3.14.70 \(\int \frac {\sqrt {\cos (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx\) [1370]

Optimal. Leaf size=401 \[ \frac {2 \left (8 A b^3+3 a^3 B-2 a b^2 B-a^2 b (9 A+C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (8 A b^4+6 a^3 b B-2 a b^3 B+3 a^4 (A-C)-a^2 b^2 (15 A+C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (4 A b^4+5 a^3 b B-a b^3 B-2 a^4 C-2 a^2 b^2 (4 A+C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \]

[Out]

2/3*(A*b^2-a*(B*b-C*a))*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^(3/2)/cos(d*x+c)^(1/2)-2/3*(4*A*b^4+5*a^3*b*
B-a*b^3*B-2*a^4*C-2*a^2*b^2*(4*A+C))*sin(d*x+c)/a^2/(a^2-b^2)^2/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)+2/3*
(8*A*b^3+3*a^3*B-2*a*b^2*B-a^2*b*(9*A+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)/a^3/(a^2-b^2)/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x
+c))^(1/2)+2/3*(8*A*b^4+6*a^3*b*B-2*a*b^3*B+3*a^4*(A-C)-a^2*b^2*(15*A+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^3/
(a^2-b^2)^2/d/((b+a*cos(d*x+c))/(a+b))^(1/2)

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Rubi [A]
time = 0.82, antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4350, 4185, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac {2 \left (3 a^3 B-a^2 b (9 A+C)-2 a b^2 B+8 A b^3\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 )}{3 a^3 d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {2 \sin (c+d x) \left (-2 a^4 C+5 a^3 b B-2 a^2 b^2 (4 A+C)-a b^3 B+4 A b^4\right )}{3 a^2 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \left (3 a^4 (A-C)+6 a^3 b B-a^2 b^2 (15 A+C)-2 a b^3 B+8 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{3 a^3 d \left (a^2-b^2\right )^2 \sqrt {\frac {a \cos (c+d x)+b}{a+b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]

[Out]

(2*(8*A*b^3 + 3*a^3*B - 2*a*b^2*B - a^2*b*(9*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2,
(2*a)/(a + b)])/(3*a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^4 + 6*a^3*b*B -
2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt
[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*
Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*(4*A*b^4 + 5*a^3*b*B - a*
b^3*B - 2*a^4*C - 2*a^2*b^2*(4*A + C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*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 4185

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*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a +
b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), I
nt[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*
(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] &
& ILtQ[n, 0])

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*} \int \frac {\sqrt {\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} \left (4 A b^2-a b B-a^2 (3 A-C)\right )+\frac {3}{2} a (A b-a B+b C) \sec (c+d x)-\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (4 A b^4+5 a^3 b B-a b^3 B-2 a^4 C-2 a^2 b^2 (4 A+C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} \left (8 A b^4+6 a^3 b B-2 a b^3 B+3 a^4 (A-C)-a^2 b^2 (15 A+C)\right )+\frac {1}{4} a \left (2 A b^3+3 a^3 B+a b^2 B-2 a^2 b (3 A+2 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (4 A b^4+5 a^3 b B-a b^3 B-2 a^4 C-2 a^2 b^2 (4 A+C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (8 A b^3+3 a^3 B-2 a b^2 B-a^2 b (9 A+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}{3 a^3 \left (a^2-b^2\right )}+\frac {\left (\left (8 A b^4+6 a^3 b B-2 a b^3 B+3 a^4 (A-C)-a^2 b^2 (15 A+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}{3 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (4 A b^4+5 a^3 b B-a b^3 B-2 a^4 C-2 a^2 b^2 (4 A+C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (8 A b^3+3 a^3 B-2 a b^2 B-a^2 b (9 A+C)\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{3 a^3 \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (8 A b^4+6 a^3 b B-2 a b^3 B+3 a^4 (A-C)-a^2 b^2 (15 A+C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{3 a^3 \left (a^2-b^2\right )^2 \sqrt {b+a \cos (c+d x)}}\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (4 A b^4+5 a^3 b B-a b^3 B-2 a^4 C-2 a^2 b^2 (4 A+C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (8 A b^3+3 a^3 B-2 a b^2 B-a^2 b (9 A+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}{3 a^3 \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (8 A b^4+6 a^3 b B-2 a b^3 B+3 a^4 (A-C)-a^2 b^2 (15 A+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}{3 a^3 \left (a^2-b^2\right )^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}\\ &=\frac {2 \left (8 A b^3+3 a^3 B-2 a b^2 B-a^2 b (9 A+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 )}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (8 A b^4+6 a^3 b B-2 a b^3 B+3 a^4 (A-C)-a^2 b^2 (15 A+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)}}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (4 A b^4+5 a^3 b B-a b^3 B-2 a^4 C-2 a^2 b^2 (4 A+C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 22.20, size = 3834, normalized size = 9.56 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]

