\(\int \frac {(a+a \cos (c+d x))^{3/2} (A+C \cos ^2(c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\) [187]

Optimal result

Integrand size = 37, antiderivative size = 266 \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \] Output:

2/231*a^2*(28*A+33*C)*sin(d*x+c)/d/cos(d*x+c)^(7/2)/(a+a*cos(d*x+c))^(1/2) 
+2/385*a^2*(112*A+143*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(1 
/2)+8/1155*a^2*(112*A+143*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c) 
)^(1/2)+16/1155*a^2*(112*A+143*C)*sin(d*x+c)/d/cos(d*x+c)^(1/2)/(a+a*cos(d 
*x+c))^(1/2)+2/33*a*A*(a+a*cos(d*x+c))^(1/2)*sin(d*x+c)/d/cos(d*x+c)^(9/2) 
+2/11*A*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(11/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.55 \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} (1652 A+1188 C+(4228 A+4147 C) \cos (c+d x)+2 (728 A+737 C) \cos (2 (c+d x))+1456 A \cos (3 (c+d x))+1859 C \cos (3 (c+d x))+224 A \cos (4 (c+d x))+286 C \cos (4 (c+d x))+224 A \cos (5 (c+d x))+286 C \cos (5 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{2310 d \cos ^{\frac {11}{2}}(c+d x)} \] Input:

Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x] 
^(13/2),x]
 

Output:

(a*Sqrt[a*(1 + Cos[c + d*x])]*(1652*A + 1188*C + (4228*A + 4147*C)*Cos[c + 
 d*x] + 2*(728*A + 737*C)*Cos[2*(c + d*x)] + 1456*A*Cos[3*(c + d*x)] + 185 
9*C*Cos[3*(c + d*x)] + 224*A*Cos[4*(c + d*x)] + 286*C*Cos[4*(c + d*x)] + 2 
24*A*Cos[5*(c + d*x)] + 286*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(2310*d* 
Cos[c + d*x]^(11/2))
 

Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {3042, 3523, 27, 3042, 3454, 27, 3042, 3459, 3042, 3251, 3042, 3251, 3042, 3250}

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.

\(\displaystyle \int \frac {(a \cos (c+d x)+a)^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\)

\(\Big \downarrow \) 3523

\(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^{3/2} (3 a A+a (6 A+11 C) \cos (c+d x))}{2 \cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^{3/2} (3 a A+a (6 A+11 C) \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (3 a A+a (6 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3454

\(\displaystyle \frac {\frac {2}{9} \int \frac {3 \sqrt {\cos (c+d x) a+a} \left ((28 A+33 C) a^2+3 (8 A+11 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} \int \frac {\sqrt {\cos (c+d x) a+a} \left ((28 A+33 C) a^2+3 (8 A+11 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a} \left ((28 A+33 C) a^2+3 (8 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3459

\(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{7} a^2 (112 A+143 C) \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a^3 (28 A+33 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{7} a^2 (112 A+143 C) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a^3 (28 A+33 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3251

\(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{7} a^2 (112 A+143 C) \left (\frac {4}{5} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (28 A+33 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{7} a^2 (112 A+143 C) \left (\frac {4}{5} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (28 A+33 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3251

\(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{7} a^2 (112 A+143 C) \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (28 A+33 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{7} a^2 (112 A+143 C) \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^3 (28 A+33 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3250

\(\displaystyle \frac {\frac {2 a^2 A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{3} \left (\frac {2 a^3 (28 A+33 C) \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {3}{7} a^2 (112 A+143 C) \left (\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4}{5} \left (\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4 a \sin (c+d x)}{3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )\right )\right )}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)}\)

Input:

Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2 
),x]
 

Output:

(2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + 
 ((2*a^2*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(9/2)) 
 + ((2*a^3*(28*A + 33*C)*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)*Sqrt[a + a* 
Cos[c + d*x]]) + (3*a^2*(112*A + 143*C)*((2*a*Sin[c + d*x])/(5*d*Cos[c + d 
*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*((2*a*Sin[c + d*x])/(3*d*Cos[c + 
d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*Sin[c + d*x])/(3*d*Sqrt[Cos[c 
+ d*x]]*Sqrt[a + a*Cos[c + d*x]])))/5))/7)/3)/(11*a)
 

