\(\int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx\) [627]

Optimal result

Integrand size = 35, antiderivative size = 487 \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\frac {(4 A b-5 a B) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{4 b^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (12 a A b-15 a^2 B-4 b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{4 b^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\left (4 a A b-5 a^2 B+b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}} \]

[Out]

2*a*(A*b-B*a)*sin(d*x+c)/b/(a^2-b^2)/d/cos(d*x+c)^(5/2)/(a+b*sec(d*x+c))^(1/2)+1/4*(4*A*b-5*B*a)*(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)/b^2/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)-1/4*(12*A*a*b-15*B*a^2-4*B*b^2)*(cos(1/2*d*x+1/2*c)^2)
^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1
/2)/b^3/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)-1/2*(4*A*a*b-5*B*a^2+B*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2
)/b^2/(a^2-b^2)/d/cos(d*x+c)^(3/2)+1/4*(12*A*a^2*b-4*A*b^3-15*B*a^3+7*B*a*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/
2)/b^3/(a^2-b^2)/d/cos(d*x+c)^(1/2)-1/4*(12*A*a^2*b-4*A*b^3-15*B*a^3+7*B*a*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/c
os(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2
)/b^3/(a^2-b^2)/d/((b+a*cos(d*x+c))/(a+b))^(1/2)

Rubi [A] (verified)

Time = 2.10 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3034, 4114, 4187, 4193, 3944, 2886, 2884, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{2 b^2 d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (A b-a B) \sin (c+d x)}{b d \left (a^2-b^2\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{4 b^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{4 b^3 d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{4 b^3 d \left (a^2-b^2\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {(4 A b-5 a B) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{4 b^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \]

[In]

Int[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)),x]

[Out]

((4*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*b^2*d*Sqrt[Cos[c
+ d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((12*a*A*b - 15*a^2*B - 4*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Ellip
ticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((12*a^2*A*b - 4
*A*b^3 - 15*a^3*B + 7*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x
]])/(4*b^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*a*(A*b - a*B)*Sin[c + d*x])/(b*(a^2 - b^2)*d
*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]) - ((4*a*A*b - 5*a^2*B + b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c +
d*x])/(2*b^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + ((12*a^2*A*b - 4*A*b^3 - 15*a^3*B + 7*a*b^2*B)*Sqrt[a + b*Sec
[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)*d*Sqrt[Cos[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 2884

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

Rule 2886

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/
(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3034

Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.)*((g_.)*sin[(e_.
) + (f_.)*(x_)])^(p_.), x_Symbol] :> Dist[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p, Int[(a + b*Csc[e + f*x])^m*((
c + d*Csc[e + f*x])^n/(g*Csc[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d
, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] && IntegerQ[n])

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 3944

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[d*Sqrt
[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/Sqrt[a + b*Csc[e + f*x]]), Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f
*x]]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4114

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

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 4187

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[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(
m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Dist[d/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(
d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + (A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a
*C*n)*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] && GtQ[n, 0]