[Out]

((b + a*Cos[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x
] + a^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (4*(9*a^2*A*b^2*Sin[c + d*x] - 5*A*b^4
*Sin[c + d*x] - 6*a^3*b*B*Sin[c + d*x] + 2*a*b^3*B*Sin[c + d*x] + 3*a^4*C*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x
]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c
+ 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) - (4*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*((2*a^2*A*Sqrt[Cos[c + d
*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (10*A*b^2*Sqrt[Cos[c + d*x]])/((a^2 - b^2)
^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^4*Sqrt[Cos[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[b +
a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a*b*B*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqr
t[Sec[c + d*x]]) - (4*b^3*B*Sqrt[Cos[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]
) - (2*a^2*C*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*b^2*C*Sqrt[C
os[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*A*b*Sqrt[Cos[c + d*x]]*Sqrt
[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (4*A*b^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*
a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqr
t[b + a*Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d
*x]]) - (8*a*b*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Sec[c
+ d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(8*A*b^
4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + 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^3 -
2*a*b^2*(3*A + B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + 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^4 + 6*a^3*b*B -
2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)
/2]))/(3*a*(a^3 - a*b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)*((-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)*(8*A*b^4 + 6*a^3*b*B -
2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + 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^3 - 2*a*b^2*(3*A +
B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + 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^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a
^4*(A - C) - a^2*b^2*(15*A + C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*(a^3 -
a*b^2)^2*(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)*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*EllipticE[I*ArcSinh[T
an[(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^3 - 2*a*b^2*(3*A + B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + C))*EllipticF[I*ArcSinh[T
an[(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^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/
2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a*(a^3 - a*b^2)^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*((8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(b
+ a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) -
a^2*b^2*(15*A + C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*C
os[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(8*A*b^3 - 2*a*b^2*(3*A + B) + 3*a^3*
(A + B - C) - a^2*b*(9*A - 3*B + 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^4 + 6*a^3*b*B - 2*a*b^3
*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(8*A*b^
4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + 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^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + 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 ...

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3772\) vs. \(2(431)=862\).
time = 0.42, size = 3773, normalized size = 9.41

