\(\int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx\) [395]

Optimal result

Integrand size = 25, antiderivative size = 237 \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {35 \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{1024 \sqrt {2} a^{9/2} d}-\frac {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}}+\frac {853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}} \] Output:

35/2048*arctan(1/2*a^(1/2)*sin(d*x+c)*2^(1/2)/cos(d*x+c)^(1/2)/(a+a*cos(d* 
x+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)*2^(1/2)/a^(9/2)/d-1/8*sin(d 
*x+c)/d/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(5/2)-19/96*sin(d*x+c)/a/d/(a+a* 
cos(d*x+c))^(7/2)/sec(d*x+c)^(3/2)-187/768*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c 
))^(5/2)/sec(d*x+c)^(1/2)+853/3072*sin(d*x+c)/a^3/d/(a+a*cos(d*x+c))^(3/2) 
/sec(d*x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 6.11 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.67 \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \cos ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\frac {1}{1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}} \sqrt {1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )} \left (\frac {35}{128} \arcsin \left (\frac {\sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{\sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )}}\right )+\frac {93 \sin \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{128 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )}}-\frac {163 \sin ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{192 \cos ^2\left (\frac {1}{2} (c+d x)\right )^{3/2}}+\frac {25 \sin ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{48 \cos ^2\left (\frac {1}{2} (c+d x)\right )^{5/2}}-\frac {\sin ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{8 \cos ^2\left (\frac {1}{2} (c+d x)\right )^{7/2}}\right )}{d (a (1+\cos (c+d x)))^{9/2}} \] Input:

Integrate[1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(7/2)),x]
 

Output:

(2*Cos[c/2 + (d*x)/2]^9*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2 
*Sin[c/2 + (d*x)/2]^2]*((35*ArcSin[Sin[c/2 + (d*x)/2]/Sqrt[Cos[(c + d*x)/2 
]^2]])/128 + (93*Sin[c/2 + (d*x)/2]*Sqrt[1 - Sec[(c + d*x)/2]^2*Sin[c/2 + 
(d*x)/2]^2])/(128*Sqrt[Cos[(c + d*x)/2]^2]) - (163*Sin[c/2 + (d*x)/2]^3*Sq 
rt[1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2])/(192*(Cos[(c + d*x)/2]^2) 
^(3/2)) + (25*Sin[c/2 + (d*x)/2]^5*Sqrt[1 - Sec[(c + d*x)/2]^2*Sin[c/2 + ( 
d*x)/2]^2])/(48*(Cos[(c + d*x)/2]^2)^(5/2)) - (Sin[c/2 + (d*x)/2]^7*Sqrt[1 
 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2])/(8*(Cos[(c + d*x)/2]^2)^(7/2) 
)))/(d*(a*(1 + Cos[c + d*x]))^(9/2))
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.10, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 4710, 3042, 3244, 27, 3042, 3456, 27, 3042, 3456, 27, 3042, 3457, 27, 3042, 3261, 218}

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.

\(\displaystyle \int \frac {1}{\sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{9/2}}dx\)

\(\Big \downarrow \) 4710

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(\cos (c+d x) a+a)^{9/2}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{9/2}}dx\)

\(\Big \downarrow \) 3244

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) (5 a-14 a \cos (c+d x))}{2 (\cos (c+d x) a+a)^{7/2}}dx}{8 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) (5 a-14 a \cos (c+d x))}{(\cos (c+d x) a+a)^{7/2}}dx}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (5 a-14 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{7/2}}dx}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 3456

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\int \frac {\sqrt {\cos (c+d x)} \left (57 a^2-130 a^2 \cos (c+d x)\right )}{2 (\cos (c+d x) a+a)^{5/2}}dx}{6 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\int \frac {\sqrt {\cos (c+d x)} \left (57 a^2-130 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^{5/2}}dx}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (57 a^2-130 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}dx}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 3456

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {\int \frac {187 a^3-666 a^3 \cos (c+d x)}{2 \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^{3/2}}dx}{4 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {\int \frac {187 a^3-666 a^3 \cos (c+d x)}{\sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^{3/2}}dx}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {\int \frac {187 a^3-666 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 3457

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {\frac {\int -\frac {105 a^4}{2 \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{2 a^2}-\frac {853 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {-\frac {105}{4} a^2 \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx-\frac {853 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {-\frac {105}{4} a^2 \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {853 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 3261

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {\frac {105 a^3 \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{2 d}-\frac {853 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}+\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\frac {\frac {187 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac {-\frac {105 a^{3/2} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} d}-\frac {853 a^3 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}}{8 a^2}}{12 a^2}+\frac {19 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}}{16 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}\right )\)

Input:

Int[1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(7/2)),x]
 

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1/8*(Cos[c + d*x]^(5/2)*Sin[c + d* 
x])/(d*(a + a*Cos[c + d*x])^(9/2)) - ((19*a*Cos[c + d*x]^(3/2)*Sin[c + d*x 
])/(6*d*(a + a*Cos[c + d*x])^(7/2)) + ((187*a^2*Sqrt[Cos[c + d*x]]*Sin[c + 
 d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ((-105*a^(3/2)*ArcTan[(Sqrt[a]*S 
in[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(2*Sq 
rt[2]*d) - (853*a^3*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Cos[c + d 
*x])^(3/2)))/(8*a^2))/(12*a^2))/(16*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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 3042

