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

Optimal. Leaf size=507 \[ \frac{\sqrt{a+b} \cot (c+d x) \left (6 a^2 (A+2 (B+6 C))+a b (27 A+72 B-56 C)+8 b^2 (3 A-3 B+C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{12 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{12 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left (4 a^2 (A+2 C)+20 a b B+15 A b^2\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{4 d}-\frac{b \tan (c+d x) (12 a B+21 A b-8 b C) \sqrt{a+b \sec (c+d x)}}{12 d}+\frac{(4 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{5/2}}{2 d} \]

[Out]

((a - b)*Sqrt[a + b]*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*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))])/(12*b*d) + (Sqrt[a + b]*(a*b*(27*A + 72*B - 56*C) + 8*b^2*(3*A - 3*B + C) + 6*a^2*(A + 2*(B + 6*C)))*Co
t[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))])/(12*d) - (Sqrt[a + b]*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*
C))*Cot[c + d*x]*EllipticPi[(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))])/(4*d) + ((5*A*b + 4*a*B)*(a + b*Sec[c +
d*x])^(3/2)*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(2*d) - (b*(21*A*b
+ 12*a*B - 8*b*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(12*d)

________________________________________________________________________________________

Rubi [A]  time = 1.06138, antiderivative size = 507, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4094, 4056, 4058, 3921, 3784, 3832, 4004} \[ \frac{\sqrt{a+b} \cot (c+d x) \left (6 a^2 (A+2 (B+6 C))+a b (27 A+72 B-56 C)+8 b^2 (3 A-3 B+C)\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{12 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (12 a^2 B+a b (27 A-56 C)-24 b^2 B\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{12 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left (4 a^2 (A+2 C)+20 a b B+15 A b^2\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{4 d}-\frac{b \tan (c+d x) (12 a B+21 A b-8 b C) \sqrt{a+b \sec (c+d x)}}{12 d}+\frac{(4 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{5/2}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)*Sqrt[a + b]*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*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))])/(12*b*d) + (Sqrt[a + b]*(a*b*(27*A + 72*B - 56*C) + 8*b^2*(3*A - 3*B + C) + 6*a^2*(A + 2*(B + 6*C)))*Co
t[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))])/(12*d) - (Sqrt[a + b]*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*
C))*Cot[c + d*x]*EllipticPi[(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))])/(4*d) + ((5*A*b + 4*a*B)*(a + b*Sec[c +
d*x])^(3/2)*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(2*d) - (b*(21*A*b
+ 12*a*B - 8*b*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(12*d)

Rule 4094

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 4056

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

Rule 4058

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 3921

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 3784

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

Rule 3832

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

Rule 4004

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 + Cs
c[e + f*x]))/(a - b))]*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + (b*B)/A, 2]], (a*A + b*B)/(a*A - b*B)]
)/(b^2*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rubi steps

\begin{align*} \int \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos (c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}+\frac{1}{2} \int \cos (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac{1}{2} (5 A b+4 a B)+(2 b B+a (A+2 C)) \sec (c+d x)-\frac{1}{2} b (3 A-4 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{(5 A b+4 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{A \cos (c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}+\frac{1}{2} \int \sqrt{a+b \sec (c+d x)} \left (\frac{1}{4} \left (15 A b^2+20 a b B+4 a^2 (A+2 C)\right )-\frac{1}{2} b (a A-4 b B-8 a C) \sec (c+d x)-\frac{1}{4} b (21 A b+12 a B-8 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{(5 A b+4 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{A \cos (c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}-\frac{b (21 A b+12 a B-8 b C) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{12 d}+\frac{1}{3} \int \frac{\frac{3}{8} a \left (15 A b^2+20 a b B+4 a^2 (A+2 C)\right )+\frac{1}{4} b \left (36 a b B+4 b^2 (3 A+C)+3 a^2 (A+12 C)\right ) \sec (c+d x)-\frac{1}{8} b \left (27 a A b+12 a^2 B-24 b^2 B-56 a b C\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{(5 A b+4 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{A \cos (c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}-\frac{b (21 A b+12 a B-8 b C) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{12 d}+\frac{1}{3} \int \frac{\frac{3}{8} a \left (15 A b^2+20 a b B+4 a^2 (A+2 C)\right )+\left (\frac{1}{8} b \left (27 a A b+12 a^2 B-24 b^2 B-56 a b C\right )+\frac{1}{4} b \left (36 a b B+4 b^2 (3 A+C)+3 a^2 (A+12 C)\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx-\frac{1}{24} \left (b \left (12 a^2 B-24 b^2 B+a b (27 A-56 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 (12 a^2 B-24 b^2 B+a b (27 A-56 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\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}}}{12 b d}+\frac{(5 A b+4 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{A \cos (c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}-\frac{b (21 A b+12 a B-8 b C) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{12 d}+\frac{1}{8} \left (a \left (15 A b^2+20 a b B+4 a^2 (A+2 C)\right )\right ) \int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx+\frac{1}{24} \left (b \left (a b (27 A+72 B-56 C)+8 b^2 (3 A-3 B+C)+6 a^2 (A+2 (B+6 C))\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{(a-b) \sqrt{a+b} \left (12 a^2 B-24 b^2 B+a b (27 A-56 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\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}}}{12 b d}+\frac{\sqrt{a+b} \left (a b (27 A+72 B-56 C)+8 b^2 (3 A-3 B+C)+6 a^2 (A+2 (B+6 C))\right ) \cot (c+d x) F\left (\sin ^{-1}\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}}}{12 d}-\frac{\sqrt{a+b} \left (15 A b^2+20 a b B+4 a^2 (A+2 C)\right ) \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\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}}}{4 d}+\frac{(5 A b+4 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{A \cos (c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{2 d}-\frac{b (21 A b+12 a B-8 b C) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{12 d}\\ \end{align*}

