\(\int \frac {A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx\) [120]

Optimal result

Integrand size = 17, antiderivative size = 120 \[ \int \frac {A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx=\frac {2 B \text {arctanh}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {a+a \cos (x)}}\right )}{a^{5/2}}+\frac {(3 A-43 B) \text {arctanh}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {a+a \cos (x)}}\right )}{16 \sqrt {2} a^{5/2}}+\frac {(A-B) \sin (x)}{4 (a+a \cos (x))^{5/2}}+\frac {(3 A-11 B) \sin (x)}{16 a (a+a \cos (x))^{3/2}} \] Output:

2*B*arctanh(a^(1/2)*sin(x)/(a+a*cos(x))^(1/2))/a^(5/2)+1/32*(3*A-43*B)*arc 
tanh(1/2*a^(1/2)*sin(x)*2^(1/2)/(a+a*cos(x))^(1/2))*2^(1/2)/a^(5/2)+1/4*(A 
-B)*sin(x)/(a+a*cos(x))^(5/2)+1/16*(3*A-11*B)*sin(x)/a/(a+a*cos(x))^(3/2)
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79 \[ \int \frac {A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx=\frac {2 (3 A-43 B) \text {arctanh}\left (\sin \left (\frac {x}{2}\right )\right ) \cos ^3\left (\frac {x}{2}\right )+64 \sqrt {2} B \text {arctanh}\left (\sqrt {2} \sin \left (\frac {x}{2}\right )\right ) \cos ^3\left (\frac {x}{2}\right )+(7 A-15 B+3 A \cos (x)-11 B \cos (x)) \tan \left (\frac {x}{2}\right )}{16 a (a (1+\cos (x)))^{3/2}} \] Input:

Integrate[(A + B*Sec[x])/(a + a*Cos[x])^(5/2),x]
 

Output:

(2*(3*A - 43*B)*ArcTanh[Sin[x/2]]*Cos[x/2]^3 + 64*Sqrt[2]*B*ArcTanh[Sqrt[2 
]*Sin[x/2]]*Cos[x/2]^3 + (7*A - 15*B + 3*A*Cos[x] - 11*B*Cos[x])*Tan[x/2]) 
/(16*a*(a*(1 + Cos[x]))^(3/2))
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.882, Rules used = {3042, 3307, 3042, 3457, 27, 3042, 3457, 27, 3042, 3464, 3042, 3128, 219, 3252, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B \sec (x)}{(a \cos (x)+a)^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \csc \left (x+\frac {\pi }{2}\right )}{\left (a \sin \left (x+\frac {\pi }{2}\right )+a\right )^{5/2}}dx\)

\(\Big \downarrow \) 3307

\(\displaystyle \int \frac {\sec (x) (A \cos (x)+B)}{(a \cos (x)+a)^{5/2}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A \sin \left (x+\frac {\pi }{2}\right )+B}{\sin \left (x+\frac {\pi }{2}\right ) \left (a \sin \left (x+\frac {\pi }{2}\right )+a\right )^{5/2}}dx\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {\int \frac {(8 a B+3 a (A-B) \cos (x)) \sec (x)}{2 (\cos (x) a+a)^{3/2}}dx}{4 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(8 a B+3 a (A-B) \cos (x)) \sec (x)}{(\cos (x) a+a)^{3/2}}dx}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {8 a B+3 a (A-B) \sin \left (x+\frac {\pi }{2}\right )}{\sin \left (x+\frac {\pi }{2}\right ) \left (\sin \left (x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {\frac {\int \frac {\left (32 B a^2+(3 A-11 B) \cos (x) a^2\right ) \sec (x)}{2 \sqrt {\cos (x) a+a}}dx}{2 a^2}+\frac {a (3 A-11 B) \sin (x)}{2 (a \cos (x)+a)^{3/2}}}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (32 B a^2+(3 A-11 B) \cos (x) a^2\right ) \sec (x)}{\sqrt {\cos (x) a+a}}dx}{4 a^2}+\frac {a (3 A-11 B) \sin (x)}{2 (a \cos (x)+a)^{3/2}}}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {32 B a^2+(3 A-11 B) \sin \left (x+\frac {\pi }{2}\right ) a^2}{\sin \left (x+\frac {\pi }{2}\right ) \sqrt {\sin \left (x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}+\frac {a (3 A-11 B) \sin (x)}{2 (a \cos (x)+a)^{3/2}}}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 3464

