\(\int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1342]

Optimal result

Integrand size = 45, antiderivative size = 359 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B+35 a^2 C\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{105 a^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{105 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d} \]

[Out]

2/7*A*cos(d*x+c)^(5/2)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+2/105*(a^2-b^2)*(25*A*a^2-6*A*b^2+21*B*a*b+35*C*a^2
)*(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)/a^2/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)+2/35*(3*A*b+7*B*a)*cos(d*x+c)^(3/2)*sin
(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d+2/105*(3*A*b^2+42*B*a*b+5*a^2*(5*A+7*C))*sin(d*x+c)*cos(d*x+c)^(1/2)*(a+b*sec
(d*x+c))^(1/2)/a/d-2/105*(6*A*b^3-63*B*a^3-21*B*a*b^2-2*a^2*b*(41*A+70*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))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/a^2
/d/((b+a*cos(d*x+c))/(a+b))^(1/2)

Rubi [A] (verified)

Time = 1.67 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4350, 4179, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{105 a d}+\frac {2 \left (a^2-b^2\right ) \left (25 a^2 A+35 a^2 C+21 a b B-6 A b^2\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{105 a^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {2 \sqrt {\cos (c+d x)} \left (-63 a^3 B-2 a^2 b (41 A+70 C)-21 a b^2 B+6 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{105 a^2 d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 (7 a B+3 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{35 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{7 d} \]

[In]

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

[Out]

(2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B + 35*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*
x)/2, (2*a)/(a + b)])/(105*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(6*A*b^3 - 63*a^3*B - 21*a*
b^2*B - 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x
]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c +
d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d) + (2*(3*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[
c + d*x]]*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d)

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]

Rule 4350

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]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {1}{7} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)} \left (\frac {1}{2} (3 A b+7 a B)+\frac {1}{2} (5 a A+7 b B+7 a C) \sec (c+d x)+\frac {1}{2} b (2 A+7 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {1}{35} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right )+\frac {1}{4} \left (44 a A b+21 a^2 B+35 b^2 B+70 a b C\right ) \sec (c+d x)+\frac {1}{4} b (16 A b+14 a B+35 b C) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}-\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} \left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 C)\right )-\frac {1}{8} a \left (84 a b B+5 a^2 (5 A+7 C)+3 b^2 (17 A+35 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{105 a} \\ & = \frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {\left (\left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B+35 a^2 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}{105 a^2}-\frac {\left (\left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 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}{105 a^2} \\ & = \frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {\left (\left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B+35 a^2 C\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{105 a^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{105 a^2 \sqrt {b+a \cos (c+d x)}} \\ & = \frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac {\left (\left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B+35 a^2 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}{105 a^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 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}{105 a^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}}} \\ & = \frac {2 \left (a^2-b^2\right ) \left (25 a^2 A-6 A b^2+21 a b B+35 a^2 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 )}{105 a^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (6 A b^3-63 a^3 B-21 a b^2 B-2 a^2 b (41 A+70 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)}}{105 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (3 A b^2+42 a b B+5 a^2 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d}+\frac {2 (3 A b+7 a B) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{7 d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 36.72 (sec) , antiderivative size = 3261, normalized size of antiderivative = 9.08 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((115*a^2*A + 12*A*b^2
+ 168*a*b*B + 140*a^2*C)*Sin[c + d*x])/(105*a) + (2*(8*A*b + 7*a*B)*Sin[2*(c + d*x)])/35 + (a*A*Sin[3*(c + d*
x)])/7))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*Cos[c + d*x]^(3/2)*((
164*a*A*b*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*A*b^3*Sqrt[Cos[c + d*x]])
/(35*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a^2*B*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]
*Sqrt[Sec[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*a*b*C
*Sqrt[Cos[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (10*a^2*A*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c
+ d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (34*A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[
c + d*x]]) + (8*a*b*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*C*Sqrt[Cos[
c + d*x]]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*b^2*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/S
qrt[b + a*Cos[c + d*x]])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x
] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*Arc
Sinh[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)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Ta
n[(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)]
- (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*T
an[(c + d*x)/2]))/(105*a^2*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c +
d*x]^(7/2)*((-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)*(-6*A*b^
3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Se
c[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A +
63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*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)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*
b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a*(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)*(-6*A*b^3
+ 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*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)*(-6*A*b^2 + a^2*(25*A + 6
3*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*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)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*
(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(35*a^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)*(-1/2*((-6*A*b^3 + 63*a^3*B + 21*a*b^2*B
+ 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(-6*A*b^3 + 63*a^3*B +
21*a*b^2*B + 2*a^2*b*(41*A + 70*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)*(-6*A*b^2 + a^2*(25*A
+ 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*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*(-6*A*b^3 + 63*a^3*B
+ 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(-6*A*b^3
+ 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*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)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*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*C
os[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)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSin
h[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)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*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)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*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])))/(105*a^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)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*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)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSin
h[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)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*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]))/(35*a^2*Sqrt[b + a*Cos[c + d*x]])))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4570\) vs. \(2(383)=766\).