Mathematica [B]  time = 25.4083, size = 4902, normalized size = 9.67 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((4*b*(3*b*B + 7*a*C)*Sin[c + d*x])/3 + (a^2*A*Sin[2*(c + d*x)])/2
+ (4*b^2*C*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])^2) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*((a^3*A)/(Sq
rt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a*A*b^2)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a^
2*b*B)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*b^3*B)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])
+ (2*a^3*C)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*a*b^2*C)/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec
[c + d*x]]) + (11*a^2*A*b*Sqrt[Sec[c + d*x]])/(4*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b^3*Sqrt[Sec[c + d*x]])/Sqrt
[b + a*Cos[c + d*x]] + (a^3*B*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] + (4*a*b^2*B*Sqrt[Sec[c + d*x]])/Sq
rt[b + a*Cos[c + d*x]] + (4*a^2*b*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*b^3*C*Sqrt[Sec[c + d
*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (9*a^2*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(4*Sqrt[b + a*Cos[c + d*x
]]) + (a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] - (2*a*b^2*B*Cos[2*(c + d*x)]*Sqrt[
Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] - (14*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c
+ d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*((a + b)*(12*a^2*B - 24*b^2*B + a*
b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c +
d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin
[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)
] - 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] +
2*a*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*
Sec[(c + d*x)/2]^2)/(a + b)] + (12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Se
c[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(6*d*(b + a*Cos[c + d*x])^3*(Sec[(c + d*x)/2]^2)^(3/2)
*Sec[c + d*x]^(5/2)*((a*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*
x]*((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec
[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8
*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[
c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((a - b)*EllipticF[ArcSin[T
an[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c +
d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*
(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(12*(b + a*Cos[c
+ d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec
[c + d*x]]*Tan[(c + d*x)/2]*((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/
2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(
21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*
x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((
a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a
- b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (12*a^2*B - 24*b^
2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*
x)/2]))/(4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[
(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(
c + d*x)/2])*((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a
+ b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(21*A*b + 6*a*(A
+ 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((
b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((a - b)*Elliptic
F[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])
*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (12*a^2*B - 24*b^2*B + a*b*(27*A
- 56*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(12*Sqr
t[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*((a + b)*(12*a^2*B
- 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[(
(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*E
llipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x
)/2]^2)/(a + b)] - 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a -
b)/(a + b)] + 2*a*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a
*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*(
Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c
+ d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(12*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^
(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*
Sec[c + d*x]]*(((12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*(Cos[c + d*x
]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x])/2 + (a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcS
in[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a +
b)]*Tan[(c + d*x)/2] - b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)
/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x
)/2] - 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]
+ 2*a*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x
])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + (3*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c +
d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sec[c + d*x]*Tan[(c + d*x)/2]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x])
+ Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + ((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*E
llipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/
(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*
Sec[(c + d*x)/2]^2)/(a + b)]) - (b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan
[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*
Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)
/(a + b)]) - (3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(
a + b)] + 2*a*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*
x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[
((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) - (b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C)
)*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*
Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*Sec[(c +
d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b
)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]) - 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*Sec[(c + d*x)/2]^2*Sqrt[((b +
a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*(((a - b)*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sq
rt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (a*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[
(c + d*x)/2]^2)*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])) - a*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))
*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]*Tan[c + d*x] + (12*a^2*B - 24*b^2*B + a*b*(27*A - 56
*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]*Tan[c + d*x]))
/(6*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)))))/2

