\(\int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [508]

Optimal result

Integrand size = 43, antiderivative size = 261 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{5/2} (283 A+326 B+400 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{128 d}+\frac {a^3 (283 A+326 B+400 C) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (787 A+950 B+1040 C) \cos (c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (79 A+110 B+80 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {a (A+2 B) \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d} \] Output:

1/128*a^(5/2)*(283*A+326*B+400*C)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c 
))^(1/2))/d+1/128*a^3*(283*A+326*B+400*C)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1 
/2)+1/960*a^3*(787*A+950*B+1040*C)*cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c) 
)^(1/2)+1/240*a^2*(79*A+110*B+80*C)*cos(d*x+c)^2*(a+a*sec(d*x+c))^(1/2)*si 
n(d*x+c)/d+1/8*a*(A+2*B)*cos(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)*sin(d*x+c)/d+ 
1/5*A*cos(d*x+c)^4*(a+a*sec(d*x+c))^(5/2)*sin(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.89 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.84 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {a^2 \cos (c+d x) \sqrt {a (1+\sec (c+d x))} \left (126 B \cos ^2(c+d x) \sin (c+d x)+420 C \cos ^2(c+d x) \sin (c+d x)+90 A \cos ^3(c+d x) \sin (c+d x)+288 B \cos ^3(c+d x) \sin (c+d x)+220 A \cos ^4(c+d x) \sin (c+d x)+105 C \sin (2 (c+d x))-3150 C \operatorname {Hypergeometric2F1}\left (\frac {1}{2},4,\frac {3}{2},1-\sec (c+d x)\right ) \tan (c+d x)-2934 B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},5,\frac {3}{2},1-\sec (c+d x)\right ) \tan (c+d x)-2830 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},6,\frac {3}{2},1-\sec (c+d x)\right ) \tan (c+d x)\right )}{315 d (1+\cos (c+d x))} \] Input:

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

Output:

-1/315*(a^2*Cos[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(126*B*Cos[c + d*x]^2* 
Sin[c + d*x] + 420*C*Cos[c + d*x]^2*Sin[c + d*x] + 90*A*Cos[c + d*x]^3*Sin 
[c + d*x] + 288*B*Cos[c + d*x]^3*Sin[c + d*x] + 220*A*Cos[c + d*x]^4*Sin[c 
 + d*x] + 105*C*Sin[2*(c + d*x)] - 3150*C*Hypergeometric2F1[1/2, 4, 3/2, 1 
 - Sec[c + d*x]]*Tan[c + d*x] - 2934*B*Hypergeometric2F1[1/2, 5, 3/2, 1 - 
Sec[c + d*x]]*Tan[c + d*x] - 2830*A*Hypergeometric2F1[1/2, 6, 3/2, 1 - Sec 
[c + d*x]]*Tan[c + d*x]))/(d*(1 + Cos[c + d*x]))
 

Rubi [A] (verified)

Time = 1.75 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.372, Rules used = {3042, 4574, 27, 3042, 4505, 27, 3042, 4505, 27, 3042, 4503, 3042, 4292, 3042, 4261, 216}

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.

\(\displaystyle \int \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4574

\(\displaystyle \frac {\int \frac {1}{2} \cos ^4(c+d x) (\sec (c+d x) a+a)^{5/2} (5 a (A+2 B)+a (3 A+10 C) \sec (c+d x))dx}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \cos ^4(c+d x) (\sec (c+d x) a+a)^{5/2} (5 a (A+2 B)+a (3 A+10 C) \sec (c+d x))dx}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (5 a (A+2 B)+a (3 A+10 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 4505

\(\displaystyle \frac {\frac {1}{4} \int \frac {1}{2} \cos ^3(c+d x) (\sec (c+d x) a+a)^{3/2} \left ((79 A+110 B+80 C) a^2+(39 A+30 B+80 C) \sec (c+d x) a^2\right )dx+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} \int \cos ^3(c+d x) (\sec (c+d x) a+a)^{3/2} \left ((79 A+110 B+80 C) a^2+(39 A+30 B+80 C) \sec (c+d x) a^2\right )dx+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{8} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left ((79 A+110 B+80 C) a^2+(39 A+30 B+80 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 4505

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{3} \int \frac {1}{2} \cos ^2(c+d x) \sqrt {\sec (c+d x) a+a} \left ((787 A+950 B+1040 C) a^3+3 (157 A+170 B+240 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \int \cos ^2(c+d x) \sqrt {\sec (c+d x) a+a} \left ((787 A+950 B+1040 C) a^3+3 (157 A+170 B+240 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a} \left ((787 A+950 B+1040 C) a^3+3 (157 A+170 B+240 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^3 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 4503

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {15}{4} a^3 (283 A+326 B+400 C) \int \cos (c+d x) \sqrt {\sec (c+d x) a+a}dx+\frac {a^4 (787 A+950 B+1040 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {15}{4} a^3 (283 A+326 B+400 C) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^4 (787 A+950 B+1040 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 4292

