3.1036 \(\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\)

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

[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)]*Sqrt[Sec[c + d*x]])/(105*a^3*d*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))*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)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*
x]^(5/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) - (2*(4*A*b^2
- 7*a*b*B - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Sec[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.17314, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661} \[ -\frac{2 \sin (c+d x) \left (-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right ) \sqrt{a+b \sec (c+d x)}}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left (a^2-b^2\right ) \sqrt{\sec (c+d x)} \left (25 a^2 A+35 a^2 C-14 a b B+8 A b^2\right ) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{105 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left (a^2 b (19 A+35 C)+63 a^3 B-14 a b^2 B+8 A b^3\right ) \sqrt{a+b \sec (c+d x)} E\left (\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right )}{105 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[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]^(7/2),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)]*Sqrt[Sec[c + d*x]])/(105*a^3*d*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))*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)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*
x]^(5/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) - (2*(4*A*b^2
- 7*a*b*B - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Sec[c + d*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 \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 \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2}{7} \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 \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (A b+7 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{4 \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 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (A b+7 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (4 A b^2-7 a b B-5 a^2 (5 A+7 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{8 \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 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (A b+7 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (4 A b^2-7 a b B-5 a^2 (5 A+7 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt{\sec (c+d x)}}+\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 )\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx}{105 a^3}+\frac{\left (8 A b^3+63 a^3 B-14 a b^2 B+a^2 b (19 A+35 C)\right ) \int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{105 a^3}\\ &=\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (A b+7 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (4 A b^2-7 a b B-5 a^2 (5 A+7 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt{\sec (c+d x)}}+\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)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{b+a \cos (c+d x)}} \, dx}{105 a^3 \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{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)} \sqrt{\sec (c+d x)}}\\ &=\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (A b+7 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (4 A b^2-7 a b B-5 a^2 (5 A+7 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt{\sec (c+d x)}}+\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}} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\frac{b}{a+b}+\frac{a \cos (c+d x)}{a+b}}} \, dx}{105 a^3 \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{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}} \sqrt{\sec (c+d x)}}\\ &=\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 ) \sqrt{\sec (c+d x)}}{105 a^3 d \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 ) 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}} \sqrt{\sec (c+d x)}}+\frac{2 A \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (A b+7 a B) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (4 A b^2-7 a b B-5 a^2 (5 A+7 C)\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 6.76679, size = 4441, normalized size = 12.34 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[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]^(7/2),x]

[Out]

(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a
*b^2*B + 35*a^2*b*C)*Cot[c])/(105*a^3*d) + ((115*a^2*A - 16*A*b^2 + 28*a*b*B + 140*a^2*C)*Cos[d*x]*Sin[c])/(10
5*a^2*d) + (2*(A*b + 7*a*B)*Cos[2*d*x]*Sin[2*c])/(35*a*d) + (A*Cos[3*d*x]*Sin[3*c])/(7*d) + ((115*a^2*A - 16*A
*b^2 + 28*a*b*B + 140*a^2*C)*Cos[c]*Sin[d*x])/(105*a^2*d) + (2*(A*b + 7*a*B)*Cos[2*c]*Sin[2*d*x])/(35*a*d) + (
A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (20*A*
AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 +
Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[
Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A
+ B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]
^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[
c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d
*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1
+ Cot[c]^2]*Sec[c + d*x]^(5/2)) - (8*A*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin
[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a
*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[
c]^2])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sq
rt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*
Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])
]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(105*a^2*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*
C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (28*b*B*AppellF1[1/2, 1/2,
1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*
Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt
[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]
+ C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcT
an[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - Ar
cTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]
]]])/(15*a*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec
[c + d*x]^(5/2)) - (4*C*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan
[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*
Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqr
t[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c
]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot
[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1
+ Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*C
os[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (38*A*b*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c
+ d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]
]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c
]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])
)))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan
[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[T
an[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*
Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[
d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[T
an[c]]]*Sqrt[1 + Tan[c]^2]]))/(105*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]
)*Sec[c + d*x]^(5/2)) - (16*A*b^3*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Ap
pellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1
+ Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt
[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*T
an[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec
[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b
*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x
+ ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + T
an[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(1
05*a^2*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (6*a
*B*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -
((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a
*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Ta
n[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt
[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sq
rt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2]
)]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt
[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*
Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*Sqrt[b + a*Cos[c + d*x]]*(A
+ 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) + (4*b^2*B*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*
(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + Ar
cTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*
(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1
+ Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x
+ ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*
x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcT
an[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a
*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*
x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*a*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos
[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (2*b*C*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*
x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))
/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan
[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[
Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^
2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c
]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) -
((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*
Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c
]^2]]))/(3*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.686, size = 4764, normalized size = 13.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)^(7/2),x)

[Out]