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 3042

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

rule 3250

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(3/2), x_Symbol] :> Simp[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sq 
rt[a + b*Sin[e + f*x]]*Sqrt[c + d*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 3251

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e 
+ f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim 
p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2)))   Int[Sqrt[a + b*Sin[e 
+ f*x]]*(c + d*Sin[e + f*x])^(n + 1), 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[n, 
-1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
 

rule 3454

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

rule 3459

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( 
f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp 
[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) 
*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* 
c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d))   Int[Sqrt[a + b*Sin[e + f*x] 
]*(c + d*Sin[e + f*x])^(n + 1), 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] && LtQ[n, 
-1]
 

rule 3523

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

Maple [A] (verified)

Time = 4.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.53

method	result	size
default	\(\frac {2 a \sqrt {2}\, \sin \left (d x +c \right ) \left (\left (896 \cos \left (d x +c \right )^{5}+448 \cos \left (d x +c \right )^{4}+336 \cos \left (d x +c \right )^{3}+280 \cos \left (d x +c \right )^{2}+245 \cos \left (d x +c \right )+105\right ) A +\cos \left (d x +c \right )^{2} \left (1144 \cos \left (d x +c \right )^{3}+572 \cos \left (d x +c \right )^{2}+429 \cos \left (d x +c \right )+165\right ) C \right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{1155 d \cos \left (d x +c \right )^{\frac {11}{2}} \left (1+\cos \left (d x +c \right )\right )}\)	\(142\)
parts	\(\frac {2 A a \sqrt {2}\, \sin \left (d x +c \right ) \left (128 \cos \left (d x +c \right )^{5}+64 \cos \left (d x +c \right )^{4}+48 \cos \left (d x +c \right )^{3}+40 \cos \left (d x +c \right )^{2}+35 \cos \left (d x +c \right )+15\right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{165 d \cos \left (d x +c \right )^{\frac {11}{2}} \left (1+\cos \left (d x +c \right )\right )}+\frac {2 C a \sqrt {2}\, \sin \left (d x +c \right ) \left (104 \cos \left (d x +c \right )^{3}+52 \cos \left (d x +c \right )^{2}+39 \cos \left (d x +c \right )+15\right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{105 d \cos \left (d x +c \right )^{\frac {7}{2}} \left (1+\cos \left (d x +c \right )\right )}\)	\(180\)

Input:

int((a+a*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x,method=_ 
RETURNVERBOSE)
 

Output:

2/1155/d*a*2^(1/2)*sin(d*x+c)*((896*cos(d*x+c)^5+448*cos(d*x+c)^4+336*cos( 
d*x+c)^3+280*cos(d*x+c)^2+245*cos(d*x+c)+105)*A+cos(d*x+c)^2*(1144*cos(d*x 
+c)^3+572*cos(d*x+c)^2+429*cos(d*x+c)+165)*C)*(a*cos(1/2*d*x+1/2*c)^2)^(1/ 
2)/cos(d*x+c)^(11/2)/(1+cos(d*x+c))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.52 \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 \, {\left (8 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{5} + 4 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{3} + 5 \, {\left (56 \, A + 33 \, C\right )} a \cos \left (d x + c\right )^{2} + 245 \, A a \cos \left (d x + c\right ) + 105 \, A a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{1155 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}} \] Input:

integrate((a+a*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, a 
lgorithm="fricas")
 

Output:

2/1155*(8*(112*A + 143*C)*a*cos(d*x + c)^5 + 4*(112*A + 143*C)*a*cos(d*x + 
 c)^4 + 3*(112*A + 143*C)*a*cos(d*x + c)^3 + 5*(56*A + 33*C)*a*cos(d*x + c 
)^2 + 245*A*a*cos(d*x + c) + 105*A*a)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d* 
x + c))*sin(d*x + c)/(d*cos(d*x + c)^7 + d*cos(d*x + c)^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((a+a*cos(d*x+c))**(3/2)*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(13/2),x 
)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (230) = 460\).