________________________________________________________________________________________

Maple [B]  time = 0.94, size = 4884, normalized size = 9.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/d*(cos(d*x+c)+1)^2*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))^2*(27*A*cos(d*x+c)^3*a*b^2+8*C*si
n(d*x+c)*cos(d*x+c)^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)*Ellipt
icF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^3+24*B*cos(d*x+c)^2*sin(d*x+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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1
/2))*b^3+12*B*cos(d*x+c)^2*sin(d*x+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((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b-24*B*cos(d*x+c)^2*sin(d*x+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((-1+cos(d*x+c))/sin(d*x+c
),((a-b)/(a+b))^(1/2))*a*b^2+72*B*cos(d*x+c)^2*sin(d*x+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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^2-56*C*sin(d*x+c)*
cos(d*x+c)^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)*EllipticE((-1+c
os(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b-56*C*sin(d*x+c)*cos(d*x+c)^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)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*
b^2+72*C*sin(d*x+c)*cos(d*x+c)^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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b+56*C*sin(d*x+c)*cos(d*x+c)^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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a
-b)/(a+b))^(1/2))*a*b^2+90*A*cos(d*x+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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*sin(d*x+c)*a*b^2+120*B*cos(d*x+c)*(c
os(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)*EllipticPi((-1+cos(d*x+c))/sin
(d*x+c),-1,((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^2*b+90*A*cos(d*x+c)^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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*sin(d*x+c
)*a*b^2+120*B*cos(d*x+c)^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)*E
llipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^2*b-72*A*cos(d*x+c)^2*(cos(d*x+c)/(c
os(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b
)/(a+b))^(1/2))*sin(d*x+c)*a*b^2+27*A*cos(d*x+c)^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)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a*b^2-24*C*sin(d*x
+c)*cos(d*x+c)^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)*EllipticF((
-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3+6*A*cos(d*x+c)^5*a^3-6*A*cos(d*x+c)^3*a^3-72*B*cos(d*x+c)*a
^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)*sin(d*x+c)*EllipticF((-1+
cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b+72*B*sin(d*x+c)*cos(d*x+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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^2-1
2*B*cos(d*x+c)^3*a^3+24*B*cos(d*x+c)^2*b^3+12*B*cos(d*x+c)^2*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*c
os(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^3-24*B
*cos(d*x+c)^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)*EllipticE((-1+
cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*b^3+48*C*cos(d*x+c)^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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))
*sin(d*x+c)*a^3-12*A*cos(d*x+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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^3+24*A*cos(d*x+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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-
b)/(a+b))^(1/2))*sin(d*x+c)*a^3+12*B*cos(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(c
os(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^3-24*B*cos(d*x+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((-1+cos(d*x+c))/sin
(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*b^3-24*C*cos(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*co
s(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^3+48*C*
cos(d*x+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)*EllipticPi((-1+co
s(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^3+24*A*sin(d*x+c)*cos(d*x+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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^
(1/2))*b^3-56*C*sin(d*x+c)*cos(d*x+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((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^2-56*C*a^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)*sin(d*x+c)*cos(d*x+c)*EllipticE((-1+cos(d*x+c))/sin(d*x
+c),((a-b)/(a+b))^(1/2))*b+72*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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*cos(d*x+c)*a^2*b+12*B*sin(d*x+c)*cos(
d*x+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((-1+cos(d*x
+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b-24*B*sin(d*x+c)*cos(d*x+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((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^2+56*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)*EllipticF((-1+cos(d*x+c))/s
in(d*x+c),((a-b)/(a+b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a*b^2-72*B*sin(d*x+c)*cos(d*x+c)^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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))
^(1/2))*a^2*b+6*A*sin(d*x+c)*cos(d*x+c)^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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b+56*C*cos(d*x+c)^3*a^2*b+8*C*cos(
d*x+c)^3*a*b^2-64*C*cos(d*x+c)*a*b^2+24*B*cos(d*x+c)^3*a*b^2-24*B*cos(d*x+c)^2*a*b^2+12*B*cos(d*x+c)^3*a^2*b-5
6*C*cos(d*x+c)^2*a^2*b+6*A*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^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)*sin(d*x+c)*cos(d*x+c)*b+27*A*EllipticE((-1+cos(d*
x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^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)*sin(d*x+c)*cos(d*x+c)*b+27*A*b^2*(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)*sin(d*x+c)*cos(d*x+c)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a-72*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)*EllipticF((-1+cos(d*x+c))/sin(d*
x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*cos(d*x+c)*a*b^2+8*C*cos(d*x+c)^2*b^3-24*B*cos(d*x+c)*b^3-27*A*cos(d*x+c)
^3*a^2*b+12*B*cos(d*x+c)^4*a^3+24*B*cos(d*x+c)*sin(d*x+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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^3+8*C*sin(d*x+c)*cos
(d*x+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)*EllipticF((-1+cos(d*
x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^3-27*A*cos(d*x+c)^2*a*b^2-6*A*cos(d*x+c)^2*a^2*b+33*A*cos(d*x+c)^4*a^2
*b+27*A*cos(d*x+c)^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)*Ellipti
cE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^2*b-12*A*cos(d*x+c)^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)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(
1/2))*sin(d*x+c)*a^3+24*A*cos(d*x+c)^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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^3+24*A*cos(d*x+c)^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)*EllipticF((-1+cos(d*x+c))/sin(d*
x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*b^3-12*B*cos(d*x+c)^2*a^2*b+56*C*cos(d*x+c)^2*a*b^2-8*C*b^3)/sin(d*x+c)^5
/(b+a*cos(d*x+c))/cos(d*x+c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right )^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )\right )} \sqrt{b \sec \left (d x + c\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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