\(\int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) [447]

Optimal result

Integrand size = 35, antiderivative size = 427 \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{315 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (10 A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}} \]

[Out]

2/315*(a^2-b^2)*(39*A*a^2*b+8*A*b^3+75*B*a^3-18*B*a*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellip
ticF(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^3/d/(a+b*se
c(d*x+c))^(1/2)+2/9*a*A*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(7/2)+2/63*(10*A*b+9*B*a)*sin(d*x+c)*(a
+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(5/2)+2/315*(49*A*a^2+3*A*b^2+72*B*a*b)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/
d/sec(d*x+c)^(3/2)+2/315*(88*A*a^2*b-4*A*b^3+75*B*a^3+9*B*a*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^2/d/sec(d
*x+c)^(1/2)+2/315*(147*A*a^4+33*A*a^2*b^2+8*A*b^4+246*B*a^3*b-18*B*a*b^3)*(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^3/d/((b+a*cos(d*x+c
))/(a+b))^(1/2)/sec(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4110, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (49 a^2 A+72 a b B+3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{315 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 \left (a^2-b^2\right ) \left (75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^3 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (9 a B+10 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]

[In]

Int[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]

[Out]

(2*(a^2 - b^2)*(39*a^2*A*b + 8*A*b^3 + 75*a^3*B - 18*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c
+ d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(147*a^4*A + 33*a^2*A*b
^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*
a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(
9*d*Sec[c + d*x]^(7/2)) + (2*(10*A*b + 9*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))
+ (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sec[c + d*x]^(3/2)) + (2
*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^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 4110

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x]
+ Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[a*(a*B*n - A*b*(m - n - 1)) + (
2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; Free
Q[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LeQ[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 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*} \text {integral}& = \frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {2}{9} \int \frac {-\frac {1}{2} a (10 A b+9 a B)-\frac {1}{2} \left (7 a^2 A+9 A b^2+18 a b B\right ) \sec (c+d x)-\frac {3}{2} b (2 a A+3 b B) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (10 A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 \int \frac {\frac {1}{4} a \left (49 a^2 A+3 A b^2+72 a b B\right )+\frac {1}{4} a \left (92 a A b+45 a^2 B+63 b^2 B\right ) \sec (c+d x)+a b (10 A b+9 a B) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{63 a} \\ & = \frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (10 A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {8 \int \frac {-\frac {3}{8} a \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right )-\frac {1}{8} a^2 \left (147 a^2 A+209 A b^2+396 a b B\right ) \sec (c+d x)-\frac {1}{4} a b \left (49 a^2 A+3 A b^2+72 a b B\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{315 a^2} \\ & = \frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (10 A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {16 \int \frac {\frac {3}{16} a \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right )+\frac {3}{16} a^2 \left (186 a^2 A b+2 A b^3+75 a^3 B+153 a b^2 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{945 a^3} \\ & = \frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (10 A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right )\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 a^3}+\frac {\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{315 a^3} \\ & = \frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (10 A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 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}{315 a^3 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{315 a^3 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = \frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (10 A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 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}{315 a^3 \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 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}{315 a^3 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ & = \frac {2 \left (a^2-b^2\right ) \left (39 a^2 A b+8 A b^3+75 a^3 B-18 a b^2 B\right ) \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 ) \sqrt {\sec (c+d x)}}{315 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (10 A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (49 a^2 A+3 A b^2+72 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (88 a^2 A b-4 A b^3+75 a^3 B+9 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sqrt {\sec (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.50 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.73 \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^{3/2} \left (8 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \left (a^2 \left (186 a^2 A b+2 A b^3+75 a^3 B+153 a b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )+\left (147 a^4 A+33 a^2 A b^2+8 A b^4+246 a^3 b B-18 a b^3 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )-b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )\right )\right )+a (b+a \cos (c+d x)) \left (\left (804 a^2 A b-32 A b^3+690 a^3 B+72 a b^2 B\right ) \sin (c+d x)+a \left (2 \left (133 a^2 A+6 A b^2+144 a b B\right ) \sin (2 (c+d x))+5 a (2 (10 A b+9 a B) \sin (3 (c+d x))+7 a A \sin (4 (c+d x)))\right )\right )\right )}{1260 a^3 d (b+a \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x)} \]

