3.5.73 \(\int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx\) [473]

Optimal. Leaf size=588 \[ -\frac {2 \left (17 a^4 A b+116 a^2 A b^3-128 A b^5-5 a^5 B-80 a^3 b^2 B+80 a b^4 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \]

[Out]

2/3*b*(A*b-B*a)*sin(d*x+c)/a/(a^2-b^2)/d/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c))^(3/2)+2/3*b*(12*A*a^2*b-8*A*b^3-9*B
*a^3+5*B*a*b^2)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c))^(1/2)-2/15*(17*A*a^4*b+116*A*a^
2*b^3-128*A*b^5-5*B*a^5-80*B*a^3*b^2+80*B*a*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin
(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c)^(1/2)/a^5/(a^2-b^2)/d/(a+b*
sec(d*x+c))^(1/2)+2/15*(3*A*a^4-71*A*a^2*b^2+48*A*b^4+50*B*a^3*b-30*B*a*b^3)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)
/a^3/(a^2-b^2)^2/d/sec(d*x+c)^(3/2)-2/15*(14*A*a^4*b-98*A*a^2*b^3+64*A*b^5-5*B*a^5+65*B*a^3*b^2-40*B*a*b^4)*si
n(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^4/(a^2-b^2)^2/d/sec(d*x+c)^(1/2)+2/15*(9*A*a^6+55*A*a^4*b^2-212*A*a^2*b^4+12
8*A*b^6-40*B*a^5*b+140*B*a^3*b^3-80*B*a*b^5)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2
*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/a^5/(a^2-b^2)^2/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/s
ec(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.19, antiderivative size = 588, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4115, 4185, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {2 b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right ) \sin (c+d x)}{3 a^2 d \left (a^2-b^2\right )^2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{15 a^3 d \left (a^2-b^2\right )^2 \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{15 a^4 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)}}-\frac {2 \left (-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\right ) \sqrt {\sec (c+d x)} \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^5 d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(17*a^4*A*b + 116*a^2*A*b^3 - 128*A*b^5 - 5*a^5*B - 80*a^3*b^2*B + 80*a*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(
a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^5*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x
]]) + (2*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*Ellipt
icE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a
+ b)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b*Sec[c
+ d*x])^(3/2)) + (2*b*(12*a^2*A*b - 8*A*b^3 - 9*a^3*B + 5*a*b^2*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sec[c
+ d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4*A - 71*a^2*A*b^2 + 48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B)*Sqr
t[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) - (2*(14*a^4*A*b - 98*a^2*A*b^
3 + 64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)
^2*d*Sqrt[Sec[c + d*x]])

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 3941

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 3943

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[Sqrt[d*C
sc[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 4115

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

Rule 4120

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 4185

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 4189

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]

