\(\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx\) [313]

Optimal result

Integrand size = 25, antiderivative size = 288 \[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {2 \left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (56 a A b+15 a^2 B+25 b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 (7 A b+5 a B) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 B (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \] Output:

2/105*(161*A*a^2*b+63*A*b^3+15*B*a^3+145*B*a*b^2)*(a+b*cos(d*x+c))^(1/2)*E 
llipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/b/d/((a+b*cos(d*x+c)) 
/(a+b))^(1/2)-2/105*(a^2-b^2)*(56*A*a*b+15*B*a^2+25*B*b^2)*((a+b*cos(d*x+c 
))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/b/d 
/(a+b*cos(d*x+c))^(1/2)+2/105*(56*A*a*b+15*B*a^2+25*B*b^2)*(a+b*cos(d*x+c) 
)^(1/2)*sin(d*x+c)/d+2/35*(7*A*b+5*B*a)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/ 
d+2/7*B*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 2.35 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.88 \[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx=\frac {2 b \left (105 a^3 A+119 a A b^2+135 a^2 b B+25 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+2 \left (161 a^2 A b+63 A b^3+15 a^3 B+145 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )+b (a+b \cos (c+d x)) \left (154 a A b+90 a^2 B+65 b^2 B+6 b (7 A b+15 a B) \cos (c+d x)+15 b^2 B \cos (2 (c+d x))\right ) \sin (c+d x)}{105 b d \sqrt {a+b \cos (c+d x)}} \] Input:

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

Output:

(2*b*(105*a^3*A + 119*a*A*b^2 + 135*a^2*b*B + 25*b^3*B)*Sqrt[(a + b*Cos[c 
+ d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*(161*a^2*A*b + 
63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a 
+ b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b 
)/(a + b)]) + b*(a + b*Cos[c + d*x])*(154*a*A*b + 90*a^2*B + 65*b^2*B + 6* 
b*(7*A*b + 15*a*B)*Cos[c + d*x] + 15*b^2*B*Cos[2*(c + d*x)])*Sin[c + d*x]) 
/(105*b*d*Sqrt[a + b*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3042, 3232, 27, 3042, 3232, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3232

\(\displaystyle \frac {2}{7} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} (7 a A+5 b B+(7 A b+5 a B) \cos (c+d x))dx+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \int (a+b \cos (c+d x))^{3/2} (7 a A+5 b B+(7 A b+5 a B) \cos (c+d x))dx+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (7 a A+5 b B+(7 A b+5 a B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (35 A a^2+40 b B a+21 A b^2+\left (15 B a^2+56 A b a+25 b^2 B\right ) \cos (c+d x)\right )dx+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {a+b \cos (c+d x)} \left (35 A a^2+40 b B a+21 A b^2+\left (15 B a^2+56 A b a+25 b^2 B\right ) \cos (c+d x)\right )dx+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (35 A a^2+40 b B a+21 A b^2+\left (15 B a^2+56 A b a+25 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {105 A a^3+135 b B a^2+119 A b^2 a+25 b^3 B+\left (15 B a^3+161 A b a^2+145 b^2 B a+63 A b^3\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {105 A a^3+135 b B a^2+119 A b^2 a+25 b^3 B+\left (15 B a^3+161 A b a^2+145 b^2 B a+63 A b^3\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {105 A a^3+135 b B a^2+119 A b^2 a+25 b^3 B+\left (15 B a^3+161 A b a^2+145 b^2 B a+63 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {2 \left (15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (15 a^2 B+56 a A b+25 b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}\right )\right )+\frac {2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}\)

Input:

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

Output:

(2*B*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + ((2*(7*A*b + 5*a*B)* 
(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (((2*(161*a^2*A*b + 63*A* 
b^3 + 15*a^3*B + 145*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x) 
/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*(a^2 - b 
^2)*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*El 
lipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]))/3 + ( 
2*(56*a*A*b + 15*a^2*B + 25*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/ 
(3*d))/5)/7
 

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 3132

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

rule 3134

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

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

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

rule 3231

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

rule 3232

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1304\) vs. \(2(273)=546\).

Time = 20.30 (sec) , antiderivative size = 1305, normalized size of antiderivative = 4.53

method	result	size
default	\(\text {Expression too large to display}\)	\(1305\)
parts	\(\text {Expression too large to display}\)	\(1491\)

Input:

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

Output:

-2/105*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*B* 
cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8*b^4+(-168*A*b^4-480*B*a*b^3-360*B* 
b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(392*A*a*b^3+168*A*b^4+360*B* 
a^2*b^2+480*B*a*b^3+280*B*b^4)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-1 
54*A*a^2*b^2-196*A*a*b^3-42*A*b^4-90*B*a^3*b-180*B*a^2*b^2-170*B*a*b^3-80* 
B*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-56*A*(sin(1/2*d*x+1/2*c)^2) 
^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1 
/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b+56*A*a*(sin(1/2*d*x+1/2*c)^2)^(1/2 
)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d* 
x+1/2*c),(-2*b/(a-b))^(1/2))*b^3+161*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/ 
(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c) 
,(-2*b/(a-b))^(1/2))*a^3*b-161*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)* 
sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b 
/(a-b))^(1/2))*a^2*b^2+63*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1 
/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b 
))^(1/2))*a*b^3-63*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+ 
1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2 
))*b^4-15*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+ 
(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4-10 
*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.95 \[ \int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx =\text {Too large to display} \] Input:

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

Output:

-2/315*(sqrt(1/2)*(-30*I*B*a^4 - 7*I*A*a^3*b + 115*I*B*a^2*b^2 + 231*I*A*a 
*b^3 + 75*I*B*b^4)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8 
/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2* 
a)/b) + sqrt(1/2)*(30*I*B*a^4 + 7*I*A*a^3*b - 115*I*B*a^2*b^2 - 231*I*A*a* 
b^3 - 75*I*B*b^4)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/ 
27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a 
)/b) + 3*sqrt(1/2)*(-15*I*B*a^3*b - 161*I*A*a^2*b^2 - 145*I*B*a*b^3 - 63*I 
*A*b^4)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9* 
a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9* 
a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) + 3*sqrt 
(1/2)*(15*I*B*a^3*b + 161*I*A*a^2*b^2 + 145*I*B*a*b^3 + 63*I*A*b^4)*sqrt(b 
)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, we 
ierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/ 
3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) - 3*(15*B*b^4*cos(d*x 
+ c)^2 + 45*B*a^2*b^2 + 77*A*a*b^3 + 25*B*b^4 + 3*(15*B*a*b^3 + 7*A*b^4)*c 
os(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^2*d)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(cos(c + d*x)*b + a),x)*a**3 + 3*int(sqrt(cos(c + d*x)*b + a)*cos( 
c + d*x),x)*a**2*b + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*b**3 
+ 3*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2,x)*a*b**2