[In]

Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]

[Out]

((a + b*Sec[c + d*x])^(3/2)*(8*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a^2*(186*a^2*A*b + 2*A*b^3 + 75*a^3*B + 153
*a*b^2*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + (147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3
*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])) + a*(b + a*Cos[
c + d*x])*((804*a^2*A*b - 32*A*b^3 + 690*a^3*B + 72*a*b^2*B)*Sin[c + d*x] + a*(2*(133*a^2*A + 6*A*b^2 + 144*a*
b*B)*Sin[2*(c + d*x)] + 5*a*(2*(10*A*b + 9*a*B)*Sin[3*(c + d*x)] + 7*a*A*Sin[4*(c + d*x)])))))/(1260*a^3*d*(b
+ a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(6589\) vs. \(2(445)=890\).

Time = 20.79 (sec) , antiderivative size = 6590, normalized size of antiderivative = 15.43

method	result	size
parts	\(\text {Expression too large to display}\)	\(6590\)
default	\(\text {Expression too large to display}\)	\(6634\)

[In]

int((a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c))/sec(d*x+c)^(9/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.16 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.56 \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {\sqrt {2} {\left (-225 i \, B a^{5} - 264 i \, A a^{4} b + 33 i \, B a^{3} b^{2} + 60 i \, A a^{2} b^{3} - 36 i \, B a b^{4} + 16 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (225 i \, B a^{5} + 264 i \, A a^{4} b - 33 i \, B a^{3} b^{2} - 60 i \, A a^{2} b^{3} + 36 i \, B a b^{4} - 16 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-147 i \, A a^{5} - 246 i \, B a^{4} b - 33 i \, A a^{3} b^{2} + 18 i \, B a^{2} b^{3} - 8 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (147 i \, A a^{5} + 246 i \, B a^{4} b + 33 i \, A a^{3} b^{2} - 18 i \, B a^{2} b^{3} + 8 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) + \frac {6 \, {\left (35 \, A a^{5} \cos \left (d x + c\right )^{4} + 5 \, {\left (9 \, B a^{5} + 10 \, A a^{4} b\right )} \cos \left (d x + c\right )^{3} + {\left (49 \, A a^{5} + 72 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (75 \, B a^{5} + 88 \, A a^{4} b + 9 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{945 \, a^{4} d} \]

[In]

integrate((a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c))/sec(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

1/945*(sqrt(2)*(-225*I*B*a^5 - 264*I*A*a^4*b + 33*I*B*a^3*b^2 + 60*I*A*a^2*b^3 - 36*I*B*a*b^4 + 16*I*A*b^5)*sq
rt(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)*(225*I*B*a^5 + 264*I*A*a^4*b - 33*I*B*a^3*b^2 - 60*I*A*a^2*b^3 + 36*I*B*a*b^4
- 16*I*A*b^5)*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)*(-147*I*A*a^5 - 246*I*B*a^4*b - 33*I*A*a^3*b^2 + 18*I*B*a^
2*b^3 - 8*I*A*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrass
PInverse(-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)*(147*I*A*a^5 + 246*I*B*a^4*b + 33*I*A*a^3*b^2 - 18*I*B*a^2*b^3 + 8*I*A*a*b^4)*sqrt(a)*weier
strassZeta(-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*(35*A*a^5*cos(d*x + c)^
4 + 5*(9*B*a^5 + 10*A*a^4*b)*cos(d*x + c)^3 + (49*A*a^5 + 72*B*a^4*b + 3*A*a^3*b^2)*cos(d*x + c)^2 + (75*B*a^5
+ 88*A*a^4*b + 9*B*a^3*b^2 - 4*A*a^2*b^3)*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^4*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c))/sec(d*x+c)**(9/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c))/sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/2)/sec(d*x + c)^(9/2), x)

Giac [F]

\[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c))/sec(d*x+c)^(9/2),x, algorithm="giac")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/2)/sec(d*x + c)^(9/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]

[In]

int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(3/2))/(1/cos(c + d*x))^(9/2),x)

[Out]

int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(3/2))/(1/cos(c + d*x))^(9/2), x)