\(\displaystyle \frac {\frac {a^2 (3 A-43 B) \int \frac {1}{\sqrt {\cos (x) a+a}}dx+32 a B \int \sqrt {\cos (x) a+a} \sec (x)dx}{4 a^2}+\frac {a (3 A-11 B) \sin (x)}{2 (a \cos (x)+a)^{3/2}}}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 (3 A-43 B) \int \frac {1}{\sqrt {\sin \left (x+\frac {\pi }{2}\right ) a+a}}dx+32 a B \int \frac {\sqrt {\sin \left (x+\frac {\pi }{2}\right ) a+a}}{\sin \left (x+\frac {\pi }{2}\right )}dx}{4 a^2}+\frac {a (3 A-11 B) \sin (x)}{2 (a \cos (x)+a)^{3/2}}}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 3128

\(\displaystyle \frac {\frac {32 a B \int \frac {\sqrt {\sin \left (x+\frac {\pi }{2}\right ) a+a}}{\sin \left (x+\frac {\pi }{2}\right )}dx-2 a^2 (3 A-43 B) \int \frac {1}{2 a-\frac {a^2 \sin ^2(x)}{\cos (x) a+a}}d\left (-\frac {a \sin (x)}{\sqrt {\cos (x) a+a}}\right )}{4 a^2}+\frac {a (3 A-11 B) \sin (x)}{2 (a \cos (x)+a)^{3/2}}}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {32 a B \int \frac {\sqrt {\sin \left (x+\frac {\pi }{2}\right ) a+a}}{\sin \left (x+\frac {\pi }{2}\right )}dx+\sqrt {2} a^{3/2} (3 A-43 B) \text {arctanh}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {a \cos (x)+a}}\right )}{4 a^2}+\frac {a (3 A-11 B) \sin (x)}{2 (a \cos (x)+a)^{3/2}}}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 3252

\(\displaystyle \frac {\frac {\sqrt {2} a^{3/2} (3 A-43 B) \text {arctanh}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {a \cos (x)+a}}\right )-64 a^2 B \int \frac {1}{a-\frac {a^2 \sin ^2(x)}{\cos (x) a+a}}d\left (-\frac {a \sin (x)}{\sqrt {\cos (x) a+a}}\right )}{4 a^2}+\frac {a (3 A-11 B) \sin (x)}{2 (a \cos (x)+a)^{3/2}}}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\sqrt {2} a^{3/2} (3 A-43 B) \text {arctanh}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {2} \sqrt {a \cos (x)+a}}\right )+64 a^{3/2} B \text {arctanh}\left (\frac {\sqrt {a} \sin (x)}{\sqrt {a \cos (x)+a}}\right )}{4 a^2}+\frac {a (3 A-11 B) \sin (x)}{2 (a \cos (x)+a)^{3/2}}}{8 a^2}+\frac {(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}\)

Input:

Int[(A + B*Sec[x])/(a + a*Cos[x])^(5/2),x]
 

Output:

((A - B)*Sin[x])/(4*(a + a*Cos[x])^(5/2)) + ((64*a^(3/2)*B*ArcTanh[(Sqrt[a 
]*Sin[x])/Sqrt[a + a*Cos[x]]] + Sqrt[2]*a^(3/2)*(3*A - 43*B)*ArcTanh[(Sqrt 
[a]*Sin[x])/(Sqrt[2]*Sqrt[a + a*Cos[x]])])/(4*a^2) + (a*(3*A - 11*B)*Sin[x 
])/(2*(a + a*Cos[x])^(3/2)))/(8*a^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3128

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

rule 3252

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f)   Subst[Int[1/(b*c + a*d - d*x^2), 
x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, 
 e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3307

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_.)*((a_) + (b_.)*sin[(e_.) + 
(f_.)*(x_)])^(m_.), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*((d + c*Sin[e + 
 f*x])^n/Sin[e + f*x]^n), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IntegerQ 
[n]
 

rule 3457

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( 
n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) 
  Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b 
*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 
2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ 
[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] 
 &&  !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

rule 3464

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + 
(f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A 
*b - a*B)/(b*c - a*d)   Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Simp[(B*c 
- A*d)/(b*c - a*d)   Int[Sqrt[a + b*Sin[e + f*x]]/(c + d*Sin[e + f*x]), x], 
 x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - 
 b^2, 0] && NeQ[c^2 - d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(321\) vs. \(2(95)=190\).

