\(\int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \, dx\) [486]

Optimal result

Integrand size = 23, antiderivative size = 264 \[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=\frac {2 a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (8 a^4+17 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^2+25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^2 d}-\frac {8 a (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b^2 d}+\frac {2 \cos (c+d x) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{7 b d} \] Output:

2/105*a*(8*a^2+19*b^2)*(a+b*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c) 
,2^(1/2)*(b/(a+b))^(1/2))/b^3/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/105*(8*a^ 
4+17*a^2*b^2-25*b^4)*((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^3/d/(a+b*cos(d*x+c))^(1/2)+2/105*(8*a^2 
+25*b^2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^2/d-8/35*a*(a+b*cos(d*x+c))^( 
3/2)*sin(d*x+c)/b^2/d+2/7*cos(d*x+c)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 1.48 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.81 \[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=\frac {4 a \left (8 a^3+8 a^2 b+19 a b^2+19 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-4 \left (8 a^4+17 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+b \left (-16 a^3+136 a b^2+\left (-4 a^2 b+145 b^3\right ) \cos (c+d x)+36 a b^2 \cos (2 (c+d x))+15 b^3 \cos (3 (c+d x))\right ) \sin (c+d x)}{210 b^3 d \sqrt {a+b \cos (c+d x)}} \] Input:

Integrate[Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]],x]
 

Output:

(4*a*(8*a^3 + 8*a^2*b + 19*a*b^2 + 19*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + 
b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 4*(8*a^4 + 17*a^2*b^2 - 25*b^4 
)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] 
 + b*(-16*a^3 + 136*a*b^2 + (-4*a^2*b + 145*b^3)*Cos[c + d*x] + 36*a*b^2*C 
os[2*(c + d*x)] + 15*b^3*Cos[3*(c + d*x)])*Sin[c + d*x])/(210*b^3*d*Sqrt[a 
 + b*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 3272, 27, 3042, 3502, 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.

\(\displaystyle \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3272

\(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (-4 a \cos ^2(c+d x)+5 b \cos (c+d x)+2 a\right )dx}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {2 \int -\frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (2 a b-\left (8 a^2+25 b^2\right ) \cos (c+d x)\right )dx}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \sqrt {a+b \cos (c+d x)} \left (2 a b-\left (8 a^2+25 b^2\right ) \cos (c+d x)\right )dx}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (2 a b+\left (-8 a^2-25 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {-\frac {\frac {2}{3} \int -\frac {b \left (2 a^2+25 b^2\right )+a \left (8 a^2+19 b^2\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {1}{3} \int \frac {b \left (2 a^2+25 b^2\right )+a \left (8 a^2+19 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {1}{3} \int \frac {b \left (2 a^2+25 b^2\right )+a \left (8 a^2+19 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (8 a^4+17 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {a \left (8 a^2+19 b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (8 a^4+17 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^2+19 b^2\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (8 a^4+17 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (8 a^4+17 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (8 a^4+17 a^2 b^2-25 b^4\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (8 a^4+17 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {\left (8 a^4+17 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {-\frac {\frac {1}{3} \left (\frac {2 \left (8 a^4+17 a^2 b^2-25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^2+19 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^2+25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}}{5 b}-\frac {8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}}{7 b}+\frac {2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}\)

Input:

Int[Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]],x]
 

Output:

(2*Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(7*b*d) + ((-8*a* 
(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*b*d) - (((-2*a*(8*a^2 + 19*b^2 
)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqr 
t[(a + b*Cos[c + d*x])/(a + b)]) + (2*(8*a^4 + 17*a^2*b^2 - 25*b^4)*Sqrt[( 
a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*S 
qrt[a + b*Cos[c + d*x]]))/3 - (2*(8*a^2 + 25*b^2)*Sqrt[a + b*Cos[c + d*x]] 
*Sin[c + d*x])/(3*d))/(5*b))/(7*b)
 

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]
 

rule 3272

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* 
x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m 
 + n))   Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d 
*(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 
 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m 
] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] 
&& NeQ[c, 0])))
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(826\) vs. \(2(249)=498\).

