3.7.68 \(\int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx\) [668]

Optimal. Leaf size=474 \[ -\frac {2 b \left (17 a^4+116 a^2 b^2-128 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \]

[Out]

2/3*b^2*sin(d*x+c)/a/(a^2-b^2)/d/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c))^(3/2)+8/3*b^2*(3*a^2-2*b^2)*sin(d*x+c)/a^2/
(a^2-b^2)^2/d/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c))^(1/2)-2/15*b*(17*a^4+116*a^2*b^2-128*b^4)*(cos(1/2*d*x+1/2*c)^
2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/
2)*sec(d*x+c)^(1/2)/a^5/(a^2-b^2)/d/(a+b*sec(d*x+c))^(1/2)+2/15*(3*a^4-71*a^2*b^2+48*b^4)*sin(d*x+c)*(a+b*sec(
d*x+c))^(1/2)/a^3/(a^2-b^2)^2/d/sec(d*x+c)^(3/2)-4/15*b*(7*a^4-49*a^2*b^2+32*b^4)*sin(d*x+c)*(a+b*sec(d*x+c))^
(1/2)/a^4/(a^2-b^2)^2/d/sec(d*x+c)^(1/2)+2/15*(9*a^6+55*a^4*b^2-212*a^2*b^4+128*b^6)*(cos(1/2*d*x+1/2*c)^2)^(1
/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/a^5/(a^2-b
^2)^2/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.91, antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3932, 4185, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{3 a^2 d \left (a^2-b^2\right )^2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{15 a^4 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)}}-\frac {2 b \left (17 a^4+116 a^2 b^2-128 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{15 a^3 d \left (a^2-b^2\right )^2 \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]

[Out]

(-2*b*(17*a^4 + 116*a^2*b^2 - 128*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)
]*Sqrt[Sec[c + d*x]])/(15*a^5*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(9*a^6 + 55*a^4*b^2 - 212*a^2*b^4 +
128*b^6)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*
Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b
*Sec[c + d*x])^(3/2)) + (8*b^2*(3*a^2 - 2*b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)*Sqrt[a
+ b*Sec[c + d*x]]) + (2*(3*a^4 - 71*a^2*b^2 + 48*b^4)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^
2)^2*d*Sec[c + d*x]^(3/2)) - (4*b*(7*a^4 - 49*a^2*b^2 + 32*b^4)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4
*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])

Rule 2732

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

Rule 2734

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

Rule 2740

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

Rule 2742

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

Rule 3932

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[b^2*Co
t[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)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*(a^2*(m + 1) - b^2*(m + n + 1) - a*b*(m + 1
)*Csc[e + f*x] + b^2*(m + n + 2)*Csc[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0]
&& LtQ[m, -1] && IntegersQ[2*m, 2*n]

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 4189

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

Rubi steps

\begin {align*} \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx &=\frac {2 b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \int \frac {-\frac {3 a^2}{2}+4 b^2+\frac {3}{2} a b \sec (c+d x)-3 b^2 \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {2 b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {4 \int \frac {\frac {1}{4} \left (3 a^4-71 a^2 b^2+48 b^4\right )-\frac {1}{2} a b \left (3 a^2-b^2\right ) \sec (c+d x)+4 b^2 \left (3 a^2-2 b^2\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {2 b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {8 \int \frac {\frac {3}{4} b \left (7 a^4-49 a^2 b^2+32 b^4\right )-\frac {1}{8} a \left (9 a^4+27 a^2 b^2-16 b^4\right ) \sec (c+d x)-\frac {1}{4} b \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{15 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {2 b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {16 \int \frac {\frac {3}{16} \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right )-\frac {3}{4} a b \left (2 a^4+11 a^2 b^2-8 b^4\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{45 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {2 b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}-\frac {\left (b \left (17 a^4+116 a^2 b^2-128 b^4\right )\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )}+\frac {\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )^2}\\ &=\frac {2 b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}-\frac {\left (b \left (17 a^4+116 a^2 b^2-128 b^4\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}{15 a^5 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{15 a^5 \left (a^2-b^2\right )^2 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=\frac {2 b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}-\frac {\left (b \left (17 a^4+116 a^2 b^2-128 b^4\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}{15 a^5 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{15 a^5 \left (a^2-b^2\right )^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}\\ &=-\frac {2 b \left (17 a^4+116 a^2 b^2-128 b^4\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)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {8 b^2 \left (3 a^2-2 b^2\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4-71 a^2 b^2+48 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {4 b \left (7 a^4-49 a^2 b^2+32 b^4\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 1.36, size = 292, normalized size = 0.62 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x) \left (\frac {2 \left (\frac {b+a \cos (c+d x)}{a+b}\right )^{3/2} \left (\left (9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )+b \left (-17 a^5+17 a^4 b-116 a^3 b^2+116 a^2 b^3+128 a b^4-128 b^5\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )\right )}{(a-b)^2}+a \left (\frac {10 b^5 \sin (c+d x)}{-a^2+b^2}-\frac {10 b^4 \left (-15 a^2+11 b^2\right ) (b+a \cos (c+d x)) \sin (c+d x)}{\left (a^2-b^2\right )^2}-28 b (b+a \cos (c+d x))^2 \sin (c+d x)+3 a (b+a \cos (c+d x))^2 \sin (2 (c+d x))\right )\right )}{15 a^5 d (a+b \sec (c+d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]