Time = 0.34 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.68

method	result	size
default	\(\frac {\sqrt {a \sin \left (\frac {x}{2}\right )^{2}}\, \left (3 A \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}+4 a}{\cos \left (\frac {x}{2}\right )}\right ) \cos \left (\frac {x}{2}\right )^{4} a -43 B \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}+4 a}{\cos \left (\frac {x}{2}\right )}\right ) a \cos \left (\frac {x}{2}\right )^{4}+32 B \ln \left (-\frac {4 \left (a \sqrt {2}\, \cos \left (\frac {x}{2}\right )-\sqrt {a}\, \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}-2 a \right )}{2 \cos \left (\frac {x}{2}\right )-\sqrt {2}}\right ) a \cos \left (\frac {x}{2}\right )^{4}+32 B \ln \left (\frac {4 a \sqrt {2}\, \cos \left (\frac {x}{2}\right )+4 \sqrt {a}\, \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}+8 a}{2 \cos \left (\frac {x}{2}\right )+\sqrt {2}}\right ) a \cos \left (\frac {x}{2}\right )^{4}+3 A \sqrt {2}\, \cos \left (\frac {x}{2}\right )^{2} \sqrt {a}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}-11 B \sqrt {a}\, \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}\, \cos \left (\frac {x}{2}\right )^{2}+2 A \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}\, \sqrt {a}-2 B \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}\, \sqrt {a}\right )}{32 a^{\frac {7}{2}} \cos \left (\frac {x}{2}\right )^{3} \sin \left (\frac {x}{2}\right ) \sqrt {a \cos \left (\frac {x}{2}\right )^{2}}}\)	\(322\)
parts	\(\frac {A \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}\, \left (3 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}+4 a}{\cos \left (\frac {x}{2}\right )}\right ) a \cos \left (\frac {x}{2}\right )^{4}+3 \sqrt {a}\, \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}\, \cos \left (\frac {x}{2}\right )^{2}+2 \sqrt {a}\, \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}\right )}{32 a^{\frac {7}{2}} \cos \left (\frac {x}{2}\right )^{3} \sin \left (\frac {x}{2}\right ) \sqrt {a \cos \left (\frac {x}{2}\right )^{2}}}-\frac {B \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}\, \left (43 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}+4 a}{\cos \left (\frac {x}{2}\right )}\right ) a \cos \left (\frac {x}{2}\right )^{4}-32 \ln \left (-\frac {4 \left (a \sqrt {2}\, \cos \left (\frac {x}{2}\right )-\sqrt {a}\, \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}-2 a \right )}{2 \cos \left (\frac {x}{2}\right )-\sqrt {2}}\right ) a \cos \left (\frac {x}{2}\right )^{4}-32 \ln \left (\frac {4 a \sqrt {2}\, \cos \left (\frac {x}{2}\right )+4 \sqrt {a}\, \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}+8 a}{2 \cos \left (\frac {x}{2}\right )+\sqrt {2}}\right ) a \cos \left (\frac {x}{2}\right )^{4}+11 \sqrt {a}\, \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}\, \cos \left (\frac {x}{2}\right )^{2}+2 \sqrt {a}\, \sqrt {2}\, \sqrt {a \sin \left (\frac {x}{2}\right )^{2}}\right )}{32 a^{\frac {7}{2}} \cos \left (\frac {x}{2}\right )^{3} \sin \left (\frac {x}{2}\right ) \sqrt {a \cos \left (\frac {x}{2}\right )^{2}}}\)	\(355\)

Input:

