3.5.71 \(\int \frac {A+B \sec (c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx\) [471]

Optimal. Leaf size=368 \[ -\frac {2 \left (9 a^2 A b-8 A b^3-3 a^3 B+2 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3 a^3 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A-15 a^2 A b^2+8 A b^4+6 a^3 b B-2 a b^3 B\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 b (A b-a B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (8 a^2 A b-4 A b^3-5 a^3 B+a b^2 B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.59, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4115, 4185, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {2 b (A b-a B) \sin (c+d x) \sqrt {\sec (c+d x)}}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac {2 b \left (-5 a^3 B+8 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{3 a^2 d \left (a^2-b^2\right )^2 \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (-3 a^3 B+9 a^2 A b+2 a b^2 B-8 A b^3\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 )}{3 a^3 d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^4 A+6 a^3 b B-15 a^2 A b^2-2 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 {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(9*a^2*A*b - 8*A*b^3 - 3*a^3*B + 2*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a
)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8
*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*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)]*Sqrt[Sec[c + d*x]]) + (2*b*(A*b - a*B)*Sqrt[Sec[c + d*x]]*Sin[c +
d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*Sqrt[Sec
[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*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 4115

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

Rubi steps

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

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Mathematica [A]
time = 2.49, size = 297, normalized size = 0.81 \begin {gather*} \frac {2 (b+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x) \left (-\frac {\left (\frac {b+a \cos (c+d x)}{a+b}\right )^{3/2} \left (-a^2 \left (-6 a^2 A b+2 A b^3+3 a^3 B+a b^2 B\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )-\left (3 a^4 A-15 a^2 A b^2+8 A b^4+6 a^3 b B-2 a b^3 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )-b F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}-\frac {a b \left (b \left (-8 a^2 A b+4 A b^3+5 a^3 B-a b^2 B\right )+a \left (-9 a^2 A b+5 A b^3+6 a^3 B-2 a b^2 B\right ) \cos (c+d x)\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2}\right )}{3 a^3 d (a+b \sec (c+d x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*(-((((b + a*Cos[c + d*x])/(a + b))^(3/2)*(-(a^2*(-6*a^2*A*b + 2*A*b
^3 + 3*a^3*B + a*b^2*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]) - (3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B
- 2*a*b^3*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])))/((a
- b)^2*(a + b))) - (a*b*(b*(-8*a^2*A*b + 4*A*b^3 + 5*a^3*B - a*b^2*B) + a*(-9*a^2*A*b + 5*A*b^3 + 6*a^3*B - 2*
a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*a^3*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. \(5168\) vs. \(2(398)=796\).
time = 24.29, size = 5169, normalized size = 14.05