method	result	size
default	\(\text {Expression too large to display}\)	\(3773\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/3/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(1+cos(d*x+c))^5*(-1+cos(d*x+c))^3*(-11*A*sin(d*x+c)*((a-b)/(a+b))^(
1/2)*a^2*b^3*(1/(1+cos(d*x+c)))^(3/2)+4*A*sin(d*x+c)*((a-b)/(a+b))^(1/2)*a*b^4*(1/(1+cos(d*x+c)))^(3/2)+8*A*si
n(d*x+c)*((a-b)/(a+b))^(1/2)*b^5*(1/(1+cos(d*x+c)))^(3/2)+6*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*El
lipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^3-3*A*((b+a*cos(d*x+c))/(1+
cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*b-
9*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*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^2+6*B*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(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^2-2*B*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*Ellipt
icE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^4+3*B*((b+a*cos(d*x+c))/(1+cos(d*
x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*b-3*B*((
b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*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^2-2*B*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*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^3-3*C*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(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^4*b-C*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a
+b))^(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^3+3*C*((b+a*co
s(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^
(1/2))*a^4*b-C*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*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^2+8*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+
c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a*b^4+3*A*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(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^4*b-15*A*((b+a*cos(d*x+c))
/(1+cos(d*x+c))/(a+b))^(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^3+6*B*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*cos(d*x+c)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/
2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*b-2*B*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*cos(d*x+c)*Ellipti
cE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^2*b^3-3*B*cos(d*x+c)*((b+a*cos(d*x+c
))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*
a^4*b-2*B*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*cos(d*x+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^2-C*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*Ellipti
cE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^3*b^2-C*cos(d*x+c)*((b+a*cos(d*x+c))
/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^
4*b-9*A*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)
/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*b+6*A*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*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^2+8*A*cos(d*x+c)*((b+a*cos(d*x+c))
/(1+cos(d*x+c))/(a+b))^(1/2)*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^3-15*A*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(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^2+8*A*cos(d*x+c)*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(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*b^4-2*B*sin(d*x+c)*((a-b)/(a+b))^
(1/2)*a*b^4*(1/(1+cos(d*x+c)))^(3/2)+5*B*sin(d*x+c)*((a-b)/(a+b))^(1/2)*a^3*b^2*(1/(1+cos(d*x+c)))^(3/2)-B*sin
(d*x+c)*((a-b)/(a+b))^(1/2)*a^2*b^3*(1/(1+cos(d*x+c)))^(3/2)-3*C*((a-b)/(a+b))^(1/2)*cos(d*x+c)*a^5*sin(d*x+c)
*(1/(1+cos(d*x+c)))^(3/2)-2*C*((a-b)/(a+b))^(1/2)*a^4*b*sin(d*x+c)*(1/(1+cos(d*x+c)))^(3/2)+C*((a-b)/(a+b))^(1
/2)*a^3*b^2*sin(d*x+c)*(1/(1+cos(d*x+c)))^(3/2)-C*((a-b)/(a+b))^(1/2)*a^2*b^3*sin(d*x+c)*(1/(1+cos(d*x+c)))^(3
/2)-3*A*sin(d*x+c)*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*a^5*(1/(1+cos(d*x+c)))^(3/2)-3*A*sin(d*x+c)*((a-b)/(a+b))^
(1/2)*a^3*b^2*(1/(1+cos(d*x+c)))^(3/2)+C*((a-b)/(a+b))^(1/2)*cos(d*x+c)*a^4*b*sin(d*x+c)*(1/(1+cos(d*x+c)))^(3
/2)+6*B*sin(d*x+c)*((a-b)/(a+b))^(1/2)*cos(d*x+c)*a^4*b*(1/(1+cos(d*x+c)))^(3/2)-3*A*sin(d*x+c)*((a-b)/(a+b))^
(1/2)*cos(d*x+c)^2*a^4*b*(1/(1+cos(d*x+c)))^(3/2)+3*A*sin(d*x+c)*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*a^3*b^2*(1/(
1+cos(d*x+c)))^(3/2)+3*A*sin(d*x+c)*((a-b)/(a+b))^(1/2)*cos(d*x+c)^2*a^2*b^3*(1/(1+cos(d*x+c)))^(3/2)+8*A*((b+
a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*Ellip...