Time = 12.58 (sec) , antiderivative size = 4571, normalized size of antiderivative = 12.73

method	result	size
default	\(\text {Expression too large to display}\)	\(4571\)

[In]

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

[Out]

2/105/d*(15*A*sin(d*x+c)*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a^4-105*C*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^2*cos(d*x+c)-6*A*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*(1/
(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*b^4-25*A*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-
(a+b)/(a-b))^(1/2))*cos(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^4-82*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^3*b+82
*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+co
s(d*x+c)))^(1/2)*a^2*b^2+6*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*b^3+82*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^3*b-51*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^2-6*A*Ellip
ticF(((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^3+63*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^4*cos(d*x+c)-63*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*Elliptic
E(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4*cos(d*x+c)-35*C*(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^4*cos(
d*x+c)+140*C*(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*b-140*C*(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*b+140*C*(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^2*b^2-84*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*
b+21*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^2+63*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*b-21*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*Ell
ipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b^2+21*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^3-6*A*
sin(d*x+c)*((a-b)/(a+b))^(1/2)*b^4*(1/(1+cos(d*x+c)))^(1/2)-105*C*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-cs
c(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^2+39*A*sin(d*x+c)*cos(d*
x+c)^2*((a-b)/(a+b))^(1/2)*a^3*b*(1/(1+cos(d*x+c)))^(1/2)+27*A*sin(d*x+c)*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^2
*b^2*(1/(1+cos(d*x+c)))^(1/2)+63*B*sin(d*x+c)*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^3*b*(1/(1+cos(d*x+c)))^(1/2)+
107*A*sin(d*x+c)*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b*(1/(1+cos(d*x+c)))^(1/2)+27*A*sin(d*x+c)*cos(d*x+c)*((a-
b)/(a+b))^(1/2)*a^2*b^2*(1/(1+cos(d*x+c)))^(1/2)-3*A*sin(d*x+c)*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^3*(1/(1+cos
(d*x+c)))^(1/2)+63*B*sin(d*x+c)*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b*(1/(1+cos(d*x+c)))^(1/2)+63*B*sin(d*x+c)*
cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^2*b^2*(1/(1+cos(d*x+c)))^(1/2)+39*A*sin(d*x+c)*cos(d*x+c)^3*((a-b)/(a+b))^(1/
2)*a^3*b*(1/(1+cos(d*x+c)))^(1/2)+175*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b*(1/(1+cos(d*x+c)))^(1/2)*sin(d*x+
c)+25*A*sin(d*x+c)*cos(d*x+c)*((a-b)/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a^4+25*A*sin(d*x+c)*((a-b)/(a+b))^(
1/2)*a^3*b*(1/(1+cos(d*x+c)))^(1/2)+82*A*sin(d*x+c)*((a-b)/(a+b))^(1/2)*a^2*b^2*(1/(1+cos(d*x+c)))^(1/2)+3*A*s
in(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^3*(1/(1+cos(d*x+c)))^(1/2)+63*B*sin(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b*(1/(1+c
os(d*x+c)))^(1/2)+42*B*sin(d*x+c)*((a-b)/(a+b))^(1/2)*a^2*b^2*(1/(1+cos(d*x+c)))^(1/2)+21*B*sin(d*x+c)*((a-b)/
(a+b))^(1/2)*a*b^3*(1/(1+cos(d*x+c)))^(1/2)+35*C*((a-b)/(a+b))^(1/2)*a^3*b*(1/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)
+140*C*((a-b)/(a+b))^(1/2)*a^2*b^2*(1/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+35*C*sin(d*x+c)*cos(d*x+c)*((a-b)/(a+b)
)^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a^4-82*A*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))
^(1/2))*cos(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*b+82*A*EllipticE(((a-b)/(a+b))^(1/2)*(c
ot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b^2
+6*A*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*(1/(a+b)*(b+a*cos(
d*x+c))/(1+cos(d*x+c)))^(1/2)*a*b^3+82*A*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^
(1/2))*cos(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^3*b-51*A*EllipticF(((a-b)/(a+b))^(1/2)*(co
t(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*a^2*b^2-
84*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*b*cos(d*x+c)+21*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^2*cos(d*x+c)+63*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+co
s(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*b*cos(d*x+c)-
21*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^2*b^2*cos(d*x+c)+21*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^3*cos(d*x+c)+140*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+c
os(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*b*cos(d*x+c)
-140*C*(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*b*cos(d*x+c)+140*C*(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^2*b^2*cos(d*x+c)+15*A*(1/(1+cos(d*x+c)))^(1/2)*((a
-b)/(a+b))^(1/2)*a^4*cos(d*x+c)^4*sin(d*x+c)+25*A*(1/(1+cos(d*x+c)))^(1/2)*((a-b)/(a+b))^(1/2)*a^4*cos(d*x+c)^
2*sin(d*x+c)+35*C*(1/(1+cos(d*x+c)))^(1/2)*((a-b)/(a+b))^(1/2)*a^4*cos(d*x+c)^2*sin(d*x+c)-6*A*EllipticF(((a-b
)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c
)))^(1/2)*a*b^3+21*B*sin(d*x+c)*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^4*(1/(1+cos(d*x+c)))^(1/2)+21*B*sin(d*x+c)*
cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^4*(1/(1+cos(d*x+c)))^(1/2)+63*B*sin(d*x+c)*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a
^4*(1/(1+cos(d*x+c)))^(1/2)+63*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^4-63*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*Ellipt
icE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^4-35*C*(1/(a+b)*(b+a*cos(d*x+c))/(1+co
s(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^4-6*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^4-25*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^4)*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/(1/(1+cos(d*x+c)))^(1/2)/(b+a*cos(d*x
+c))/a^2/((a-b)/(a+b))^(1/2)/(1+cos(d*x+c))