Rubi steps

\begin {align*} \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx &=\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (-3 a^2 A+8 A b^2-5 a b B\right )+\frac {3}{2} a (A b-a B) \sec (c+d x)-3 b (A b-a B) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {4 \int \frac {\frac {1}{4} \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right )-\frac {1}{4} a \left (6 a^2 A b-2 A b^3-3 a^3 B-a b^2 B\right ) \sec (c+d x)+b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {8 \int \frac {\frac {3}{8} \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right )-\frac {1}{8} a \left (9 a^4 A+27 a^2 A b^2-16 A b^4-30 a^3 b B+10 a b^3 B\right ) \sec (c+d x)-\frac {1}{4} b \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{15 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {16 \int \frac {\frac {3}{16} \left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right )-\frac {3}{16} a \left (8 a^4 A b+44 a^2 A b^3-32 A b^5-5 a^5 B-35 a^3 b^2 B+20 a b^4 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{45 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}-\frac {\left (17 a^4 A b+116 a^2 A b^3-128 A b^5-5 a^5 B-80 a^3 b^2 B+80 a b^4 B\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )}+\frac {\left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )^2}\\ &=\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}-\frac {\left (\left (17 a^4 A b+116 a^2 A b^3-128 A b^5-5 a^5 B-80 a^3 b^2 B+80 a b^4 B\right ) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{15 a^5 \left (a^2-b^2\right )^2 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}-\frac {\left (\left (17 a^4 A b+116 a^2 A b^3-128 A b^5-5 a^5 B-80 a^3 b^2 B+80 a b^4 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{15 a^5 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) \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^5 \left (a^2-b^2\right )^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}\\ &=-\frac {2 \left (17 a^4 A b+116 a^2 A b^3-128 A b^5-5 a^5 B-80 a^3 b^2 B+80 a b^4 B\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 ) \sqrt {\sec (c+d x)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (12 a^2 A b-8 A b^3-9 a^3 B+5 a b^2 B\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A-71 a^2 A b^2+48 A b^4+50 a^3 b B-30 a b^3 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (14 a^4 A b-98 a^2 A b^3+64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 3.97, size = 392, normalized size = 0.67 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x) \left (-\frac {2 \left (\frac {b+a \cos (c+d x)}{a+b}\right )^{3/2} \left (a^2 \left (8 a^4 A b+44 a^2 A b^3-32 A b^5-5 a^5 B-35 a^3 b^2 B+20 a b^4 B\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )-\left (9 a^6 A+55 a^4 A b^2-212 a^2 A b^4+128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )-b F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}+a \left (\frac {10 b^4 (A b-a B) \sin (c+d x)}{-a^2+b^2}-\frac {10 b^3 \left (-15 a^2 A b+11 A b^3+12 a^3 B-8 a b^2 B\right ) (b+a \cos (c+d x)) \sin (c+d x)}{\left (a^2-b^2\right )^2}-2 (14 A b-5 a B) (b+a \cos (c+d x))^2 \sin (c+d x)+3 a A (b+a \cos (c+d x))^2 \sin (2 (c+d x))\right )\right )}{15 a^5 d (a+b \sec (c+d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*((-2*((b + a*Cos[c + d*x])/(a + b))^(3/2)*(a^2*(8*a^4*A*b + 44*a^2*A*
b^3 - 32*A*b^5 - 5*a^5*B - 35*a^3*b^2*B + 20*a*b^4*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] - (9*a^6*A + 55*a^
4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*((a + b)*EllipticE[(c + d*x)/2,
(2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])))/((a - b)^2*(a + b)) + a*((10*b^4*(A*b - a*B)*Sin[
c + d*x])/(-a^2 + b^2) - (10*b^3*(-15*a^2*A*b + 11*A*b^3 + 12*a^3*B - 8*a*b^2*B)*(b + a*Cos[c + d*x])*Sin[c +
d*x])/(a^2 - b^2)^2 - 2*(14*A*b - 5*a*B)*(b + a*Cos[c + d*x])^2*Sin[c + d*x] + 3*a*A*(b + a*Cos[c + d*x])^2*Si
n[2*(c + d*x)])))/(15*a^5*d*(a + b*Sec[c + d*x])^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(8250\) vs. \(2(606)=1212\).
time = 16.87, size = 8251, normalized size = 14.03

