3.1363 \(\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=461 \[ -\frac{2 \left (6 a^2 b (2 A+5 C)-5 a^3 B-40 a b^2 B+48 A b^3\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{15 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (a^2 (-(A-5 C))-5 a b B+6 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left (a^2-b^2\right )}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (-a^2 (9 A b-15 b C)+5 a^3 B-20 a b^2 B+24 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left (a^2-b^2\right )}-\frac{2 \sqrt{\cos (c+d x)} \left (-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)+25 a^3 b B-40 a b^3 B+48 A b^4\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{15 a^4 d \left (a^2-b^2\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}} \]

[Out]

(-2*(48*A*b^3 - 5*a^3*B - 40*a*b^2*B + 6*a^2*b*(2*A + 5*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c +
d*x)/2, (2*a)/(a + b)])/(15*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(48*A*b^4 + 25*a^3*b*B - 4
0*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)
]*Sqrt[a + b*Sec[c + d*x]])/(15*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a
*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(24*A*b^3 + 5*a^3*B - 20
*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)
*d) - (2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*
(a^2 - b^2)*d)

________________________________________________________________________________________

Rubi [A]  time = 1.5673, antiderivative size = 461, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4265, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ -\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (a^2 (-(A-5 C))-5 a b B+6 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left (a^2-b^2\right )}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (-a^2 (9 A b-15 b C)+5 a^3 B-20 a b^2 B+24 A b^3\right ) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left (a^2-b^2\right )}-\frac{2 \left (6 a^2 b (2 A+5 C)-5 a^3 B-40 a b^2 B+48 A b^3\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{15 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \left (-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)+25 a^3 b B-40 a b^3 B+48 A b^4\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{15 a^4 d \left (a^2-b^2\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(48*A*b^3 - 5*a^3*B - 40*a*b^2*B + 6*a^2*b*(2*A + 5*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c +
d*x)/2, (2*a)/(a + b)])/(15*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(48*A*b^4 + 25*a^3*b*B - 4
0*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)
]*Sqrt[a + b*Sec[c + d*x]])/(15*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a
*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(24*A*b^3 + 5*a^3*B - 20
*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)
*d) - (2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*
(a^2 - b^2)*d)

Rule 4265

Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]

Rule 4100

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*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a +
b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), I
nt[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*
(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] &
& ILtQ[n, 0])

Rule 4104

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 +
1)*(d*Csc[e + f*x])^n)/(a*f*n), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rule 4035

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

Rule 3856

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)], x_Symbol] :> Dist[Sqrt[a +
b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]]), Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; Free
Q[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2655

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

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

Rule 3858

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(Sqrt[d*
Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]])/Sqrt[a + b*Csc[e + f*x]], Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2663

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

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

Rubi steps

\begin{align*} \int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}-\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} \left (6 A b^2-5 a b B-a^2 (A-5 C)\right )+\frac{1}{2} a (A b-a B+b C) \sec (c+d x)-2 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}-\frac{2 \left (6 A b^2-5 a b B-a^2 (A-5 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} \left (24 A b^3+5 a^3 B-20 a b^2 B-3 a^2 b (3 A-5 C)\right )+\frac{1}{4} a \left (2 A b^2-5 a b B+a^2 (3 A+5 C)\right ) \sec (c+d x)-\frac{1}{2} b \left (6 A b^2-5 a b B-a^2 (A-5 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{5 a^2 \left (a^2-b^2\right )}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (24 A b^3+5 a^3 B-20 a b^2 B-a^2 (9 A b-15 b C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}-\frac{2 \left (6 A b^2-5 a b B-a^2 (A-5 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}-\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{8} \left (48 A b^4+25 a^3 b B-40 a b^3 B-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)\right )+\frac{1}{8} a \left (12 A b^3-5 a^3 B-10 a b^2 B+3 a^2 b (A+5 C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{15 a^3 \left (a^2-b^2\right )}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (24 A b^3+5 a^3 B-20 a b^2 B-a^2 (9 A b-15 b C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}-\frac{2 \left (6 A b^2-5 a b B-a^2 (A-5 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}-\frac{\left (\left (48 A b^3-5 a^3 B-40 a b^2 B+6 a^2 b (2 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx}{15 a^4}-\frac{\left (\left (48 A b^4+25 a^3 b B-40 a b^3 B-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{15 a^4 \left (a^2-b^2\right )}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (24 A b^3+5 a^3 B-20 a b^2 B-a^2 (9 A b-15 b C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}-\frac{2 \left (6 A b^2-5 a b B-a^2 (A-5 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}-\frac{\left (\left (48 A b^3-5 a^3 B-40 a b^2 B+6 a^2 b (2 A+5 C)\right ) \sqrt{b+a \cos (c+d x)}\right ) \int \frac{1}{\sqrt{b+a \cos (c+d x)}} \, dx}{15 a^4 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\left (\left (48 A b^4+25 a^3 b B-40 a b^3 B-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{b+a \cos (c+d x)} \, dx}{15 a^4 \left (a^2-b^2\right ) \sqrt{b+a \cos (c+d x)}}\\ &=\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (24 A b^3+5 a^3 B-20 a b^2 B-a^2 (9 A b-15 b C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}-\frac{2 \left (6 A b^2-5 a b B-a^2 (A-5 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}-\frac{\left (\left (48 A b^3-5 a^3 B-40 a b^2 B+6 a^2 b (2 A+5 C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{15 a^4 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\left (\left (48 A b^4+25 a^3 b B-40 a b^3 B-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}\right ) \int \sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}} \, dx}{15 a^4 \left (a^2-b^2\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}\\ &=-\frac{2 \left (48 A b^3-5 a^3 B-40 a b^2 B+6 a^2 b (2 A+5 C)\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{15 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \left (48 A b^4+25 a^3 b B-40 a b^3 B-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)\right ) \sqrt{\cos (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right ) \sqrt{a+b \sec (c+d x)}}{15 a^4 \left (a^2-b^2\right ) d \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}+\frac{2 \left (A b^2-a (b B-a C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (24 A b^3+5 a^3 B-20 a b^2 B-a^2 (9 A b-15 b C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}-\frac{2 \left (6 A b^2-5 a b B-a^2 (A-5 C)\right ) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}\\ \end{align*}