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

rule 3244

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e 
+ f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[1/(a*b* 
(2*m + 1))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* 
Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) 
 + b*c*(m + n))*Sin[e + f*x], x], 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] && LtQ[m, -1] 
&& GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
 

rule 3261

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e 
_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f)   Subst[Int[1/(2*b^2 - (a*c 
 - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S 
in[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 3456

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[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( 
a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1))   Int[(a + b*Sin[e + f*x])^(m 
+ 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + 
 b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre 
eQ[{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] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In 
tegerQ[2*n] || EqQ[c, 0])
 

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 4710

Int[(csc[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Csc[a 
+ b*x])^m*(c*Sin[a + b*x])^m   Int[ActivateTrig[u]/(c*Sin[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u, x]
 

Maple [A] (verified)

Time = 4.41 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {a \left (\cos \left (d x +c \right )+1\right )}\, \left (\tan \left (d x +c \right ) \sec \left (d x +c \right )^{2} \left (853 \cos \left (d x +c \right )^{3}+819 \cos \left (d x +c \right )^{2}+455 \cos \left (d x +c \right )+105\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \left (-105 \cos \left (d x +c \right )-420-630 \sec \left (d x +c \right )-420 \sec \left (d x +c \right )^{2}-105 \sec \left (d x +c \right )^{3}\right )\right )}{6144 d \left (\cos \left (d x +c \right )^{5}+5 \cos \left (d x +c \right )^{4}+10 \cos \left (d x +c \right )^{3}+10 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )+1\right ) \sec \left (d x +c \right )^{\frac {7}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, a^{5}}\)	\(224\)

Input:

int(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/6144/d*2^(1/2)*(a*(cos(d*x+c)+1))^(1/2)/(cos(d*x+c)^5+5*cos(d*x+c)^4+10* 
cos(d*x+c)^3+10*cos(d*x+c)^2+5*cos(d*x+c)+1)/sec(d*x+c)^(7/2)/(cos(d*x+c)/ 
(cos(d*x+c)+1))^(1/2)*(tan(d*x+c)*sec(d*x+c)^2*(853*cos(d*x+c)^3+819*cos(d 
*x+c)^2+455*cos(d*x+c)+105)*2^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+arcs 
in(cot(d*x+c)-csc(d*x+c))*(-105*cos(d*x+c)-420-630*sec(d*x+c)-420*sec(d*x+ 
c)^2-105*sec(d*x+c)^3))/a^5
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {105 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - \frac {2 \, {\left (853 \, \cos \left (d x + c\right )^{4} + 819 \, \cos \left (d x + c\right )^{3} + 455 \, \cos \left (d x + c\right )^{2} + 105 \, \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{6144 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \] Input:

integrate(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(7/2),x, algorithm="fricas")
 

Output:

-1/6144*(105*sqrt(2)*(cos(d*x + c)^5 + 5*cos(d*x + c)^4 + 10*cos(d*x + c)^ 
3 + 10*cos(d*x + c)^2 + 5*cos(d*x + c) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt(a* 
cos(d*x + c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) - 2*(853*cos( 
d*x + c)^4 + 819*cos(d*x + c)^3 + 455*cos(d*x + c)^2 + 105*cos(d*x + c))*s 
qrt(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^5*d*cos(d*x + 
c)^5 + 5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d*x + c)^3 + 10*a^5*d*cos(d*x 
 + c)^2 + 5*a^5*d*cos(d*x + c) + a^5*d)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate(1/(a+a*cos(d*x+c))**(9/2)/sec(d*x+c)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {9}{2}} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(7/2),x, algorithm="maxima")
 

Output:

integrate(1/((a*cos(d*x + c) + a)^(9/2)*sec(d*x + c)^(7/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(7/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {1}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{9/2}} \,d x \] Input:

int(1/((1/cos(c + d*x))^(7/2)*(a + a*cos(c + d*x))^(9/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/((1/cos(c + d*x))^(7/2)*(a + a*cos(c + d*x))^(9/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )+1}}{\cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{4}+5 \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{4}+10 \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4}+10 \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4}+5 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4}+\sec \left (d x +c \right )^{4}}d x \right )}{a^{5}} \] Input:

int(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(7/2),x)
 

Output:

(sqrt(a)*int((sqrt(sec(c + d*x))*sqrt(cos(c + d*x) + 1))/(cos(c + d*x)**5* 
sec(c + d*x)**4 + 5*cos(c + d*x)**4*sec(c + d*x)**4 + 10*cos(c + d*x)**3*s 
ec(c + d*x)**4 + 10*cos(c + d*x)**2*sec(c + d*x)**4 + 5*cos(c + d*x)*sec(c 
 + d*x)**4 + sec(c + d*x)**4),x))/a**5