method	result	size
default	\(\text {Expression too large to display}\)	\(8251\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 2.47, size = 1531, normalized size = 2.60 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45*(sqrt(2)*(-15*I*B*a^7*b^2 + 42*I*A*a^6*b^3 - 185*I*B*a^5*b^4 + 242*I*A*a^4*b^5 + 340*I*B*a^3*b^6 - 520*I*
A*a^2*b^7 - 160*I*B*a*b^8 + 256*I*A*b^9 + (-15*I*B*a^9 + 42*I*A*a^8*b - 185*I*B*a^7*b^2 + 242*I*A*a^6*b^3 + 34
0*I*B*a^5*b^4 - 520*I*A*a^4*b^5 - 160*I*B*a^3*b^6 + 256*I*A*a^2*b^7)*cos(d*x + c)^2 - 2*(15*I*B*a^8*b - 42*I*A
*a^7*b^2 + 185*I*B*a^6*b^3 - 242*I*A*a^5*b^4 - 340*I*B*a^4*b^5 + 520*I*A*a^3*b^6 + 160*I*B*a^2*b^7 - 256*I*A*a
*b^8)*cos(d*x + c))*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a
*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a) + sqrt(2)*(15*I*B*a^7*b^2 - 42*I*A*a^6*b^3 + 185*I*B*a^5*b^4 - 24
2*I*A*a^4*b^5 - 340*I*B*a^3*b^6 + 520*I*A*a^2*b^7 + 160*I*B*a*b^8 - 256*I*A*b^9 + (15*I*B*a^9 - 42*I*A*a^8*b +
185*I*B*a^7*b^2 - 242*I*A*a^6*b^3 - 340*I*B*a^5*b^4 + 520*I*A*a^4*b^5 + 160*I*B*a^3*b^6 - 256*I*A*a^2*b^7)*co
s(d*x + c)^2 - 2*(-15*I*B*a^8*b + 42*I*A*a^7*b^2 - 185*I*B*a^6*b^3 + 242*I*A*a^5*b^4 + 340*I*B*a^4*b^5 - 520*I
*A*a^3*b^6 - 160*I*B*a^2*b^7 + 256*I*A*a*b^8)*cos(d*x + c))*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a
^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqrt(2)*(-9*I*A*a^7*b
^2 + 40*I*B*a^6*b^3 - 55*I*A*a^5*b^4 - 140*I*B*a^4*b^5 + 212*I*A*a^3*b^6 + 80*I*B*a^2*b^7 - 128*I*A*a*b^8 + (-
9*I*A*a^9 + 40*I*B*a^8*b - 55*I*A*a^7*b^2 - 140*I*B*a^6*b^3 + 212*I*A*a^5*b^4 + 80*I*B*a^4*b^5 - 128*I*A*a^3*b
^6)*cos(d*x + c)^2 + 2*(-9*I*A*a^8*b + 40*I*B*a^7*b^2 - 55*I*A*a^6*b^3 - 140*I*B*a^5*b^4 + 212*I*A*a^4*b^5 + 8
0*I*B*a^3*b^6 - 128*I*A*a^2*b^7)*cos(d*x + c))*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b
- 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c)
+ 3*I*a*sin(d*x + c) + 2*b)/a)) - 3*sqrt(2)*(9*I*A*a^7*b^2 - 40*I*B*a^6*b^3 + 55*I*A*a^5*b^4 + 140*I*B*a^4*b^
5 - 212*I*A*a^3*b^6 - 80*I*B*a^2*b^7 + 128*I*A*a*b^8 + (9*I*A*a^9 - 40*I*B*a^8*b + 55*I*A*a^7*b^2 + 140*I*B*a^
6*b^3 - 212*I*A*a^5*b^4 - 80*I*B*a^4*b^5 + 128*I*A*a^3*b^6)*cos(d*x + c)^2 + 2*(9*I*A*a^8*b - 40*I*B*a^7*b^2 +
55*I*A*a^6*b^3 + 140*I*B*a^5*b^4 - 212*I*A*a^4*b^5 - 80*I*B*a^3*b^6 + 128*I*A*a^2*b^7)*cos(d*x + c))*sqrt(a)*
weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)
/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a)) + 6*(3*(A*a^9 - 2*A*a^
7*b^2 + A*a^5*b^4)*cos(d*x + c)^4 + (5*B*a^9 - 8*A*a^8*b - 10*B*a^7*b^2 + 16*A*a^6*b^3 + 5*B*a^5*b^4 - 8*A*a^4
*b^5)*cos(d*x + c)^3 + 5*(2*B*a^8*b - 5*A*a^7*b^2 - 16*B*a^6*b^3 + 25*A*a^5*b^4 + 10*B*a^4*b^5 - 16*A*a^3*b^6)
*cos(d*x + c)^2 + (5*B*a^7*b^2 - 14*A*a^6*b^3 - 65*B*a^5*b^4 + 98*A*a^4*b^5 + 40*B*a^3*b^6 - 64*A*a^2*b^7)*cos
(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/((a^12 - 2*a^10*b^2 + a^8*
b^4)*d*cos(d*x + c)^2 + 2*(a^11*b - 2*a^9*b^3 + a^7*b^5)*d*cos(d*x + c) + (a^10*b^2 - 2*a^8*b^4 + a^6*b^6)*d)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6192 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B/cos(c + d*x))/((a + b/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(5/2)),x)

[Out]

int((A + B/cos(c + d*x))/((a + b/cos(c + d*x))^(5/2)*(1/cos(c + d*x))^(5/2)), x)

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