3.1335 \(\int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=360 \[ \frac{2 \left (a^2-b^2\right ) \left (25 a^2 A+35 a^2 C-14 a b B+8 A b^2\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{2 a}{a+b}\right )}{105 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{105 a^2 d}+\frac{2 \sqrt{\cos (c+d x)} \left (a^2 b (19 A+35 C)+63 a^3 B-14 a b^2 B+8 A b^3\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{105 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d} \]

[Out]

(2*(a^2 - b^2)*(25*a^2*A + 8*A*b^2 - 14*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^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 63*a^3*B - 14*a*
b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]]
)/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b^2 - 7*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^2*d) + (2*(A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c +
d*x]]*Sin[c + d*x])/(35*a*d) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)

________________________________________________________________________________________

Rubi [A]  time = 1.29943, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4265, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ -\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{105 a^2 d}+\frac{2 \left (a^2-b^2\right ) \left (25 a^2 A+35 a^2 C-14 a b B+8 A b^2\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{105 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \left (a^2 b (19 A+35 C)+63 a^3 B-14 a b^2 B+8 A b^3\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{105 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a^2 - b^2)*(25*a^2*A + 8*A*b^2 - 14*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^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 63*a^3*B - 14*a*
b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]]
)/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b^2 - 7*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^2*d) + (2*(A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c +
d*x]]*Sin[c + d*x])/(35*a*d) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)

Rule 4265

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]

Rule 4094

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 4104

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 4035

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 3856

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 2655

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

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

Rule 3858

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(Sqrt[d*
Csc[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 2663

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

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

Rubi steps

\begin{align*} \int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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{7}{2}}(c+d x)} \, dx\\ &=\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d}+\frac{1}{7} \left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} (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 (4 A+7 C) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 (A b+7 a B) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d}-\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} \left (4 A b^2-7 a b B-5 a^2 (5 A+7 C)\right )-\frac{1}{4} a (23 A b+21 a B+35 b C) \sec (c+d x)-\frac{1}{2} b (A b+7 a B) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx}{35 a}\\ &=-\frac{2 \left (4 A b^2-7 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^2 d}+\frac{2 (A b+7 a B) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \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 (8 A b^3+63 a^3 B-14 a b^2 B+a^2 b (19 A+35 C)\right )+\frac{1}{8} a \left (2 A b^2+49 a b B+5 a^2 (5 A+7 C)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx}{105 a^2}\\ &=-\frac{2 \left (4 A b^2-7 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^2 d}+\frac{2 (A b+7 a B) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d}+\frac{\left (\left (a^2-b^2\right ) \left (25 a^2 A+8 A b^2-14 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^3}+\frac{\left (\left (8 A b^3+63 a^3 B-14 a b^2 B+a^2 b (19 A+35 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^3}\\ &=-\frac{2 \left (4 A b^2-7 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^2 d}+\frac{2 (A b+7 a B) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d}+\frac{\left (\left (a^2-b^2\right ) \left (25 a^2 A+8 A b^2-14 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^3 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (8 A b^3+63 a^3 B-14 a b^2 B+a^2 b (19 A+35 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^3 \sqrt{b+a \cos (c+d x)}}\\ &=-\frac{2 \left (4 A b^2-7 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^2 d}+\frac{2 (A b+7 a B) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d}+\frac{\left (\left (a^2-b^2\right ) \left (25 a^2 A+8 A b^2-14 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^3 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left (\left (8 A b^3+63 a^3 B-14 a b^2 B+a^2 b (19 A+35 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^3 \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}\\ &=\frac{2 \left (a^2-b^2\right ) \left (25 a^2 A+8 A b^2-14 a b B+35 a^2 C\right ) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{105 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (8 A b^3+63 a^3 B-14 a b^2 B+a^2 b (19 A+35 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^3 d \sqrt{\frac{b+a \cos (c+d x)}{a+b}}}-\frac{2 \left (4 A b^2-7 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^2 d}+\frac{2 (A b+7 a B) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d}\\ \end{align*}

Mathematica [C]  time = 22.9347, size = 3071, normalized size = 8.53 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(((115*a^2*A - 16*A*b^2 + 28*a*b*B + 140*a^2*C)*Sin[c + d*x])/(21
0*a^2) + ((A*b + 7*a*B)*Sin[2*(c + d*x)])/(35*a) + (A*Sin[3*(c + d*x)])/14))/d - (2*Cos[c + d*x]^(3/2)*((19*A*
b*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*A*b^3*Sqrt[Cos[c + d*x]])/(105*a^
2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*a*B*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[S
ec[c + d*x]]) - (2*b^2*B*Sqrt[Cos[c + d*x]])/(15*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (b*C*Sqrt[Co
s[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*a*A*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(
21*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(105*a*Sqrt[b + a*Cos[c + d*x]]
) + (7*b*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (a*C*Sqrt[Cos[c + d*x]]*Sqrt
[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]))*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sqrt[a + b*Sec[c + d*x]]
*((-I)*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*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)*(8*
A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*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)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B
+ a^2*b*(19*A + 35*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^3*d*(b + a*C
os[c + d*x])*Sqrt[Sec[c + d*x]]*(-(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)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*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)*(8*A*b^
2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*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)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a
^2*b*(19*A + 35*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^2*(b + a*Cos[c +
d*x])^(3/2)) + (Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(8*A*b^
3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*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)*(8*A*b^2 - 2*a*b*(3*A + 7
*B) + a^2*(25*A + 63*B + 35*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sq
rt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C)
)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(35*a^3*Sqrt[b + a*Cos[c + d*x]]) - (2*Co
s[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-((8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 3
5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2))/2 - I*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*
(19*A + 35*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)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A +
63*B + 35*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*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*
C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19
*A + 35*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)*(8*A*b^3 +
63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*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*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)] + ((I/2)*a*(a + b)*(8*A*b^2 - 2*a*
b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*EllipticF[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*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)*(8*A*b^2 - 2*a*b*(3*A +
7*B) + a^2*(25*A + 63*B + 35*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)*(8*A*b^3 + 63*a^3*B
- 14*a*b^2*B + a^2*b*(19*A + 35*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^3*Sqrt[b + a*Cos[
c + d*x]]) - (Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(8*A*b^3 + 63*a^3*B - 14*
a*b^2*B + a^2*b*(19*A + 35*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqr
t[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A +
63*B + 35*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)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*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^3*Sqrt[b + a*Cos[c + d*x]])))