Mathematica [C]  time = 24.839, size = 3870, normalized size = 8.39 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(-9*A*b + 5*a*B)*Sin[c
+ d*x])/(15*a^3) + (4*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(
b + a*Cos[c + d*x])) + (2*A*Sin[2*(c + d*x)])/(5*a^2)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(
a + b*Sec[c + d*x])^(3/2)) - (4*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*((6*a*A*Sqrt[Cos[c + d*x]])/(5*(a^2 -
b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^2*Sqrt[Cos[c + d*x]])/(5*a*(a^2 - b^2)*Sqrt[b + a*
Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (32*A*b^4*Sqrt[Cos[c + d*x]])/(5*a^3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*
Sqrt[Sec[c + d*x]]) - (10*b*B*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])
+ (16*b^3*B*Sqrt[Cos[c + d*x]])/(3*a^2*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*C*Sqrt[
Cos[c + d*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*b^2*C*Sqrt[Cos[c + d*x]])/(a*(a^
2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(5*(a^2
- b^2)*Sqrt[b + a*Cos[c + d*x]]) - (8*A*b^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(5*a^2*(a^2 - b^2)*Sqrt[b +
a*Cos[c + d*x]]) + (2*a*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) + (
4*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) - (2*b*C*Sqrt[Cos[c
+ d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]))*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A
+ B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*
C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[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)] + I*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A +
5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[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)] + (48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2
*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(15*a^4*(
a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(
(-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-48*A*b^4 - 25*a^3*
b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[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)] + I*a*(a + b)*(-48*A*b^3 +
4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[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)] + (48*A*b^4
+ 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]
^2)^(3/2)*Tan[(c + d*x)/2]))/(15*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^(3/2)) + (2*Sqrt[Cos[c + d*x]]*(Cos[(c +
d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*
A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[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)] + I*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*
(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[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)] + (48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a
^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(5*
a^4*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(((4
8*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c +
d*x)/2]^2)^(5/2))/2 - I*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*
C))*EllipticE[I*ArcSinh[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] + I*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*
(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[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] - a*(48*A*b^4 + 25*a^3*b*B - 40*a*b
^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] + (
3*(48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(
c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(
4*A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[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)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A +
10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[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)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(-48*
A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*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)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*
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])))/(15*a^4*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c
+ d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*
b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[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)] + I*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6
*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*S
ec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (48*A*b^4 + 25*a^3*b*B - 40*a*b^3*
B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*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]))/(5*a^4*
(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]])))

________________________________________________________________________________________

Maple [B]  time = 0.613, size = 2418, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*(cos(d*x+c)+1)^5*(-1+cos(d*x+c))^3*(-3*A*cos(d*x+c
)^2*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)-12*A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(
1/2)/sin(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*b-36*A*EllipticF((-1
+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(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^2-48*A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(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*b^3+24*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos
(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^2-48*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*b^4*
(1/(cos(d*x+c)+1))^(3/2)+9*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/
(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4+15*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ellipti
cE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4-9*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)
*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)-15*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))
^(3/2)-5*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)-9*A*((a-b)/(a+b))^(1/2)*sin(
d*x+c)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)-24*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a*b^3*(1/(cos(d*x+c)+1))^(3/2)-5*B*(
(a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)+20*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^2*(1/(c
os(d*x+c)+1))^(3/2)+40*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a*b^3*(1/(cos(d*x+c)+1))^(3/2)-15*C*((a-b)/(a+b))^(1/2
)*a^3*b*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)-30*C*((a-b)/(a+b))^(1/2)*a^2*b^2*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/
2)-5*B*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)-3*A*cos(d*x+c)^3*((a-b)/(a+b))
^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)-25*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(
(-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b+40*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos
(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^3+30*B*El
lipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(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*b+40*B*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(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^2-30*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ellipt
icE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^2-30*C*EllipticF((-1+cos(d*x+c)
)*((a-b)/(a+b))^(1/2)/sin(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*b-3
*A*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)-5*B*cos(d*x+c)^2*((a-b)/(a+b))^(
1/2)*sin(d*x+c)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)-3*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b*(1/(cos(d*x
+c)+1))^(3/2)-18*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^2*(1/(cos(d*x+c)+1))^(3/2)-24*A*cos(d*x+c)*
((a-b)/(a+b))^(1/2)*sin(d*x+c)*a*b^3*(1/(cos(d*x+c)+1))^(3/2)+15*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a
^3*b*(1/(cos(d*x+c)+1))^(3/2)+20*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^2*(1/(cos(d*x+c)+1))^(3/2)-
15*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+3*A*cos(d*x+c)^2*((a-b)/(a+b))^(
1/2)*sin(d*x+c)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)+6*A*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^2*(1/(cos
(d*x+c)+1))^(3/2)+5*B*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(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^4-15*C*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(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^4-9*A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b
))^(1/2)/sin(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^4-48*A*(1/(a+b)*(b
+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(
1/2))*b^4)*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(3/2)/a^4/(b+a*cos(d*x+c))/(a-b)/sin(d*x+c)^6

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac{5}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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