\(\int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx\) [252]

Optimal result

Integrand size = 25, antiderivative size = 214 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {2 \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{7/2} d}-\frac {177 \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {\cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {17 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {49 \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^2 d (a+a \cos (c+d x))^{3/2}} \] Output:

2*arcsin(a^(1/2)*sin(d*x+c)/(a+a*cos(d*x+c))^(1/2))/a^(7/2)/d-177/128*arct 
an(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)) 
*2^(1/2)/a^(7/2)/d-1/6*cos(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(7/2 
)-17/48*cos(d*x+c)^(3/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^(5/2)-49/64*cos(d 
*x+c)^(1/2)*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^(3/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.62 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=-\frac {\sqrt {a (1+\cos (c+d x))} \left (1176 \arcsin \left (\sqrt {1-\cos (c+d x)}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )+4248 \arcsin \left (\sqrt {\cos (c+d x)}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )-2124 \sqrt {2} \arctan \left (\frac {\sqrt {\cos (c+d x)}}{\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )+362 \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)+247 \sqrt {1-\cos (c+d x)} \cos ^{\frac {5}{2}}(c+d x)+147 \sqrt {-((-1+\cos (c+d x)) \cos (c+d x))}\right ) \sin (c+d x)}{192 a^4 d \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^4} \] Input:

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

Output:

-1/192*(Sqrt[a*(1 + Cos[c + d*x])]*(1176*ArcSin[Sqrt[1 - Cos[c + d*x]]]*Co 
s[(c + d*x)/2]^6 + 4248*ArcSin[Sqrt[Cos[c + d*x]]]*Cos[(c + d*x)/2]^6 - 21 
24*Sqrt[2]*ArcTan[Sqrt[Cos[c + d*x]]/Sqrt[Sin[(c + d*x)/2]^2]]*Cos[(c + d* 
x)/2]^6 + 362*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2) + 247*Sqrt[1 - Cos 
[c + d*x]]*Cos[c + d*x]^(5/2) + 147*Sqrt[-((-1 + Cos[c + d*x])*Cos[c + d*x 
])])*Sin[c + d*x])/(a^4*d*Sqrt[1 - Cos[c + d*x]]*(1 + Cos[c + d*x])^4)
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3244, 27, 3042, 3456, 27, 3042, 3456, 27, 3042, 3461, 3042, 3253, 223, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3244

\(\displaystyle -\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) (5 a-12 a \cos (c+d x))}{2 (\cos (c+d x) a+a)^{5/2}}dx}{6 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) (5 a-12 a \cos (c+d x))}{(\cos (c+d x) a+a)^{5/2}}dx}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (5 a-12 a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}}dx}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 3456

\(\displaystyle -\frac {\frac {\int \frac {3 \sqrt {\cos (c+d x)} \left (17 a^2-32 a^2 \cos (c+d x)\right )}{2 (\cos (c+d x) a+a)^{3/2}}dx}{4 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {3 \int \frac {\sqrt {\cos (c+d x)} \left (17 a^2-32 a^2 \cos (c+d x)\right )}{(\cos (c+d x) a+a)^{3/2}}dx}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (17 a^2-32 a^2 \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}}dx}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 3456

\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {49 a^3-128 a^3 \cos (c+d x)}{2 \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{2 a^2}+\frac {49 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {49 a^3-128 a^3 \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{4 a^2}+\frac {49 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \left (\frac {\int \frac {49 a^3-128 a^3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}+\frac {49 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 3461

\(\displaystyle -\frac {\frac {3 \left (\frac {177 a^3 \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx-128 a^2 \int \frac {\sqrt {\cos (c+d x) a+a}}{\sqrt {\cos (c+d x)}}dx}{4 a^2}+\frac {49 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3 \left (\frac {177 a^3 \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-128 a^2 \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a^2}+\frac {49 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 3253

\(\displaystyle -\frac {\frac {3 \left (\frac {177 a^3 \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx+\frac {256 a^2 \int \frac {1}{\sqrt {1-\frac {a \sin ^2(c+d x)}{\cos (c+d x) a+a}}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}}{4 a^2}+\frac {49 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {\frac {3 \left (\frac {177 a^3 \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {256 a^{5/2} \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{4 a^2}+\frac {49 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 3261