Fricas [C] (verification not implemented)

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

Time = 0.14 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.73 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {6 \, {\left (15 \, A a^{4} \cos \left (d x + c\right )^{2} + 5 \, {\left (5 \, A + 7 \, C\right )} a^{4} + 42 \, B a^{3} b + 3 \, A a^{2} b^{2} + 3 \, {\left (7 \, B a^{4} + 8 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-15 i \, {\left (5 \, A + 7 \, C\right )} a^{4} - 126 i \, B a^{3} b + i \, {\left (11 \, A - 35 \, C\right )} a^{2} b^{2} + 42 i \, B a b^{3} - 12 i \, A b^{4}\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 (15 i \, {\left (5 \, A + 7 \, C\right )} a^{4} + 126 i \, B a^{3} b - i \, {\left (11 \, A - 35 \, C\right )} a^{2} b^{2} - 42 i \, B a b^{3} + 12 i \, A b^{4}\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 (-63 i \, B a^{4} - 2 i \, {\left (41 \, A + 70 \, C\right )} a^{3} b - 21 i \, B a^{2} b^{2} + 6 i \, A a b^{3}\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 (63 i \, B a^{4} + 2 i \, {\left (41 \, A + 70 \, C\right )} a^{3} b + 21 i \, B a^{2} b^{2} - 6 i \, A a b^{3}\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 )}{315 \, a^{3} d} \]

[In]

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

[Out]

1/315*(6*(15*A*a^4*cos(d*x + c)^2 + 5*(5*A + 7*C)*a^4 + 42*B*a^3*b + 3*A*a^2*b^2 + 3*(7*B*a^4 + 8*A*a^3*b)*cos
(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + sqrt(2)*(-15*I*(5*A + 7*C
)*a^4 - 126*I*B*a^3*b + I*(11*A - 35*C)*a^2*b^2 + 42*I*B*a*b^3 - 12*I*A*b^4)*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*(5*A + 7*C)*a^4 + 126*I*B*a^3*b - I*(11*A - 35*C)*a^2*b^2 - 42*I*B*a*b^3 + 12*I*A*b^4)*sqrt(a)*weierst
rassPInverse(-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)*(-63*I*B*a^4 - 2*I*(41*A + 70*C)*a^3*b - 21*I*B*a^2*b^2 + 6*I*A*a*b^3)*sqrt(a)*weierstra
ssZeta(-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, 8/2
7*(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)*(63*I*B*a^4 + 2*I*(
41*A + 70*C)*a^3*b + 21*I*B*a^2*b^2 - 6*I*A*a*b^3)*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)))/(a^3*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

[In]

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

[Out]

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