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {15}{4} a^3 (283 A+326 B+400 C) \left (\frac {1}{2} \int \sqrt {\sec (c+d x) a+a}dx+\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (787 A+950 B+1040 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {15}{4} a^3 (283 A+326 B+400 C) \left (\frac {1}{2} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (787 A+950 B+1040 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 4261

\(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (\frac {15}{4} a^3 (283 A+326 B+400 C) \left (\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {a \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{\sec (c+d x) a+a}+a}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}\right )+\frac {a^4 (787 A+950 B+1040 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {5 a^2 (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}+\frac {1}{8} \left (\frac {a^3 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}+\frac {1}{6} \left (\frac {a^4 (787 A+950 B+1040 C) \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}+\frac {15}{4} a^3 (283 A+326 B+400 C) \left (\frac {\sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )\right )\right )}{10 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}\)

Input:

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

Output:

(A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d) + ((5*a^2 
*(A + 2*B)*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + 
 ((a^3*(79*A + 110*B + 80*C)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c 
 + d*x])/(3*d) + ((a^4*(787*A + 950*B + 1040*C)*Cos[c + d*x]*Sin[c + d*x]) 
/(2*d*Sqrt[a + a*Sec[c + d*x]]) + (15*a^3*(283*A + 326*B + 400*C)*((Sqrt[a 
]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*Sin[c + 
d*x])/(d*Sqrt[a + a*Sec[c + d*x]])))/4)/6)/8)/(10*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 216

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

rule 3042

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

rule 4261

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

rule 4292

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) 
 + (a_)], x_Symbol] :> Simp[a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n*Sqrt[a 
+ b*Csc[e + f*x]])), x] + Simp[a*((2*n + 1)/(2*b*d*n))   Int[Sqrt[a + b*Csc 
[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && 
 EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)] && IntegerQ[2*n]
 

rule 4503

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

rule 4505

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

rule 4574

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*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e 
 + f*x])^n/(f*n)), x] - Simp[1/(b*d*n)   Int[(a + b*Csc[e + f*x])^m*(d*Csc[ 
e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] 
, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] 
&&  !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
 

Maple [A] (warning: unable to verify)

Time = 1.46 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.62

\[\frac {a^{2} \left (\left (4245 \cos \left (d x +c \right )+4245\right ) A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{\sqrt {\csc \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot \left (d x +c \right )^{2}-1}}\right )+\left (4890 \cos \left (d x +c \right )+4890\right ) B \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{\sqrt {\csc \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot \left (d x +c \right )^{2}-1}}\right )+\left (6000 \cos \left (d x +c \right )+6000\right ) C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{\sqrt {\csc \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot \left (d x +c \right )^{2}-1}}\right )+\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (384 \cos \left (d x +c \right )^{4}+1392 \cos \left (d x +c \right )^{3}+2264 \cos \left (d x +c \right )^{2}+2830 \cos \left (d x +c \right )+4245\right ) A +\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (480 \cos \left (d x +c \right )^{3}+1840 \cos \left (d x +c \right )^{2}+3260 \cos \left (d x +c \right )+4890\right ) B +\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (640 \cos \left (d x +c \right )^{2}+2720 \cos \left (d x +c \right )+6000\right ) C \right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{1920 d \left (\cos \left (d x +c \right )+1\right )}\]

Input:

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

Output:

1/1920/d*a^2*((4245*cos(d*x+c)+4245)*A*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)* 
arctanh(2^(1/2)/(csc(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c)+cot(d*x+c)^2-1)^(1/2 
)*(csc(d*x+c)-cot(d*x+c)))+(4890*cos(d*x+c)+4890)*B*(-cos(d*x+c)/(cos(d*x+ 
c)+1))^(1/2)*arctanh(2^(1/2)/(csc(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c)+cot(d*x 
+c)^2-1)^(1/2)*(csc(d*x+c)-cot(d*x+c)))+(6000*cos(d*x+c)+6000)*C*(-cos(d*x 
+c)/(cos(d*x+c)+1))^(1/2)*arctanh(2^(1/2)/(csc(d*x+c)^2-2*csc(d*x+c)*cot(d 
*x+c)+cot(d*x+c)^2-1)^(1/2)*(csc(d*x+c)-cot(d*x+c)))+sin(d*x+c)*cos(d*x+c) 
*(384*cos(d*x+c)^4+1392*cos(d*x+c)^3+2264*cos(d*x+c)^2+2830*cos(d*x+c)+424 
5)*A+sin(d*x+c)*cos(d*x+c)*(480*cos(d*x+c)^3+1840*cos(d*x+c)^2+3260*cos(d* 
x+c)+4890)*B+sin(d*x+c)*cos(d*x+c)*(640*cos(d*x+c)^2+2720*cos(d*x+c)+6000) 
*C)*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)
 

Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.88 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left [\frac {15 \, {\left ({\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (384 \, A a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (29 \, A + 10 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (283 \, A + 230 \, B + 80 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (283 \, A + 326 \, B + 272 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3840 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {15 \, {\left ({\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (384 \, A a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (29 \, A + 10 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (283 \, A + 230 \, B + 80 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (283 \, A + 326 \, B + 272 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{1920 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \] Input:

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

Output:

[1/3840*(15*((283*A + 326*B + 400*C)*a^2*cos(d*x + c) + (283*A + 326*B + 4 
00*C)*a^2)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + 
 c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(co 
s(d*x + c) + 1)) + 2*(384*A*a^2*cos(d*x + c)^5 + 48*(29*A + 10*B)*a^2*cos( 
d*x + c)^4 + 8*(283*A + 230*B + 80*C)*a^2*cos(d*x + c)^3 + 10*(283*A + 326 
*B + 272*C)*a^2*cos(d*x + c)^2 + 15*(283*A + 326*B + 400*C)*a^2*cos(d*x + 
c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) 
+ d), -1/1920*(15*((283*A + 326*B + 400*C)*a^2*cos(d*x + c) + (283*A + 326 
*B + 400*C)*a^2)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*co 
s(d*x + c)/(sqrt(a)*sin(d*x + c))) - (384*A*a^2*cos(d*x + c)^5 + 48*(29*A 
+ 10*B)*a^2*cos(d*x + c)^4 + 8*(283*A + 230*B + 80*C)*a^2*cos(d*x + c)^3 + 
 10*(283*A + 326*B + 272*C)*a^2*cos(d*x + c)^2 + 15*(283*A + 326*B + 400*C 
)*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/ 
(d*cos(d*x + c) + d)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1965 vs. \(2 (233) = 466\).

Time = 1.49 (sec) , antiderivative size = 1965, normalized size of antiderivative = 7.53 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3840*(15*(283*A*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 326*B*sqrt(-a)*a^2*sgn 
(cos(d*x + c)) + 400*C*sqrt(-a)*a^2*sgn(cos(d*x + c)))*log(abs((sqrt(-a)*t 
an(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a*(2*sqrt(2 
) + 3))) - 15*(283*A*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 326*B*sqrt(-a)*a^2*s 
gn(cos(d*x + c)) + 400*C*sqrt(-a)*a^2*sgn(cos(d*x + c)))*log(abs((sqrt(-a) 
*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 + a*(2*sqrt 
(2) - 3))) + 4*(4245*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan( 
1/2*d*x + 1/2*c)^2 + a))^18*A*sqrt(-a)*a^3*sgn(cos(d*x + c)) + 4890*sqrt(2 
)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^18 
*B*sqrt(-a)*a^3*sgn(cos(d*x + c)) + 6000*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1 
/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^18*C*sqrt(-a)*a^3*sgn(cos(d*x 
 + c)) - 114615*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d 
*x + 1/2*c)^2 + a))^16*A*sqrt(-a)*a^4*sgn(cos(d*x + c)) - 132030*sqrt(2)*( 
sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^16*B* 
sqrt(-a)*a^4*sgn(cos(d*x + c)) - 162000*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/ 
2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^16*C*sqrt(-a)*a^4*sgn(cos(d*x 
+ c)) + 1298820*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d 
*x + 1/2*c)^2 + a))^14*A*sqrt(-a)*a^5*sgn(cos(d*x + c)) + 1319880*sqrt(2)* 
(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^14*B 
*sqrt(-a)*a^5*sgn(cos(d*x + c)) + 1801920*sqrt(2)*(sqrt(-a)*tan(1/2*d*x...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*a**2*(int(sqrt(sec(c + d*x) + 1)*cos(c + d*x)**5*sec(c + d*x)**4,x 
)*c + int(sqrt(sec(c + d*x) + 1)*cos(c + d*x)**5*sec(c + d*x)**3,x)*b + 2* 
int(sqrt(sec(c + d*x) + 1)*cos(c + d*x)**5*sec(c + d*x)**3,x)*c + int(sqrt 
(sec(c + d*x) + 1)*cos(c + d*x)**5*sec(c + d*x)**2,x)*a + 2*int(sqrt(sec(c 
 + d*x) + 1)*cos(c + d*x)**5*sec(c + d*x)**2,x)*b + int(sqrt(sec(c + d*x) 
+ 1)*cos(c + d*x)**5*sec(c + d*x)**2,x)*c + 2*int(sqrt(sec(c + d*x) + 1)*c 
os(c + d*x)**5*sec(c + d*x),x)*a + int(sqrt(sec(c + d*x) + 1)*cos(c + d*x) 
**5*sec(c + d*x),x)*b + int(sqrt(sec(c + d*x) + 1)*cos(c + d*x)**5,x)*a)