method	result	size
default	\(\text {Expression too large to display}\)	\(5169\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.63, size = 1186, normalized size = 3.22 \begin {gather*} \frac {\sqrt {2} {\left (-9 i \, B a^{5} b^{2} + 24 i \, A a^{4} b^{3} + 9 i \, B a^{3} b^{4} - 36 i \, A a^{2} b^{5} - 4 i \, B a b^{6} + 16 i \, A b^{7} + {\left (-9 i \, B a^{7} + 24 i \, A a^{6} b + 9 i \, B a^{5} b^{2} - 36 i \, A 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 \, {\left (9 i \, B a^{6} b - 24 i \, A a^{5} b^{2} - 9 i \, B a^{4} b^{3} + 36 i \, A a^{3} b^{4} + 4 i \, B a^{2} b^{5} - 16 i \, A a b^{6}\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 ) + \sqrt {2} {\left (9 i \, B a^{5} b^{2} - 24 i \, A a^{4} b^{3} - 9 i \, B a^{3} b^{4} + 36 i \, A a^{2} b^{5} + 4 i \, B a b^{6} - 16 i \, A b^{7} + {\left (9 i \, B a^{7} - 24 i \, A a^{6} b - 9 i \, B a^{5} b^{2} + 36 i \, A 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 \, {\left (-9 i \, B a^{6} b + 24 i \, A a^{5} b^{2} + 9 i \, B a^{4} b^{3} - 36 i \, A a^{3} b^{4} - 4 i \, B a^{2} b^{5} + 16 i \, A a b^{6}\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 (-3 i \, A a^{5} b^{2} - 6 i \, B a^{4} b^{3} + 15 i \, A a^{3} b^{4} + 2 i \, B a^{2} b^{5} - 8 i \, A a b^{6} + {\left (-3 i \, A a^{7} - 6 i \, B a^{6} b + 15 i \, A 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 \, {\left (-3 i \, A a^{6} b - 6 i \, B a^{5} b^{2} + 15 i \, A 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 )\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 (3 i \, A a^{5} b^{2} + 6 i \, B a^{4} b^{3} - 15 i \, A a^{3} b^{4} - 2 i \, B a^{2} b^{5} + 8 i \, A a b^{6} + {\left (3 i \, A a^{7} + 6 i \, B a^{6} b - 15 i \, A 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 \, {\left (3 i \, A a^{6} b + 6 i \, B a^{5} b^{2} - 15 i \, A 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 )\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 ({\left (6 \, B a^{6} b - 9 \, A a^{5} b^{2} - 2 \, B a^{4} b^{3} + 5 \, A a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (5 \, B a^{5} b^{2} - 8 \, A a^{4} b^{3} - B a^{3} b^{4} + 4 \, A a^{2} b^{5}\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 )}}}{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))/(a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/9*(sqrt(2)*(-9*I*B*a^5*b^2 + 24*I*A*a^4*b^3 + 9*I*B*a^3*b^4 - 36*I*A*a^2*b^5 - 4*I*B*a*b^6 + 16*I*A*b^7 + (-
9*I*B*a^7 + 24*I*A*a^6*b + 9*I*B*a^5*b^2 - 36*I*A*a^4*b^3 - 4*I*B*a^3*b^4 + 16*I*A*a^2*b^5)*cos(d*x + c)^2 - 2
*(9*I*B*a^6*b - 24*I*A*a^5*b^2 - 9*I*B*a^4*b^3 + 36*I*A*a^3*b^4 + 4*I*B*a^2*b^5 - 16*I*A*a*b^6)*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) + sqrt(2)*(9*I*B*a^5*b^2 - 24*I*A*a^4*b^3 - 9*I*B*a^3*b^4 + 36*I*A*a^2*b^5 + 4*I*B*a*
b^6 - 16*I*A*b^7 + (9*I*B*a^7 - 24*I*A*a^6*b - 9*I*B*a^5*b^2 + 36*I*A*a^4*b^3 + 4*I*B*a^3*b^4 - 16*I*A*a^2*b^5
)*cos(d*x + c)^2 - 2*(-9*I*B*a^6*b + 24*I*A*a^5*b^2 + 9*I*B*a^4*b^3 - 36*I*A*a^3*b^4 - 4*I*B*a^2*b^5 + 16*I*A*
a*b^6)*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)*(-3*I*A*a^5*b^2 - 6*I*B*a^4*b^3 + 15*I*A*a^3*b^4 + 2
*I*B*a^2*b^5 - 8*I*A*a*b^6 + (-3*I*A*a^7 - 6*I*B*a^6*b + 15*I*A*a^5*b^2 + 2*I*B*a^4*b^3 - 8*I*A*a^3*b^4)*cos(d
*x + c)^2 + 2*(-3*I*A*a^6*b - 6*I*B*a^5*b^2 + 15*I*A*a^4*b^3 + 2*I*B*a^3*b^4 - 8*I*A*a^2*b^5)*cos(d*x + c))*sq
rt(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*a^5*b^2 + 6*I*B*a^4*b^3 - 15*I*A*a^3*b^4 - 2*I*B*a^2*b^5 + 8*I*A*a*b^6 + (3*I*A*a^7 + 6*I*B*a^6*b - 15*I*A*
a^5*b^2 - 2*I*B*a^4*b^3 + 8*I*A*a^3*b^4)*cos(d*x + c)^2 + 2*(3*I*A*a^6*b + 6*I*B*a^5*b^2 - 15*I*A*a^4*b^3 - 2*
I*B*a^3*b^4 + 8*I*A*a^2*b^5)*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*((6*B*a^6*b - 9*A*a^5*b^2 - 2*B*a^4*b^3 + 5*A*a^3*b^4)*cos(d*x + c)^2 + (5*B*
a^5*b^2 - 8*A*a^4*b^3 - B*a^3*b^4 + 4*A*a^2*b^5)*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^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*c
os(d*x + c) + (a^8*b^2 - 2*a^6*b^4 + a^4*b^6)*d)

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Sympy [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))/(a+b*sec(d*x+c))**(5/2)/sec(d*x+c)**(1/2),x)

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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