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.79, size = 1323, normalized size = 3.30 \begin {gather*} \frac {6 \, {\left (2 \, C a^{6} b - 5 \, B a^{5} b^{2} + 2 \, {\left (4 \, A + C\right )} a^{4} b^{3} + B a^{3} b^{4} - 4 \, A a^{2} b^{5} + {\left (3 \, C a^{7} - 6 \, B a^{6} b + {\left (9 \, A + C\right )} a^{5} b^{2} + 2 \, B a^{4} b^{3} - 5 \, A a^{3} b^{4}\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 ) + {\left (\sqrt {2} {\left (-9 i \, B a^{7} + 6 i \, {\left (4 \, A + C\right )} a^{6} b + 9 i \, B a^{5} b^{2} - 2 i \, {\left (18 \, A + C\right )} a^{4} b^{3} - 4 i \, B a^{3} b^{4} + 16 i \, A a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (9 i \, B a^{6} b - 6 i \, {\left (4 \, A + C\right )} a^{5} b^{2} - 9 i \, B a^{4} b^{3} + 2 i \, {\left (18 \, A + C\right )} a^{3} b^{4} + 4 i \, B a^{2} b^{5} - 16 i \, A a b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-9 i \, B a^{5} b^{2} + 6 i \, {\left (4 \, A + C\right )} a^{4} b^{3} + 9 i \, B a^{3} b^{4} - 2 i \, {\left (18 \, A + C\right )} a^{2} b^{5} - 4 i \, B a b^{6} + 16 i \, A b^{7}\right )}\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 ) + {\left (\sqrt {2} {\left (9 i \, B a^{7} - 6 i \, {\left (4 \, A + C\right )} a^{6} b - 9 i \, B a^{5} b^{2} + 2 i \, {\left (18 \, A + C\right )} a^{4} b^{3} + 4 i \, B a^{3} b^{4} - 16 i \, A a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (-9 i \, B a^{6} b + 6 i \, {\left (4 \, A + C\right )} a^{5} b^{2} + 9 i \, B a^{4} b^{3} - 2 i \, {\left (18 \, A + C\right )} a^{3} b^{4} - 4 i \, B a^{2} b^{5} + 16 i \, A a b^{6}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (9 i \, B a^{5} b^{2} - 6 i \, {\left (4 \, A + C\right )} a^{4} b^{3} - 9 i \, B a^{3} b^{4} + 2 i \, {\left (18 \, A + C\right )} a^{2} b^{5} + 4 i \, B a b^{6} - 16 i \, A b^{7}\right )}\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 \, {\left (\sqrt {2} {\left (-3 i \, {\left (A - C\right )} a^{7} - 6 i \, B a^{6} b + i \, {\left (15 \, A + C\right )} a^{5} b^{2} + 2 i \, B a^{4} b^{3} - 8 i \, A a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-3 i \, {\left (A - C\right )} a^{6} b - 6 i \, B a^{5} b^{2} + i \, {\left (15 \, A + C\right )} a^{4} b^{3} + 2 i \, B a^{3} b^{4} - 8 i \, A a^{2} b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-3 i \, {\left (A - C\right )} a^{5} b^{2} - 6 i \, B a^{4} b^{3} + i \, {\left (15 \, A + C\right )} a^{3} b^{4} + 2 i \, B a^{2} b^{5} - 8 i \, A a b^{6}\right )}\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 \, {\left (\sqrt {2} {\left (3 i \, {\left (A - C\right )} a^{7} + 6 i \, B a^{6} b - i \, {\left (15 \, A + C\right )} a^{5} b^{2} - 2 i \, B a^{4} b^{3} + 8 i \, A a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (3 i \, {\left (A - C\right )} a^{6} b + 6 i \, B a^{5} b^{2} - i \, {\left (15 \, A + C\right )} a^{4} b^{3} - 2 i \, B a^{3} b^{4} + 8 i \, A a^{2} b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (3 i \, {\left (A - C\right )} a^{5} b^{2} + 