Integrand size = 33, antiderivative size = 223 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {2 \left (7 b^2 B+9 a (2 A b+a B)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (7 a^2 A+5 A b^2+10 a b B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 \left (7 a^2 A+5 A b^2+10 a b B\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 \left (7 b^2 B+9 a (2 A b+a B)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (9 A b+11 a B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b B \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d} \] Output:
2/15*(7*b^2*B+9*a*(2*A*b+B*a))*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/2 1*(7*A*a^2+5*A*b^2+10*B*a*b)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+2/21 *(7*A*a^2+5*A*b^2+10*B*a*b)*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/45*(7*b^2*B+9* a*(2*A*b+B*a))*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/63*b*(9*A*b+11*B*a)*cos(d*x +c)^(5/2)*sin(d*x+c)/d+2/9*b*B*cos(d*x+c)^(5/2)*(a+b*cos(d*x+c))*sin(d*x+c )/d
Time = 2.92 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.75 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {84 \left (18 a A b+9 a^2 B+7 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+60 \left (7 a^2 A+5 A b^2+10 a b B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} \left (7 \left (72 a A b+36 a^2 B+43 b^2 B\right ) \cos (c+d x)+5 \left (84 a^2 A+78 A b^2+156 a b B+18 b (A b+2 a B) \cos (2 (c+d x))+7 b^2 B \cos (3 (c+d x))\right )\right ) \sin (c+d x)}{630 d} \] Input:
Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x ]
Output:
(84*(18*a*A*b + 9*a^2*B + 7*b^2*B)*EllipticE[(c + d*x)/2, 2] + 60*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*( 72*a*A*b + 36*a^2*B + 43*b^2*B)*Cos[c + d*x] + 5*(84*a^2*A + 78*A*b^2 + 15 6*a*b*B + 18*b*(A*b + 2*a*B)*Cos[2*(c + d*x)] + 7*b^2*B*Cos[3*(c + d*x)])) *Sin[c + d*x])/(630*d)
Time = 0.95 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.91, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3042, 3469, 27, 3042, 3502, 27, 3042, 3227, 3042, 3115, 3042, 3119, 3120}
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.
\(\displaystyle \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
\(\Big \downarrow \) 3469 |
\(\displaystyle \frac {2}{9} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \left (b (9 A b+11 a B) \cos ^2(c+d x)+\left (7 B b^2+9 a (2 A b+a B)\right ) \cos (c+d x)+a (9 a A+5 b B)\right )dx+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int \cos ^{\frac {3}{2}}(c+d x) \left (b (9 A b+11 a B) \cos ^2(c+d x)+\left (7 B b^2+9 a (2 A b+a B)\right ) \cos (c+d x)+a (9 a A+5 b B)\right )dx+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (b (9 A b+11 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (7 B b^2+9 a (2 A b+a B)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (9 a A+5 b B)\right )dx+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 3502 |
\(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \left (9 \left (7 A a^2+10 b B a+5 A b^2\right )+7 \left (7 B b^2+9 a (2 A b+a B)\right ) \cos (c+d x)\right )dx+\frac {2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \cos ^{\frac {3}{2}}(c+d x) \left (9 \left (7 A a^2+10 b B a+5 A b^2\right )+7 \left (7 B b^2+9 a (2 A b+a B)\right ) \cos (c+d x)\right )dx+\frac {2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (9 \left (7 A a^2+10 b B a+5 A b^2\right )+7 \left (7 B b^2+9 a (2 A b+a B)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 \left (7 a^2 A+10 a b B+5 A b^2\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+7 \left (9 a (a B+2 A b)+7 b^2 B\right ) \int \cos ^{\frac {5}{2}}(c+d x)dx\right )+\frac {2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 \left (7 a^2 A+10 a b B+5 A b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+7 \left (9 a (a B+2 A b)+7 b^2 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )+\frac {2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 \left (7 a^2 A+10 a b B+5 A b^2\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+7 \left (9 a (a B+2 A b)+7 b^2 B\right ) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 \left (7 a^2 A+10 a b B+5 A b^2\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+7 \left (9 a (a B+2 A b)+7 b^2 B\right ) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 \left (7 a^2 A+10 a b B+5 A b^2\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+7 \left (9 a (a B+2 A b)+7 b^2 B\right ) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 \left (7 a^2 A+10 a b B+5 A b^2\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+7 \left (9 a (a B+2 A b)+7 b^2 B\right ) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}\) |
Input:
Int[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]
Output:
(2*b*B*Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(9*d) + ((2*b *(9*A*b + 11*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (9*(7*a^2*A + 5 *A*b^2 + 10*a*b*B)*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)) + 7*(7*b^2*B + 9*a*(2*A*b + a*B))*((6*EllipticE [(c + d*x)/2, 2])/(5*d) + (2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)))/7)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^( n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*( m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c - b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin [e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !(IGt Q[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs. \(2(206)=412\).
