\(\int \frac {\sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) [1037]

Optimal result

Integrand size = 45, antiderivative size = 457 \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {2 \left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{315 a^4 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^4 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}} \]

[Out]

-2/315*(a^2-b^2)*(16*A*b^3-75*B*a^3-24*B*a*b^2+6*a^2*b*(6*A+7*C))*(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^4/
d/(a+b*sec(d*x+c))^(1/2)+2/9*A*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(7/2)+2/63*(A*b+9*B*a)*sin(d*x+c
)*(a+b*sec(d*x+c))^(1/2)/a/d/sec(d*x+c)^(5/2)-2/315*(6*A*b^2-9*B*a*b-7*a^2*(7*A+9*C))*sin(d*x+c)*(a+b*sec(d*x+
c))^(1/2)/a^2/d/sec(d*x+c)^(3/2)+2/315*(8*A*b^3+75*B*a^3-12*B*a*b^2+a^2*b*(13*A+21*C))*sin(d*x+c)*(a+b*sec(d*x
+c))^(1/2)/a^3/d/sec(d*x+c)^(1/2)-2/315*(16*A*b^4-57*B*a^3*b-24*B*a*b^3+6*a^2*b^2*(4*A+7*C)-21*a^4*(7*A+9*C))*
(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^4/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 1.80 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4179, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {2 \sin (c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{315 a^4 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^4 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (9 a B+A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{63 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]

[In]

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

[Out]

(-2*(a^2 - b^2)*(16*A*b^3 - 75*a^3*B - 24*a*b^2*B + 6*a^2*b*(6*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*El
lipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^4*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(16*A*b^4 -
57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sq
rt[a + b*Sec[c + d*x]])/(315*a^4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Se
c[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(A*b + 9*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(
63*a*d*Sec[c + d*x]^(5/2)) - (2*(6*A*b^2 - 9*a*b*B - 7*a^2*(7*A + 9*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])
/(315*a^2*d*Sec[c + d*x]^(3/2)) + (2*(8*A*b^3 + 75*a^3*B - 12*a*b^2*B + a^2*b*(13*A + 21*C))*Sqrt[a + b*Sec[c
+ d*x]]*Sin[c + d*x])/(315*a^3*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 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 4179

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

Rule 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 \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 b+9 a B)+\frac {1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)+\frac {3}{2} b (2 A+3 C) \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {4 \int \frac {\frac {1}{4} \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right )-\frac {1}{4} a (47 A b+45 a B+63 b C) \sec (c+d x)-b (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 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 \int \frac {\frac {3}{8} \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right )+\frac {1}{8} a \left (2 A b^2+207 a b B+21 a^2 (7 A+9 C)\right ) \sec (c+d x)-\frac {1}{4} b \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\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 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {16 \int \frac {\frac {3}{16} \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right )+\frac {3}{16} a \left (4 A b^3-75 a^3 B-6 a b^2 B-3 a^2 b (37 A+49 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{945 a^3} \\ & = \frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {\left (\left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\right )\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 a^4}-\frac {\left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{315 a^4} \\ & = \frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {\left (\left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\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^4 \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{315 a^4 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = \frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}}-\frac {\left (\left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\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^4 \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\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^4 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ & = -\frac {2 \left (a^2-b^2\right ) \left (16 A b^3-75 a^3 B-24 a b^2 B+6 a^2 b (6 A+7 C)\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^4 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^4 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 A \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (A b+9 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{63 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a^3 d \sqrt {\sec (c+d x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.62 (sec) , antiderivative size = 5993, normalized size of antiderivative = 13.11 \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(8877\) vs. \(2(475)=950\).

Time = 25.46 (sec) , antiderivative size = 8878, normalized size of antiderivative = 19.43

method	result	size
parts	\(\text {Expression too large to display}\)	\(8878\)
default	\(\text {Expression too large to display}\)	\(8934\)

[In]

int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/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 = 0.17 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {\sqrt {2} {\left (-225 i \, B a^{5} - 3 i \, {\left (13 \, A + 21 \, C\right )} a^{4} b + 96 i \, B a^{3} b^{2} - 12 i \, {\left (3 \, A + 7 \, C\right )} a^{2} b^{3} + 48 i \, B a b^{4} - 32 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} + 3 i \, {\left (13 \, A + 21 \, C\right )} a^{4} b - 96 i \, B a^{3} b^{2} + 12 i \, {\left (3 \, A + 7 \, C\right )} a^{2} b^{3} - 48 i \, B a b^{4} + 32 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 (-21 i \, {\left (7 \, A + 9 \, C\right )} a^{5} - 57 i \, B a^{4} b + 6 i \, {\left (4 \, A + 7 \, C\right )} a^{3} b^{2} - 24 i \, B a^{2} b^{3} + 16 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 (21 i \, {\left (7 \, A + 9 \, C\right )} a^{5} + 57 i \, B a^{4} b - 6 i \, {\left (4 \, A + 7 \, C\right )} a^{3} b^{2} + 24 i \, B a^{2} b^{3} - 16 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} + A a^{4} b\right )} \cos \left (d x + c\right )^{3} + {\left (7 \, {\left (7 \, A + 9 \, C\right )} a^{5} + 9 \, B a^{4} b - 6 \, A a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (75 \, B a^{5} + {\left (13 \, A + 21 \, C\right )} a^{4} b - 12 \, B a^{3} b^{2} + 8 \, 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^{5} d} \]

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

1/945*(sqrt(2)*(-225*I*B*a^5 - 3*I*(13*A + 21*C)*a^4*b + 96*I*B*a^3*b^2 - 12*I*(3*A + 7*C)*a^2*b^3 + 48*I*B*a*
b^4 - 32*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*c
os(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a) + sqrt(2)*(225*I*B*a^5 + 3*I*(13*A + 21*C)*a^4*b - 96*I*B*a^3*b^2 +
12*I*(3*A + 7*C)*a^2*b^3 - 48*I*B*a*b^4 + 32*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)*(-21*I*(7*A + 9*C)
*a^5 - 57*I*B*a^4*b + 6*I*(4*A + 7*C)*a^3*b^2 - 24*I*B*a^2*b^3 + 16*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, 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)*(21*I*(7*A + 9*C)*a^5 + 57*I*B*a^
4*b - 6*I*(4*A + 7*C)*a^3*b^2 + 24*I*B*a^2*b^3 - 16*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, 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 + A*a^4*b)*cos(d*x
+ c)^3 + (7*(7*A + 9*C)*a^5 + 9*B*a^4*b - 6*A*a^3*b^2)*cos(d*x + c)^2 + (75*B*a^5 + (13*A + 21*C)*a^4*b - 12*B
*a^3*b^2 + 8*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^5*d)

Sympy [F(-1)]

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

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)**2)*(a+b*sec(d*x+c))**(1/2)/sec(d*x+c)**(9/2),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(9/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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