Rule 4193

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d
_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Dist[C/d^2, Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a +
b*Csc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[a + b*Csc[e + f*x]]), x] /; Fre
eQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx \\ & = \frac {2 a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (\frac {3}{2} a (A b-a B)-\frac {1}{2} b (A b-a B) \sec (c+d x)-\frac {1}{2} \left (4 a A b-5 a^2 B+b^2 B\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{b \left (a^2-b^2\right )} \\ & = \frac {2 a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\left (4 a A b-5 a^2 B+b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (-\frac {1}{4} a \left (4 a A b-5 a^2 B+b^2 B\right )+\frac {1}{2} b \left (2 a A b-a^2 B-b^2 B\right ) \sec (c+d x)+\frac {1}{4} \left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{b^2 \left (a^2-b^2\right )} \\ & = \frac {2 a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\left (4 a A b-5 a^2 B+b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{8} a \left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right )-\frac {1}{4} a b \left (4 a A b-5 a^2 B+b^2 B\right ) \sec (c+d x)-\frac {1}{8} \left (a^2-b^2\right ) \left (12 a A b-15 a^2 B-4 b^2 B\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{b^3 \left (a^2-b^2\right )} \\ & = \frac {2 a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\left (4 a A b-5 a^2 B+b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{8} a \left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right )-\frac {1}{4} a b \left (4 a A b-5 a^2 B+b^2 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{b^3 \left (a^2-b^2\right )}-\frac {\left (\left (12 a A b-15 a^2 B-4 b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{8 b^3} \\ & = \frac {2 a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\left (4 a A b-5 a^2 B+b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {\left ((4 A b-5 a B) \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}{8 b^2}+\frac {\left (\left (-12 a^2 A b+4 A b^3+15 a^3 B-7 a b^2 B\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}{8 b^3 \left (a^2-b^2\right )}-\frac {\left (\left (12 a A b-15 a^2 B-4 b^2 B\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}} \, dx}{8 b^3 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \\ & = \frac {2 a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\left (4 a A b-5 a^2 B+b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {\left ((4 A b-5 a B) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{8 b^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (12 a A b-15 a^2 B-4 b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{8 b^3 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-12 a^2 A b+4 A b^3+15 a^3 B-7 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{8 b^3 \left (a^2-b^2\right ) \sqrt {b+a \cos (c+d x)}} \\ & = -\frac {\left (12 a A b-15 a^2 B-4 b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{4 b^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\left (4 a A b-5 a^2 B+b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {\left ((4 A b-5 a B) \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}{8 b^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-12 a^2 A b+4 A b^3+15 a^3 B-7 a b^2 B\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}{8 b^3 \left (a^2-b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}} \\ & = \frac {(4 A b-5 a B) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{4 b^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (12 a A b-15 a^2 B-4 b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{4 b^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\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)}}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\left (4 a A b-5 a^2 B+b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 37.16 (sec) , antiderivative size = 141550, normalized size of antiderivative = 290.66 \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)),x]

[Out]

Result too large to show

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.48 (sec) , antiderivative size = 3532, normalized size of antiderivative = 7.25

method	result	size
default	\(\text {Expression too large to display}\)	\(3532\)

[In]