6 i \, B a^{4} b^{3} - i \, {\left (15 \, A + C\right )} a^{3} b^{4} - 2 i \, B a^{2} b^{5} + 8 i \, A a b^{6}\right )}\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 )}{9 \, {\left ({\left (a^{10} - 2 \, a^{8} b^{2} + a^{6} b^{4}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{2} - 2 \, a^{6} b^{4} + a^{4} b^{6}\right )} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/9*(6*(2*C*a^6*b - 5*B*a^5*b^2 + 2*(4*A + C)*a^4*b^3 + B*a^3*b^4 - 4*A*a^2*b^5 + (3*C*a^7 - 6*B*a^6*b + (9*A
+ C)*a^5*b^2 + 2*B*a^4*b^3 - 5*A*a^3*b^4)*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)*(-9*I*B*a^7 + 6*I*(4*A + C)*a^6*b + 9*I*B*a^5*b^2 - 2*I*(18*A + C)*a^4*b^3 - 4*I*
B*a^3*b^4 + 16*I*A*a^2*b^5)*cos(d*x + c)^2 - 2*sqrt(2)*(9*I*B*a^6*b - 6*I*(4*A + C)*a^5*b^2 - 9*I*B*a^4*b^3 +
2*I*(18*A + C)*a^3*b^4 + 4*I*B*a^2*b^5 - 16*I*A*a*b^6)*cos(d*x + c) + sqrt(2)*(-9*I*B*a^5*b^2 + 6*I*(4*A + C)*
a^4*b^3 + 9*I*B*a^3*b^4 - 2*I*(18*A + C)*a^2*b^5 - 4*I*B*a*b^6 + 16*I*A*b^7))*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)*(9*I*B*a^7 - 6*I*(4*A + C)*a^6*b - 9*I*B*a^5*b^2 + 2*I*(18*A + C)*a^4*b^3 + 4*I*B*a^3*b^4 - 16*I*A*a^2*b^5
)*cos(d*x + c)^2 - 2*sqrt(2)*(-9*I*B*a^6*b + 6*I*(4*A + C)*a^5*b^2 + 9*I*B*a^4*b^3 - 2*I*(18*A + C)*a^3*b^4 -
4*I*B*a^2*b^5 + 16*I*A*a*b^6)*cos(d*x + c) + sqrt(2)*(9*I*B*a^5*b^2 - 6*I*(4*A + C)*a^4*b^3 - 9*I*B*a^3*b^4 +
2*I*(18*A + C)*a^2*b^5 + 4*I*B*a*b^6 - 16*I*A*b^7))*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)*(-3*I*(A - C)*a^7 -
6*I*B*a^6*b + I*(15*A + C)*a^5*b^2 + 2*I*B*a^4*b^3 - 8*I*A*a^3*b^4)*cos(d*x + c)^2 + 2*sqrt(2)*(-3*I*(A - C)*a
^6*b - 6*I*B*a^5*b^2 + I*(15*A + C)*a^4*b^3 + 2*I*B*a^3*b^4 - 8*I*A*a^2*b^5)*cos(d*x + c) + sqrt(2)*(-3*I*(A -
C)*a^5*b^2 - 6*I*B*a^4*b^3 + I*(15*A + C)*a^3*b^4 + 2*I*B*a^2*b^5 - 8*I*A*a*b^6))*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)*(3*I*(A - C)*a^7 + 6*I*B*a^6
*b - I*(15*A + C)*a^5*b^2 - 2*I*B*a^4*b^3 + 8*I*A*a^3*b^4)*cos(d*x + c)^2 + 2*sqrt(2)*(3*I*(A - C)*a^6*b + 6*I
*B*a^5*b^2 - I*(15*A + C)*a^4*b^3 - 2*I*B*a^3*b^4 + 8*I*A*a^2*b^5)*cos(d*x + c) + sqrt(2)*(3*I*(A - C)*a^5*b^2
+ 6*I*B*a^4*b^3 - I*(15*A + C)*a^3*b^4 - 2*I*B*a^2*b^5 + 8*I*A*a*b^6))*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^10 - 2*a^8*b^2 + a^6*b^4)*d*cos(d*x + c)^2 + 2*
(a^9*b - 2*a^7*b^3 + a^5*b^5)*d*cos(d*x + c) + (a^8*b^2 - 2*a^6*b^4 + a^4*b^6)*d)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)**2)*cos(d*x+c)**(1/2)/(a+b*sec(d*x+c))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3435 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)*cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sqrt(cos(d*x + c))/(b*sec(d*x + c) + a)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\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 (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + b/cos(c + d*x))^(5/2),x)

[Out]

int((cos(c + d*x)^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + b/cos(c + d*x))^(5/2), x)

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