Time = 9.60 (sec) , antiderivative size = 827, normalized size of antiderivative = 3.13

method	result	size
default	\(-\frac {2 \sqrt {\left (2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} b^{4}+144 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a \,b^{3}-600 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b^{4}-4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{2} b^{2}-288 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a \,b^{3}+640 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{4}-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{3} b +6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} b^{2}+230 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a \,b^{3}-360 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{4}-8 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{4}-17 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{2} b^{2}+25 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) b^{4}+8 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{4}-8 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{3} b +19 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a^{2} b^{2}-19 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right ) a \,b^{3}+8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} b -2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}-86 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{3}+80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}\right )}{105 b^{3} \sqrt {-2 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (a +b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}\, d}\)	\(827\)

Input:

int(cos(d*x+c)^3*(a+cos(d*x+c)*b)^(1/2),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*co 
s(1/2*d*x+1/2*c)^9*b^4+144*cos(1/2*d*x+1/2*c)^7*a*b^3-600*cos(1/2*d*x+1/2* 
c)^7*b^4-4*cos(1/2*d*x+1/2*c)^5*a^2*b^2-288*cos(1/2*d*x+1/2*c)^5*a*b^3+640 
*cos(1/2*d*x+1/2*c)^5*b^4-8*cos(1/2*d*x+1/2*c)^3*a^3*b+6*cos(1/2*d*x+1/2*c 
)^3*a^2*b^2+230*cos(1/2*d*x+1/2*c)^3*a*b^3-360*cos(1/2*d*x+1/2*c)^3*b^4-8* 
(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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^4-17*(sin(1/2*d*x+1/2*c 
)^2)^(1/2)*((2*b*cos(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^2*b^2+25*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2 
*b*cos(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^4+8*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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^ 
4-8*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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+19*(sin(1/2*d*x 
+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)*EllipticE(co 
s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2-19*(sin(1/2*d*x+1/2*c)^2)^(1/ 
2)*((2*b*cos(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+8*cos(1/2*d*x+1/2*c)*a^3*b-2*cos(1/2*d*x+1/2* 
c)*a^2*b^2-86*cos(1/2*d*x+1/2*c)*a*b^3+80*cos(1/2*d*x+1/2*c)*b^4)/b^3/(-2* 
b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.80 \[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=-\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (-16 i \, a^{4} - 32 i \, a^{2} b^{2} + 75 i \, b^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {\frac {1}{2}} {\left (16 i \, a^{4} + 32 i \, a^{2} b^{2} - 75 i \, b^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (-8 i \, a^{3} b - 19 i \, a b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (8 i \, a^{3} b + 19 i \, a b^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (15 \, b^{4} \cos \left (d x + c\right )^{2} + 3 \, a b^{3} \cos \left (d x + c\right ) - 4 \, a^{2} b^{2} + 25 \, b^{4}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )\right )}}{315 \, b^{4} d} \] Input:

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

-2/315*(sqrt(1/2)*(-16*I*a^4 - 32*I*a^2*b^2 + 75*I*b^4)*sqrt(b)*weierstras 
sPInverse(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*c 
os(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + sqrt(1/2)*(16*I*a^4 + 32*I*a^ 
2*b^2 - 75*I*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)*(-8*I*a^3*b - 19*I*a*b^3)*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)*(8*I*a^3*b + 19*I*a*b^3)*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^4*cos(d*x + 
c)^2 + 3*a*b^3*cos(d*x + c) - 4*a^2*b^2 + 25*b^4)*sqrt(b*cos(d*x + c) + a) 
*sin(d*x + c))/(b^4*d)
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**3*(a+b*cos(d*x+c))**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*cos(d*x + c) + a)*cos(d*x + c)^3, x)
 

Giac [F]

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

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*cos(d*x + c) + a)*cos(d*x + c)^3, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \, dx=\int {\cos \left (c+d\,x\right )}^3\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x \] Input:

int(cos(c + d*x)^3*(a + b*cos(c + d*x))^(1/2),x)
 

Output:

int(cos(c + d*x)^3*(a + b*cos(c + d*x))^(1/2), x)
 

Reduce [F]

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

int(cos(d*x+c)^3*(a+b*cos(d*x+c))^(1/2),x)
 

Output:

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