\(\displaystyle -\frac {\frac {3 \left (\frac {-\frac {354 a^4 \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{d}-\frac {256 a^{5/2} \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{4 a^2}+\frac {49 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}\right )}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\frac {3 \left (\frac {49 a^2 \sin (c+d x) \sqrt {\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac {\frac {177 \sqrt {2} a^{5/2} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{d}-\frac {256 a^{5/2} \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{4 a^2}\right )}{8 a^2}+\frac {17 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}}{12 a^2}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}\)

Input:

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

Output:

-1/6*(Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^(7/2)) - (( 
17*a*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Cos[c + d*x])^(5/2)) + ( 
3*(((-256*a^(5/2)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]) 
/d + (177*Sqrt[2]*a^(5/2)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[ 
c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/d)/(4*a^2) + (49*a^2*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)
 

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[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 3253

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

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 3461

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + 
(f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim 
p[(A*b - a*B)/b   Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) 
, x], x] + Simp[B/b   Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[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 [A] (verified)

Time = 3.59 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.16

method	result	size
default	\(-\frac {\left (\left (-384 \cos \left (d x +c \right )^{3}-1152 \cos \left (d x +c \right )^{2}-1152 \cos \left (d x +c \right )-384\right ) \sqrt {2}\, \arctan \left (\tan \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+\sin \left (d x +c \right ) \left (247 \cos \left (d x +c \right )^{2}+362 \cos \left (d x +c \right )+147\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (-531 \cos \left (d x +c \right )^{3}-1593 \cos \left (d x +c \right )^{2}-1593 \cos \left (d x +c \right )-531\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{192 d \left (\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, a^{4}}\)	\(249\)

Input:

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

Output:

-1/192/d*((-384*cos(d*x+c)^3-1152*cos(d*x+c)^2-1152*cos(d*x+c)-384)*2^(1/2 
)*arctan(tan(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+sin(d*x+c)*(247*cos 
(d*x+c)^2+362*cos(d*x+c)+147)*2^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+(- 
531*cos(d*x+c)^3-1593*cos(d*x+c)^2-1593*cos(d*x+c)-531)*arcsin(cot(d*x+c)- 
csc(d*x+c)))*cos(d*x+c)^(1/2)*(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/(cos(d*x+c)^4 
+4*cos(d*x+c)^3+6*cos(d*x+c)^2+4*cos(d*x+c)+1)/(cos(d*x+c)/(cos(d*x+c)+1)) 
^(1/2)/a^4
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.44 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=-\frac {531 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, {\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) + 2 \, \sqrt {a \cos \left (d x + c\right ) + a} {\left (247 \, \cos \left (d x + c\right )^{2} + 362 \, \cos \left (d x + c\right ) + 147\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 768 \, {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )}\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \] Input:

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

Output:

-1/384*(531*sqrt(2)*(cos(d*x + c)^4 + 4*cos(d*x + c)^3 + 6*cos(d*x + c)^2 
+ 4*cos(d*x + c) + 1)*sqrt(a)*arctan(1/2*sqrt(2)*sqrt(a*cos(d*x + c) + a)* 
sqrt(a)*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 + a*cos(d*x + c) 
)) + 2*sqrt(a*cos(d*x + c) + a)*(247*cos(d*x + c)^2 + 362*cos(d*x + c) + 1 
47)*sqrt(cos(d*x + c))*sin(d*x + c) - 768*(cos(d*x + c)^4 + 4*cos(d*x + c) 
^3 + 6*cos(d*x + c)^2 + 4*cos(d*x + c) + 1)*sqrt(a)*arctan(sqrt(a*cos(d*x 
+ c) + a)*sqrt(a)*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 + a*co 
s(d*x + c))))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos 
(d*x + c)^2 + 4*a^4*d*cos(d*x + c) + a^4*d)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )+1}d x \right )}{a^{4}} \] Input:

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

Output:

(sqrt(a)*int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x)**3)/( 
cos(c + d*x)**4 + 4*cos(c + d*x)**3 + 6*cos(c + d*x)**2 + 4*cos(c + d*x) + 
 1),x))/a**4