-b)/(a+b))^(1/2))*b^3*cos(d*x+c)^2-48*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/
(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*b^3*cos(d*x+c)^2+144*C*(1/(a+b)*(b+
a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(
a+b))^(1/2))*a*b^2*cos(d*x+c)^2-48*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1
))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a*b^2*cos(d*x+c)^2-8*A*a^3*cos(d*x+c)^4*sin(d*x+
c)-8*A*a^3*cos(d*x+c)^3*sin(d*x+c)-16*A*a^3*cos(d*x+c)^2*sin(d*x+c)-24*C*a^3*cos(d*x+c)^2*sin(d*x+c)-33*A*b^3*
cos(d*x+c)*sin(d*x+c)-34*A*a^2*b*cos(d*x+c)^3*sin(d*x+c)-34*A*a^2*b*cos(d*x+c)^2*sin(d*x+c)-59*A*a*b^2*cos(d*x
+c)^2*sin(d*x+c)-16*A*a^2*b*cos(d*x+c)*sin(d*x+c)+16*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-
csc(d*x+c),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3+30*A*(cos(d*x+c)/(cos(d*x+
c)+1))^(1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1)
)^(1/2)*b^3+120*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*(
1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b+240*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticPi(cot(d*
x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b-76*A*(cos(d*x+c)
/(cos(d*x+c)+1))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x
+c)+1))^(1/2)*a^2*b+16*A*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2
)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*cos(d*x+c)^2+30*A*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((
a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^3*cos(d
*x+c)^2+32*A*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(
b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*cos(d*x+c)+60*A*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1
/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^3*cos(d*x+c)-152*A*El
lipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))
/(cos(d*x+c)+1))^(1/2)*a^2*b*cos(d*x+c)+66*A*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/
(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^3*cos(d*x+c)+48*C*EllipticE(cot(d*x+c)
-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(
1/2)*a^3*cos(d*x+c)+26*A*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2
)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^2-144*C*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1
/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b+33*A*(cos(d*x+c)/
(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(
a+b))^(1/2))*a*b^2+24*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Elli
pticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b+33*A*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2
))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^3*cos(d*x+c)^2+24*C*Ell
ipticE(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/
(cos(d*x+c)+1))^(1/2)*a^3*cos(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/(b+a*cos(d*x+c))/(cos(d*x+c)+1)

Fricas [F]

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

[In]

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

[Out]

integral((C*b^2*cos(d*x + c)^3*sec(d*x + c)^4 + 2*C*a*b*cos(d*x + c)^3*sec(d*x + c)^3 + 2*A*a*b*cos(d*x + c)^3
*sec(d*x + c) + A*a^2*cos(d*x + c)^3 + (C*a^2 + A*b^2)*cos(d*x + c)^3*sec(d*x + c)^2)*sqrt(b*sec(d*x + c) + a)
, x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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