[Out]

((b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*((2*((b + a*Cos[c + d*x])/(a + b))^(3/2)*((9*a^6 + 55*a^4*b^2 - 212*a
^2*b^4 + 128*b^6)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + b*(-17*a^5 + 17*a^4*b - 116*a^3*b^2 + 116*a^2*b^3 +
128*a*b^4 - 128*b^5)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]))/(a - b)^2 + a*((10*b^5*Sin[c + d*x])/(-a^2 + b^2)
- (10*b^4*(-15*a^2 + 11*b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2 - 28*b*(b + a*Cos[c + d*x])^2*S
in[c + d*x] + 3*a*(b + a*Cos[c + d*x])^2*Sin[2*(c + d*x)])))/(15*a^5*d*(a + b*Sec[c + d*x])^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4585\) vs. \(2(492)=984\).
time = 0.31, size = 4586, normalized size = 9.68

method	result	size
default	\(\text {Expression too large to display}\)	\(4586\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15/d*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(-72*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c
)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^2*sin(d*x+
c)*a^5*b^2-116*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c)
)*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^2*sin(d*x+c)*a^4*b^3+96*((b+a*cos(d*x+c))/(1
+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-
(a+b)/(a-b))^(1/2))*cos(d*x+c)^2*sin(d*x+c)*a^3*b^4+128*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+co
s(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^2*s
in(d*x+c)*a^2*b^5+55*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(
d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^2*sin(d*x+c)*a^5*b^2-212*((b+a*cos(d*x
+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*
x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^2*sin(d*x+c)*a^3*b^4+128*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(
1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x
+c)^2*sin(d*x+c)*a*b^6-26*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1
+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^6*b-89*((b+a*cos(d*x
+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*
x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^5*b^2-188*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/
(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c
)*sin(d*x+c)*a^4*b^3-20*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+c
os(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^3*b^4+224*((b+a*cos(d*
x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d
*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^2*b^5+128*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1
/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+
c)*sin(d*x+c)*a*b^6+9*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos
(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^6*b+55*((b+a*cos(d*x+c))
/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c)
,(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^5*b^2+55*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+co
s(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*sin
(d*x+c)*a^4*b^3-212*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d
*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^3*b^4-212*((b+a*cos(d*x+c)
)/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c
),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*sin(d*x+c)*a^2*b^5+128*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+
cos(d*x+c)))^(1/2)*EllipticE((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)*s
in(d*x+c)*a*b^6-17*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*
x+c))*((a-b)/(a+b))^(1/2)/sin(d*x+c),(-(a+b)/(a-b))^(1/2))*cos(d*x+c)^2*sin(d*x+c)*a^6*b-48*((a-b)/(a+b))^(1/2
)*cos(d*x+c)^3*a^2*b^5+128*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(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^7*sin(d*x+c)-9*((a-b)/(a+b))^(1/2)*cos(d*
x+c)^2*a^7-128*((a-b)/(a+b))^(1/2)*cos(d*x+c)*b^7+3*((a-b)/(a+b))^(1/2)*cos(d*x+c)^5*a^7+6*((a-b)/(a+b))^(1/2)
*cos(d*x+c)^3*a^7-9*((a-b)/(a+b))^(1/2)*a^5*b^2+5*((a-b)/(a+b))^(1/2)*a^4*b^3-50*((a-b)/(a+b))^(1/2)*a^3*b^4-1
48*((a-b)/(a+b))^(1/2)*a^2*b^5+64*((a-b)/(a+b))^(1/2)*a*b^6+128*((a-b)/(a+b))^(1/2)*b^7-9*((b+a*cos(d*x+c))/(1
+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(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^6*b*sin(d*x+c)-17*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(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^5*b^2*sin(d*x+c)-72*((b+a*co
s(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))*((a-b)/(a+b))^(1/2)/s
in(d*x+c),(-(a+b)/(a-b))^(1/2))*a^4*b^3*sin(d*x+c)-116*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^(1/2)*(1/(1+cos
(d*x+c)))^(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^4*sin(d*x