Time = 0.20 (sec) , antiderivative size = 620, normalized size of antiderivative = 2.33 \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:

integrate((a+a*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, a 
lgorithm="maxima")
 

Output:

4/1155*(11*(105*sqrt(2)*a^(3/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 245*sqrt 
(2)*a^(3/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 273*sqrt(2)*a^(3/2)*sin( 
d*x + c)^5/(cos(d*x + c) + 1)^5 - 171*sqrt(2)*a^(3/2)*sin(d*x + c)^7/(cos( 
d*x + c) + 1)^7 + 38*sqrt(2)*a^(3/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)* 
C*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^3/((sin(d*x + c)/(cos(d*x + c) 
 + 1) + 1)^(9/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2)*(3*sin(d*x + 
 c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + sin(d 
*x + c)^6/(cos(d*x + c) + 1)^6 + 1)) + 7*(165*sqrt(2)*a^(3/2)*sin(d*x + c) 
/(cos(d*x + c) + 1) - 495*sqrt(2)*a^(3/2)*sin(d*x + c)^3/(cos(d*x + c) + 1 
)^3 + 1056*sqrt(2)*a^(3/2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1254*sqrt 
(2)*a^(3/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 781*sqrt(2)*a^(3/2)*sin( 
d*x + c)^9/(cos(d*x + c) + 1)^9 - 299*sqrt(2)*a^(3/2)*sin(d*x + c)^11/(cos 
(d*x + c) + 1)^11 + 46*sqrt(2)*a^(3/2)*sin(d*x + c)^13/(cos(d*x + c) + 1)^ 
13)*A*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^5/((sin(d*x + c)/(cos(d*x 
+ c) + 1) + 1)^(13/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(5*sin 
(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 
+ 10*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*sin(d*x + c)^8/(cos(d*x + c) 
+ 1)^8 + sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 1)))/d
 

Giac [F(-1)]

Timed out. \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((a+a*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, a 
lgorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 3.80 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.34 \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {\sqrt {a+a\,\cos \left (c+d\,x\right )}\,\left (-\frac {16\,C\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{3\,d}-\frac {16\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,A+23\,C\right )}{15\,d}+\frac {48\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\left (28\,A+27\,C\right )}{35\,d}+\frac {16\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,\left (112\,A+143\,C\right )}{105\,d}+\frac {32\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,\left (112\,A+143\,C\right )}{1155\,d}\right )}{20\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+20\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )+10\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )+10\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )+2\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )+2\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )} \] Input:

int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2))/cos(c + d*x)^(13/2 
),x)
 

Output:

((a + a*cos(c + d*x))^(1/2)*((48*a*exp((c*11i)/2 + (d*x*11i)/2)*sin((3*c)/ 
2 + (3*d*x)/2)*(28*A + 27*C))/(35*d) - (16*a*exp((c*11i)/2 + (d*x*11i)/2)* 
sin(c/2 + (d*x)/2)*(12*A + 23*C))/(15*d) - (16*C*a*exp((c*11i)/2 + (d*x*11 
i)/2)*sin((5*c)/2 + (5*d*x)/2))/(3*d) + (16*a*exp((c*11i)/2 + (d*x*11i)/2) 
*sin((7*c)/2 + (7*d*x)/2)*(112*A + 143*C))/(105*d) + (32*a*exp((c*11i)/2 + 
 (d*x*11i)/2)*sin((11*c)/2 + (11*d*x)/2)*(112*A + 143*C))/(1155*d)))/(20*c 
os(c + d*x)^(1/2)*exp((c*11i)/2 + (d*x*11i)/2)*cos(c/2 + (d*x)/2) + 20*cos 
(c + d*x)^(1/2)*exp((c*11i)/2 + (d*x*11i)/2)*cos((3*c)/2 + (3*d*x)/2) + 10 
*cos(c + d*x)^(1/2)*exp((c*11i)/2 + (d*x*11i)/2)*cos((5*c)/2 + (5*d*x)/2) 
+ 10*cos(c + d*x)^(1/2)*exp((c*11i)/2 + (d*x*11i)/2)*cos((7*c)/2 + (7*d*x) 
/2) + 2*cos(c + d*x)^(1/2)*exp((c*11i)/2 + (d*x*11i)/2)*cos((9*c)/2 + (9*d 
*x)/2) + 2*cos(c + d*x)^(1/2)*exp((c*11i)/2 + (d*x*11i)/2)*cos((11*c)/2 + 
(11*d*x)/2))
 

Reduce [F]

\[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\sqrt {a}\, a \left (\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{7}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) c \right ) \] Input:

int((a+a*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x)
 

Output:

sqrt(a)*a*(int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x)**7 
,x)*a + int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x)**6,x) 
*a + int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x)**5,x)*c 
+ int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x)))/cos(c + d*x)**4,x)*c)