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Maple [B]  time = 0.837, size = 2829, normalized size = 7.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x)

[Out]

-2/105/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)^(1/2)*(cos(d*x+c)+1)^2*(-1+cos(d*x+c))^3*(25*A*cos(d*x
+c)^2*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)+19*A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))
^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*b-2*A*EllipticF((-
1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(
1/2)*a^2*b^2+8*A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*
cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a*b^3-19*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos
(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b+19*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)
+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^2-8*A*(1/(a+b)
*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b)
)^(1/2))*a*b^3+8*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*b^4*(1/(cos(d*x+c)+1))^(3/2)+25*A*cos(d*x+c)*((a-b)/(a+b))^(
1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)+35*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)
+1))^(3/2)+63*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)+25*A*((a-b)/(a+b))^(1/2
)*sin(d*x+c)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)+19*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^2*(1/(cos(d*x+c)+1))^(3/
2)-4*A*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a*b^3*(1/(cos(d*x+c)+1))^(3/2)+63*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b
*(1/(cos(d*x+c)+1))^(3/2)+7*B*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^2*(1/(cos(d*x+c)+1))^(3/2)-14*B*((a-b)/(a+b
))^(1/2)*sin(d*x+c)*a*b^3*(1/(cos(d*x+c)+1))^(3/2)+35*C*((a-b)/(a+b))^(1/2)*a^3*b*sin(d*x+c)*(1/(cos(d*x+c)+1)
)^(3/2)+35*C*((a-b)/(a+b))^(1/2)*a^2*b^2*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3/2)+15*A*cos(d*x+c)^4*((a-b)/(a+b))^(
1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)+21*B*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+
c)+1))^(3/2)+21*B*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)+15*A*cos(d*x+c)^3*(
(a-b)/(a+b))^(1/2)*sin(d*x+c)*a^4*(1/(cos(d*x+c)+1))^(3/2)+35*C*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^
4*(1/(cos(d*x+c)+1))^(3/2)+63*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-
b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b+14*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*El
lipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^2-14*B*(1/(a+b)*(b+a*cos(d*
x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b
^3-49*B*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c
))/(cos(d*x+c)+1))^(1/2)*a^3*b-14*B*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1
/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b^2-35*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1
/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b+35*C*(1/(a+b)*(b+a*co
s(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))
*a^2*b^2+35*C*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos
(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*b+18*A*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b*(1/(cos(d*x+c)+1))
^(3/2)+28*B*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)+44*A*cos(d*x+c)*((a-b)/
(a+b))^(1/2)*sin(d*x+c)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)-A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^2*b^2*(1/
(cos(d*x+c)+1))^(3/2)+4*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a*b^3*(1/(cos(d*x+c)+1))^(3/2)+28*B*cos(d*
x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)-7*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*sin(d*x+
c)*a^2*b^2*(1/(cos(d*x+c)+1))^(3/2)+70*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b*sin(d*x+c)*(1/(cos(d*x+c)+1))^(3
/2)+18*A*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*sin(d*x+c)*a^3*b*(1/(cos(d*x+c)+1))^(3/2)-A*cos(d*x+c)^2*((a-b)/(a+b
))^(1/2)*sin(d*x+c)*a^2*b^2*(1/(cos(d*x+c)+1))^(3/2)-63*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Elli
pticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4+63*B*EllipticF((-1+cos(d*x+c))*
((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^4-35*C*
EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a+b)*(b+a*cos(d*x+c))/(cos(
d*x+c)+1))^(1/2)*a^4-25*A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*(1/(a
+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^4+8*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((
-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*b^4)/a^3/((a-b)/(a+b))^(1/2)/(b+a*cos(d*x+
c))/(1/(cos(d*x+c)+1))^(3/2)/sin(d*x+c)^6

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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} \cos \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{2} + B \cos \left (d x + c\right )^{3} \sec \left (d x + c\right ) + A \cos \left (d x + c\right )^{3}\right )} \sqrt{b \sec \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral((C*cos(d*x + c)^3*sec(d*x + c)^2 + B*cos(d*x + c)^3*sec(d*x + c) + A*cos(d*x + c)^3)*sqrt(b*sec(d*x +
c) + a)*sqrt(cos(d*x + c)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)*(a+b*sec(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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} \cos \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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