int((A+B*sec(x))/(a+a*cos(x))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/32/a^(7/2)/cos(1/2*x)^3*(a*sin(1/2*x)^2)^(1/2)*(3*A*2^(1/2)*ln(2*(2*a^(1 
/2)*(a*sin(1/2*x)^2)^(1/2)+2*a)/cos(1/2*x))*cos(1/2*x)^4*a-43*B*2^(1/2)*ln 
(2*(2*a^(1/2)*(a*sin(1/2*x)^2)^(1/2)+2*a)/cos(1/2*x))*a*cos(1/2*x)^4+32*B* 
ln(-4/(2*cos(1/2*x)-2^(1/2))*(a*2^(1/2)*cos(1/2*x)-a^(1/2)*2^(1/2)*(a*sin( 
1/2*x)^2)^(1/2)-2*a))*a*cos(1/2*x)^4+32*B*ln(4/(2*cos(1/2*x)+2^(1/2))*(a*2 
^(1/2)*cos(1/2*x)+a^(1/2)*2^(1/2)*(a*sin(1/2*x)^2)^(1/2)+2*a))*a*cos(1/2*x 
)^4+3*A*2^(1/2)*cos(1/2*x)^2*a^(1/2)*(a*sin(1/2*x)^2)^(1/2)-11*B*a^(1/2)*2 
^(1/2)*(a*sin(1/2*x)^2)^(1/2)*cos(1/2*x)^2+2*A*2^(1/2)*(a*sin(1/2*x)^2)^(1 
/2)*a^(1/2)-2*B*2^(1/2)*(a*sin(1/2*x)^2)^(1/2)*a^(1/2))/sin(1/2*x)/(a*cos( 
1/2*x)^2)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (95) = 190\).

Time = 0.09 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.95 \[ \int \frac {A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx=-\frac {\sqrt {2} {\left ({\left (3 \, A - 43 \, B\right )} \cos \left (x\right )^{3} + 3 \, {\left (3 \, A - 43 \, B\right )} \cos \left (x\right )^{2} + 3 \, {\left (3 \, A - 43 \, B\right )} \cos \left (x\right ) + 3 \, A - 43 \, B\right )} \sqrt {a} \log \left (-\frac {a \cos \left (x\right )^{2} + 2 \, \sqrt {2} \sqrt {a \cos \left (x\right ) + a} \sqrt {a} \sin \left (x\right ) - 2 \, a \cos \left (x\right ) - 3 \, a}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1}\right ) - 32 \, {\left (B \cos \left (x\right )^{3} + 3 \, B \cos \left (x\right )^{2} + 3 \, B \cos \left (x\right ) + B\right )} \sqrt {a} \log \left (\frac {a \cos \left (x\right )^{3} - 7 \, a \cos \left (x\right )^{2} - 4 \, \sqrt {a \cos \left (x\right ) + a} \sqrt {a} {\left (\cos \left (x\right ) - 2\right )} \sin \left (x\right ) + 8 \, a}{\cos \left (x\right )^{3} + \cos \left (x\right )^{2}}\right ) - 4 \, {\left ({\left (3 \, A - 11 \, B\right )} \cos \left (x\right ) + 7 \, A - 15 \, B\right )} \sqrt {a \cos \left (x\right ) + a} \sin \left (x\right )}{64 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} + 3 \, a^{3} \cos \left (x\right ) + a^{3}\right )}} \] Input:

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

Output:

-1/64*(sqrt(2)*((3*A - 43*B)*cos(x)^3 + 3*(3*A - 43*B)*cos(x)^2 + 3*(3*A - 
 43*B)*cos(x) + 3*A - 43*B)*sqrt(a)*log(-(a*cos(x)^2 + 2*sqrt(2)*sqrt(a*co 
s(x) + a)*sqrt(a)*sin(x) - 2*a*cos(x) - 3*a)/(cos(x)^2 + 2*cos(x) + 1)) - 
32*(B*cos(x)^3 + 3*B*cos(x)^2 + 3*B*cos(x) + B)*sqrt(a)*log((a*cos(x)^3 - 
7*a*cos(x)^2 - 4*sqrt(a*cos(x) + a)*sqrt(a)*(cos(x) - 2)*sin(x) + 8*a)/(co 
s(x)^3 + cos(x)^2)) - 4*((3*A - 11*B)*cos(x) + 7*A - 15*B)*sqrt(a*cos(x) + 
 a)*sin(x))/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 + 3*a^3*cos(x) + a^3)
 

Sympy [F]

\[ \int \frac {A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx=\int \frac {A + B \sec {\left (x \right )}}{\left (a \left (\cos {\left (x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((A+B*sec(x))/(a+a*cos(x))**(5/2),x)
 

Output:

Integral((A + B*sec(x))/(a*(cos(x) + 1))**(5/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50224 vs. \(2 (95) = 190\).