+c)+96*((b+a*cos(d*x+c))/(1+cos(d*x+c))/(a+b))^...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^(5/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.79, size = 1036, normalized size = 2.19 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (-21 i \, a^{6} b^{3} - 121 i \, a^{4} b^{5} + 260 i \, a^{2} b^{7} - 128 i \, b^{9} + {\left (-21 i \, a^{8} b - 121 i \, a^{6} b^{3} + 260 i \, a^{4} b^{5} - 128 i \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (-21 i \, a^{7} b^{2} - 121 i \, a^{5} b^{4} + 260 i \, a^{3} b^{6} - 128 i \, a b^{8}\right )} \cos \left (d x + c\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 ) + 2 \, \sqrt {2} {\left (21 i \, a^{6} b^{3} + 121 i \, a^{4} b^{5} - 260 i \, a^{2} b^{7} + 128 i \, b^{9} + {\left (21 i \, a^{8} b + 121 i \, a^{6} b^{3} - 260 i \, a^{4} b^{5} + 128 i \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (21 i \, a^{7} b^{2} + 121 i \, a^{5} b^{4} - 260 i \, a^{3} b^{6} + 128 i \, a b^{8}\right )} \cos \left (d x + c\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 \, \sqrt {2} {\left (-9 i \, a^{7} b^{2} - 55 i \, a^{5} b^{4} + 212 i \, a^{3} b^{6} - 128 i \, a b^{8} + {\left (-9 i \, a^{9} - 55 i \, a^{7} b^{2} + 212 i \, a^{5} b^{4} - 128 i \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (-9 i \, a^{8} b - 55 i \, a^{6} b^{3} + 212 i \, a^{4} b^{5} - 128 i \, a^{2} b^{7}\right )} \cos \left (d x + c\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 \, \sqrt {2} {\left (9 i \, a^{7} b^{2} + 55 i \, a^{5} b^{4} - 212 i \, a^{3} b^{6} + 128 i \, a b^{8} + {\left (9 i \, a^{9} + 55 i \, a^{7} b^{2} - 212 i \, a^{5} b^{4} + 128 i \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (9 i \, a^{8} b + 55 i \, a^{6} b^{3} - 212 i \, a^{4} b^{5} + 128 i \, a^{2} b^{7}\right )} \cos \left (d x + c\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 ) - \frac {6 \, {\left (3 \, {\left (a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \cos \left (d x + c\right )^{4} - 8 \, {\left (a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \cos \left (d x + c\right )^{3} - 5 \, {\left (5 \, a^{7} b^{2} - 25 \, a^{5} b^{4} + 16 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (7 \, a^{6} b^{3} - 49 \, a^{4} b^{5} + 32 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{45 \, {\left ({\left (a^{12} - 2 \, a^{10} b^{2} + a^{8} b^{4}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{11} b - 2 \, a^{9} b^{3} + a^{7} b^{5}\right )} d \cos \left (d x + c\right ) + {\left (a^{10} b^{2} - 2 \, a^{8} b^{4} + a^{6} b^{6}\right )} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/45*(2*sqrt(2)*(-21*I*a^6*b^3 - 121*I*a^4*b^5 + 260*I*a^2*b^7 - 128*I*b^9 + (-21*I*a^8*b - 121*I*a^6*b^3 + 2
60*I*a^4*b^5 - 128*I*a^2*b^7)*cos(d*x + c)^2 + 2*(-21*I*a^7*b^2 - 121*I*a^5*b^4 + 260*I*a^3*b^6 - 128*I*a*b^8)
*cos(d*x + c))*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) + 2*sqrt(2)*(21*I*a^6*b^3 + 121*I*a^4*b^5 - 260*I*a^2*b^7 + 128*I*b^9
+ (21*I*a^8*b + 121*I*a^6*b^3 - 260*I*a^4*b^5 + 128*I*a^2*b^7)*cos(d*x + c)^2 + 2*(21*I*a^7*b^2 + 121*I*a^5*b^
4 - 260*I*a^3*b^6 + 128*I*a*b^8)*cos(d*x + c))*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)*(-9*I*a^7*b^2 - 55*I*a^5*b
^4 + 212*I*a^3*b^6 - 128*I*a*b^8 + (-9*I*a^9 - 55*I*a^7*b^2 + 212*I*a^5*b^4 - 128*I*a^3*b^6)*cos(d*x + c)^2 +
2*(-9*I*a^8*b - 55*I*a^6*b^3 + 212*I*a^4*b^5 - 128*I*a^2*b^7)*cos(d*x + c))*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)*(9*I*a^7*b^2 + 55*I*a^5*b^4 - 212*I*
a^3*b^6 + 128*I*a*b^8 + (9*I*a^9 + 55*I*a^7*b^2 - 212*I*a^5*b^4 + 128*I*a^3*b^6)*cos(d*x + c)^2 + 2*(9*I*a^8*b
+ 55*I*a^6*b^3 - 212*I*a^4*b^5 + 128*I*a^2*b^7)*cos(d*x + c))*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^
2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(
3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a)) - 6*(3*(a^9 - 2*a^7*b^2 + a^5*b^4)*cos(d*x + c)^4 - 8*(a^8*b
- 2*a^6*b^3 + a^4*b^5)*cos(d*x + c)^3 - 5*(5*a^7*b^2 - 25*a^5*b^4 + 16*a^3*b^6)*cos(d*x + c)^2 - 2*(7*a^6*b^3
- 49*a^4*b^5 + 32*a^2*b^7)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c
)))/((a^12 - 2*a^10*b^2 + a^8*b^4)*d*cos(d*x + c)^2 + 2*(a^11*b - 2*a^9*b^3 + a^7*b^5)*d*cos(d*x + c) + (a^10*
b^2 - 2*a^8*b^4 + a^6*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(1/sec(d*x+c)**(5/2)/(a+b*sec(d*x+c))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6191 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(1/sec(d*x+c)^(5/2)/(a+b*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^(5/2)), x)

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

[Out]

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

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