-2/105/d/((a-b)/(a+b))^(1/2)/a^3*(-8*A*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))
^(1/2))*b^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-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)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*cos(d*x+c)*a^3*b-25*A*a^3*b*((a-b)/(a+b))^(1/2)-19*A*a^2*b^2*((a-b
)/(a+b))^(1/2)+4*A*a*b^3*((a-b)/(a+b))^(1/2)-63*B*a^3*b*((a-b)/(a+b))^(1/2)-7*B*a^2*b^2*((a-b)/(a+b))^(1/2)+14
*B*a*b^3*((a-b)/(a+b))^(1/2)-35*C*a^3*b*((a-b)/(a+b))^(1/2)-35*C*a^2*b^2*((a-b)/(a+b))^(1/2)+63*B*EllipticE((-
1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1
))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+35*C*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(
a+b)/(a-b))^(1/2))*a^4*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+25*
A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*(1/(a+b)*(b+a*cos(d*x+c))
/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+18*A*cos(d*x+c)^4*((a-b)/(a+b))^(1/2)*a^3*b-A*cos(d
*x+c)^3*((a-b)/(a+b))^(1/2)*a^2*b^2+28*B*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^3*b+26*A*cos(d*x+c)^2*((a-b)/(a+b)
)^(1/2)*a^3*b+4*A*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a*b^3-7*B*cos(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^2*b^2+70*C*cos
(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^3*b-19*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b+20*A*cos(d*x+c)*((a-b)/(a+b))^(1
/2)*a^2*b^2-8*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^3+35*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b+14*B*cos(d*x+c)
*((a-b)/(a+b))^(1/2)*a^2*b^2-14*B*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^3-35*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^3
*b+35*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^2*b^2-8*A*b^4*((a-b)/(a+b))^(1/2)-25*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)
*a^4-35*C*cos(d*x+c)*((a-b)/(a+b))^(1/2)*a^4+15*A*cos(d*x+c)^5*((a-b)/(a+b))^(1/2)*a^4+10*A*cos(d*x+c)^3*((a-b
)/(a+b))^(1/2)*a^4+35*C*cos(d*x+c)^3*((a-b)/(a+b))^(1/2)*a^4+21*B*cos(d*x+c)^4*((a-b)/(a+b))^(1/2)*a^4+42*B*co
s(d*x+c)^2*((a-b)/(a+b))^(1/2)*a^4+8*A*cos(d*x+c)*((a-b)/(a+b))^(1/2)*b^4-63*B*cos(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))*a^4*(1/(a+b)*(b+a*cos(
d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+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)*(1/(cos(d*x+c)+1))^(1/
2)*sin(d*x+c)*cos(d*x+c)*a^4-8*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(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))*sin(d*x+c)*cos(d*x+c)*b^4-63*B*Elli
pticF((-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)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*cos(d*x+c)*a^4+63*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))
^(1/2)*(1/(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))
*sin(d*x+c)*cos(d*x+c)*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)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*cos(d*x+c)*a^4-19*A*Ellip
ticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b*(1/(a+b)*(b+a*cos(d*x+c))/(cos
(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+2*A*EllipticF((-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*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*si
n(d*x+c)-8*A*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^3*(1/(a+b)*(b+
a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+19*A*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*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*
x+c)+1))^(1/2)*sin(d*x+c)-19*A*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*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+8*A*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*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1
))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+49*B*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(
a+b)/(a-b))^(1/2))*a^3*b*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+1
4*B*EllipticF((-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*(1/(a+b)*(b+a*cos(d
*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-63*B*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*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1)
)^(1/2)*sin(d*x+c)-14*B*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
*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+14*B*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*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/
2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-35*C*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(
a-b))^(1/2))*a^3*b*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+35*C*El
lipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b*(1/(a+b)*(b+a*cos(d*x+c))/(
cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-35*C*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/si
n(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^2*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2
)*sin(d*x+c)+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)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*cos(d*x+c)*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)*
(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*cos(d*x+c)*a*b^3+19*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(
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))*sin(d*x+c)
*cos(d*x+c)*a^3*b-19*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(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))*sin(d*x+c)*cos(d*x+c)*a^2*b^2+8*A*(1/(a+b)*(b
+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/si
n(d*x+c),(-(a+b)/(a-b))^(1/2))*sin(d*x+c)*cos(d*x+c)*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)*(1/(cos(d*x+c)+1))^(1/2)*sin(
d*x+c)*cos(d*x+c)*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)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*cos(d*x+c)*a^2*b^2-63*B*(1/(
a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(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))*sin(d*x+c)*cos(d*x+c)*a^3*b-14*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1
))^(1/2)*(1/(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
))*sin(d*x+c)*cos(d*x+c)*a^2*b^2+14*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(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))*sin(d*x+c)*cos(d*x+c)*a*b^3-35
*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))/(c
os(d*x+c)+1))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*cos(d*x+c)*a^3*b+35*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d
*x+c)+1))^(1/2)*(1/(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))*sin(d*x+c)*cos(d*x+c)*a^3*b-35*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(1/(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))*sin(d*x+c)*cos(d*x+c)*a^2
*b^2)*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)^4*(1/cos(d*x+c))^(7/2)/sin(d*x+c)/(b+a*cos(d*x+c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)^(7/2),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)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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{7}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)^(7/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)^(7/2),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)^(7/2), x)