Time = 13.74 (sec) , antiderivative size = 50224, normalized size of antiderivative = 418.53 \[ \int \frac {A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/32*(512*((2*sin(2*x) + sin(x))*cos(5/2*x) + cos(5/2*x)*sin(4*x) + 2*cos( 
5/2*x)*sin(3*x) + (2*cos(2*x) + cos(x))*sin(5/2*x) + cos(4*x)*sin(5/2*x) + 
 2*cos(3*x)*sin(5/2*x))*cos(5*x)^2 + 2560*(5*(2*sin(2*x) + sin(x))*cos(5/2 
*x) + cos(5/2*x)*sin(5*x) + 5*cos(5/2*x)*sin(4*x) + 10*cos(5/2*x)*sin(3*x) 
 - (10*cos(2*x) + 5*cos(x) + 1)*sin(5/2*x) - cos(5*x)*sin(5/2*x) - 5*cos(4 
*x)*sin(5/2*x) - 10*cos(3*x)*sin(5/2*x))*cos(8/5*arctan2(sin(5/2*x), cos(5 
/2*x)))^2 + 10240*(5*(2*sin(2*x) + sin(x))*cos(5/2*x) + cos(5/2*x)*sin(5*x 
) + 5*cos(5/2*x)*sin(4*x) + 10*cos(5/2*x)*sin(3*x) - (10*cos(2*x) + 5*cos( 
x) + 1)*sin(5/2*x) - cos(5*x)*sin(5/2*x) - 5*cos(4*x)*sin(5/2*x) - 10*cos( 
3*x)*sin(5/2*x))*cos(6/5*arctan2(sin(5/2*x), cos(5/2*x)))^2 + 10240*(5*(2* 
sin(2*x) + sin(x))*cos(5/2*x) + cos(5/2*x)*sin(5*x) + 5*cos(5/2*x)*sin(4*x 
) + 10*cos(5/2*x)*sin(3*x) - (10*cos(2*x) + 5*cos(x) + 1)*sin(5/2*x) - cos 
(5*x)*sin(5/2*x) - 5*cos(4*x)*sin(5/2*x) - 10*cos(3*x)*sin(5/2*x))*cos(4/5 
*arctan2(sin(5/2*x), cos(5/2*x)))^2 + 2560*(5*(2*sin(2*x) + sin(x))*cos(5/ 
2*x) + cos(5/2*x)*sin(5*x) + 5*cos(5/2*x)*sin(4*x) + 10*cos(5/2*x)*sin(3*x 
) - (10*cos(2*x) + 5*cos(x) + 1)*sin(5/2*x) - cos(5*x)*sin(5/2*x) - 5*cos( 
4*x)*sin(5/2*x) - 10*cos(3*x)*sin(5/2*x))*cos(2/5*arctan2(sin(5/2*x), cos( 
5/2*x)))^2 - 512*((2*sin(2*x) + sin(x))*cos(5/2*x) + cos(5/2*x)*sin(4*x) + 
 2*cos(5/2*x)*sin(3*x) + (2*cos(2*x) + cos(x))*sin(5/2*x) + cos(4*x)*sin(5 
/2*x) + 2*cos(3*x)*sin(5/2*x))*sin(5*x)^2 + 2560*cos(4*x)^2*sin(5/2*x) ...
 

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx=\int \frac {A+\frac {B}{\cos \left (x\right )}}{{\left (a+a\,\cos \left (x\right )\right )}^{5/2}} \,d x \] Input:

int((A + B/cos(x))/(a + a*cos(x))^(5/2),x)
 

Output:

int((A + B/cos(x))/(a + a*cos(x))^(5/2), x)
 

Reduce [F]

\[ \int \frac {A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\cos \left (x \right )+1}}{\cos \left (x \right )^{3}+3 \cos \left (x \right )^{2}+3 \cos \left (x \right )+1}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (x \right )+1}\, \sec \left (x \right )}{\cos \left (x \right )^{3}+3 \cos \left (x \right )^{2}+3 \cos \left (x \right )+1}d x \right ) b \right )}{a^{3}} \] Input:

int((A+B*sec(x))/(a+a*cos(x))^(5/2),x)
 

Output:

(sqrt(a)*(int(sqrt(cos(x) + 1)/(cos(x)**3 + 3*cos(x)**2 + 3*cos(x) + 1),x) 
*a + int((sqrt(cos(x) + 1)*sec(x))/(cos(x)**3 + 3*cos(x)**2 + 3*cos(x) + 1 
),x)*b))/a**3