int((A+B*sec(d*x+c))/cos(d*x+c)^(7/2)/(a+b*sec(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/4/d*(4*A*((a-b)/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*b^3*cos(d*x+c)*sin(d*x+c)-24*A*(1/(a+b)*(b+a*cos(d*x+c
))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b*cos
(d*x+c)^2-5*B*((a-b)/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a^2*b*cos(d*x+c)^2*sin(d*x+c)+2*B*((a-b)/(a+b))^(1/
2)*(1/(1+cos(d*x+c)))^(1/2)*a*b^2*cos(d*x+c)^2*sin(d*x+c)+24*A*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(
d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b*cos(d*x+c)^3-
8*B*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*co
s(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^2*cos(d*x+c)^3-16*A*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*Ellipt
icF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2*cos(d*x+c)^2+12*A*(1/(a+b)*(b+a*co
s(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^
2*b*cos(d*x+c)^2+20*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c
)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b*cos(d*x+c)^2+2*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*Ell
ipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a*b^2*cos(d*x+c)^2+7*B*(1/(a+b)*(b+a*
cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*
a*b^2*cos(d*x+c)^2-8*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x
+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*a*b^2*cos(d*x+c)^2-15*B*(1/(1+cos(d*x+c)))^(1/2)*((a-b)/(a+
b))^(1/2)*a^3*cos(d*x+c)^2*sin(d*x+c)+4*A*(1/(1+cos(d*x+c)))^(1/2)*((a-b)/(a+b))^(1/2)*a*b^2*cos(d*x+c)*sin(d*
x+c)-5*B*(1/(1+cos(d*x+c)))^(1/2)*((a-b)/(a+b))^(1/2)*a^2*b*cos(d*x+c)*sin(d*x+c)-3*B*(1/(1+cos(d*x+c)))^(1/2)
*((a-b)/(a+b))^(1/2)*a*b^2*cos(d*x+c)*sin(d*x+c)+2*B*((a-b)/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*b^3*cos(d*x+
c)*sin(d*x+c)+24*A*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(
1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^2*cos(d*x+c)^3-24*A*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+
c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b*cos(d*x+c)^3-16*A*E
llipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*
x+c)))^(1/2)*a*b^2*cos(d*x+c)^3+12*A*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2
))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b*cos(d*x+c)^3-30*B*EllipticPi(((a-b)/(a+b))^(1/2)*(cot
(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b*c
os(d*x+c)^3+20*B*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos
(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b*cos(d*x+c)^3+2*B*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-
(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^2*cos(d*x+c)^3+7*B*EllipticE(((a-b)/(a
+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^2
*cos(d*x+c)^3+24*A*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(
1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^2*cos(d*x+c)^2-30*B*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x
+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b*cos(d
*x+c)^2+12*A*(1/(1+cos(d*x+c)))^(1/2)*((a-b)/(a+b))^(1/2)*a^2*b*cos(d*x+c)^2*sin(d*x+c)-30*B*(1/(a+b)*(b+a*cos
(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+
b))^(1/2))*a^3*cos(d*x+c)^2+2*B*(1/(1+cos(d*x+c)))^(1/2)*((a-b)/(a+b))^(1/2)*a*b^2*sin(d*x+c)+4*A*((a-b)/(a+b)
)^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a*b^2*cos(d*x+c)^2*sin(d*x+c)-8*B*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-
csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^3*cos(d*x+c)^
2+4*B*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1
+cos(d*x+c)))^(1/2)*b^3*cos(d*x+c)^2-4*A*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^
(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^3*cos(d*x+c)^3-8*B*EllipticPi(((a-b)/(a+b))^(1/2)*(co
t(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^3*co
s(d*x+c)^3+30*B*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(
d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*cos(d*x+c)^3+4*B*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+
b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^3*cos(d*x+c)^3-4*A*EllipticE(((a-b)/(a+b))^
(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^3*cos(d*
x+c)^2-30*B*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*(1/(a+b)
*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*cos(d*x+c)^3-15*B*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*
x+c)),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*cos(d*x+c)^3+2*B*(1/(1+cos(d*x
+c)))^(1/2)*((a-b)/(a+b))^(1/2)*b^3*sin(d*x+c)+30*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(
((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3*cos(d*x+c)^2-15*B*(1/(a+b)*(b+a*cos(d*x+
c))/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^3*cos(
d*x+c)^2+24*A*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d
*x+c)),(a+b)/(a-b),I/((a-b)/(a+b))^(1/2))*a^2*b*cos(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/cos(d*x+c)^(3/2)/(1/(1+co
s(d*x+c)))^(1/2)/(b+a*cos(d*x+c))/(a+b)/((a-b)/(a+b))^(1/2)/b^3/(1+cos(d*x+c))

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sec(d*x+c))/cos(d*x+c)^(7/2)/(a+b*sec(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sec(d*x+c))/cos(d*x+c)**(7/2)/(a+b*sec(d*x+c))**(3/2),x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sec(d*x+c))/cos(d*x+c)^(7/2)/(a+b*sec(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate((A+B*sec(d*x+c))/cos(d*x+c)^(7/2)/(a+b*sec(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((B*sec(d*x + c) + A)/((b*sec(d*x + c) + a)^(3/2)*cos(d*x + c)^(7/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int((A + B/cos(c + d*x))/(cos(c + d*x)^(7/2)*(a + b/cos(c + d*x))^(3/2)),x)

[Out]

int((A + B/cos(c + d*x))/(cos(c + d*x)^(7/2)*(a + b/cos(c + d*x))^(3/2)), x)