Time = 20.77 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.74
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-1120 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} b^{2}+\left (720 A \,b^{2}+1440 B a b +2240 B \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1008 A a b -1080 A \,b^{2}-504 a^{2} B -2160 B a b -2072 B \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (420 a^{2} A +1008 A a b +840 A \,b^{2}+504 a^{2} B +1680 B a b +952 B \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-210 a^{2} A -252 A a b -240 A \,b^{2}-126 a^{2} B -480 B a b -168 B \,b^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+105 a^{2} A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+75 A \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-378 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a b +150 B a b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-189 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}-147 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}\right )}{315 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(610\) |
parts | \(\text {Expression too large to display}\) | \(820\) |
Input:
int(cos(d*x+c)^(3/2)*(a+cos(d*x+c)*b)^2*(A+B*cos(d*x+c)),x,method=_RETURNV ERBOSE)
Output:
-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*B*co s(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10*b^2+(720*A*b^2+1440*B*a*b+2240*B*b^ 2)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-1008*A*a*b-1080*A*b^2-504*B*a ^2-2160*B*a*b-2072*B*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(420*A*a ^2+1008*A*a*b+840*A*b^2+504*B*a^2+1680*B*a*b+952*B*b^2)*sin(1/2*d*x+1/2*c) ^4*cos(1/2*d*x+1/2*c)+(-210*A*a^2-252*A*a*b-240*A*b^2-126*B*a^2-480*B*a*b- 168*B*b^2)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+105*a^2*A*(sin(1/2*d*x+ 1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2 *c),2^(1/2))+75*A*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2 -1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-378*A*(sin(1/2*d*x+1/2*c)^ 2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^( 1/2))*a*b+150*B*a*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1 )^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-189*B*(sin(1/2*d*x+1/2*c)^2) ^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/ 2))*a^2-147*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2 )*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^2)/(-2*sin(1/2*d*x+1/2*c)^4+sin( 1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2 )/d
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.22 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {2 \, {\left (35 \, B b^{2} \cos \left (d x + c\right )^{3} + 105 \, A a^{2} + 150 \, B a b + 75 \, A b^{2} + 45 \, {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} + 7 \, {\left (9 \, B a^{2} + 18 \, A a b + 7 \, B b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (7 i \, A a^{2} + 10 i \, B a b + 5 i \, A b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-7 i \, A a^{2} - 10 i \, B a b - 5 i \, A b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-9 i \, B a^{2} - 18 i \, A a b - 7 i \, B b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (9 i \, B a^{2} + 18 i \, A a b + 7 i \, B b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{315 \, d} \] Input:
integrate(cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorith m="fricas")
Output:
1/315*(2*(35*B*b^2*cos(d*x + c)^3 + 105*A*a^2 + 150*B*a*b + 75*A*b^2 + 45* (2*B*a*b + A*b^2)*cos(d*x + c)^2 + 7*(9*B*a^2 + 18*A*a*b + 7*B*b^2)*cos(d* x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - 15*sqrt(2)*(7*I*A*a^2 + 10*I*B*a *b + 5*I*A*b^2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*sqrt(2)*(-7*I*A*a^2 - 10*I*B*a*b - 5*I*A*b^2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*sqrt(2)*(-9*I*B*a^2 - 18*I*A*a*b - 7*I*B*b^2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*sqrt(2)*(9*I*B*a^2 + 18*I*A*a*b + 7*I*B*b^2)*weie rstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c ))))/d
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(3/2)*(a+b*cos(d*x+c))**2*(A+B*cos(d*x+c)),x)
Output:
Timed out
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorith m="maxima")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2*cos(d*x + c)^(3/2), x)
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorith m="giac")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2*cos(d*x + c)^(3/2), x)
Time = 25.12 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.18 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\frac {2\,A\,a^2\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}-\frac {2\,B\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,b^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,A\,a\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,B\,a\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2,x)
Output:
(2*A*a^2*(cos(c + d*x)^(1/2)*sin(c + d*x) + ellipticF(c/2 + (d*x)/2, 2)))/ (3*d) - (2*B*a^2*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/ 4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*A*b^2*cos(c + d*x)^( 9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (2*B*b^2*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1 /2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (4*A*a*b *cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^ 2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (4*B*a*b*cos(c + d*x)^(9/2)*sin(c + d*x )*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2) )
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx=\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{3}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) b^{3}+3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a \,b^{2}+3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} b \] Input:
int(cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)),x)
Output:
int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a**3 + int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*b**3 + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a*b**2 + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a**2*b