Integrand size = 33, antiderivative size = 255 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {2 \left (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b \left (27 a A b+22 a^2 B+7 b^2 B\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b^2 (9 A b+13 a B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b B \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d} \] Output:
2/15*(15*A*a^3+27*A*a*b^2+27*B*a^2*b+7*B*b^3)*EllipticE(sin(1/2*d*x+1/2*c) ,2^(1/2))/d+2/21*(21*A*a^2*b+5*A*b^3+7*B*a^3+15*B*a*b^2)*InverseJacobiAM(1 /2*d*x+1/2*c,2^(1/2))/d+2/21*(21*A*a^2*b+5*A*b^3+7*B*a^3+15*B*a*b^2)*cos(d *x+c)^(1/2)*sin(d*x+c)/d+2/45*b*(27*A*a*b+22*B*a^2+7*B*b^2)*cos(d*x+c)^(3/ 2)*sin(d*x+c)/d+2/63*b^2*(9*A*b+13*B*a)*cos(d*x+c)^(5/2)*sin(d*x+c)/d+2/9* b*B*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d
Time = 2.70 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.77 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {84 \left (15 a^3 A+27 a A b^2+27 a^2 b B+7 b^3 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+60 \left (21 a^2 A b+5 A b^3+7 a^3 B+15 a b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} \left (7 b \left (108 a A b+108 a^2 B+43 b^2 B\right ) \cos (c+d x)+5 \left (252 a^2 A b+78 A b^3+84 a^3 B+234 a b^2 B+18 b^2 (A b+3 a B) \cos (2 (c+d x))+7 b^3 B \cos (3 (c+d x))\right )\right ) \sin (c+d x)}{630 d} \] Input:
Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x ]
Output:
(84*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*EllipticE[(c + d*x)/2, 2] + 60*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*EllipticF[(c + d*x)/ 2, 2] + Sqrt[Cos[c + d*x]]*(7*b*(108*a*A*b + 108*a^2*B + 43*b^2*B)*Cos[c + d*x] + 5*(252*a^2*A*b + 78*A*b^3 + 84*a^3*B + 234*a*b^2*B + 18*b^2*(A*b + 3*a*B)*Cos[2*(c + d*x)] + 7*b^3*B*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d )
Time = 1.33 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.96, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {3042, 3469, 27, 3042, 3512, 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.
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3469 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (b (9 A b+13 a B) \cos ^2(c+d x)+\left (7 B b^2+9 a (2 A b+a B)\right ) \cos (c+d x)+3 a (3 a A+b B)\right )dx+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (b (9 A b+13 a B) \cos ^2(c+d x)+\left (7 B b^2+9 a (2 A b+a B)\right ) \cos (c+d x)+3 a (3 a A+b B)\right )dx+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b (9 A b+13 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (7 B b^2+9 a (2 A b+a B)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a (3 a A+b B)\right )dx+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \cos ^2(c+d x)+9 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \cos (c+d x)\right )dx+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \sqrt {\cos (c+d x)} \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \cos ^2(c+d x)+9 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \cos (c+d x)\right )dx+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (21 (3 a A+b B) a^2+7 b \left (22 B a^2+27 A b a+7 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+9 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {\cos (c+d x)} \left (7 \left (15 A a^3+27 b B a^2+27 A b^2 a+7 b^3 B\right )+15 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \cos (c+d x)\right )dx+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\cos (c+d x)} \left (7 \left (15 A a^3+27 b B a^2+27 A b^2 a+7 b^3 B\right )+15 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \cos (c+d x)\right )dx+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (7 \left (15 A a^3+27 b B a^2+27 A b^2 a+7 b^3 B\right )+15 \left (7 B a^3+21 A b a^2+15 b^2 B a+5 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+7 \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (7 \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (7 \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (7 \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {14 \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {14 b \left (22 a^2 B+27 a A b+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (\frac {14 \left (15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+15 \left (7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )+\frac {2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\) |
Input:
Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]
Output:
(2*b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + ((2 *b^2*(9*A*b + 13*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + ((14*b*(27* a*A*b + 22*a^2*B + 7*b^2*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3*(( 14*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*EllipticE[(c + d*x)/2, 2 ])/d + 15*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d))))/5)/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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(744\) vs. \(2(238)=476\).
Time = 21.04 (sec) , antiderivative size = 745, normalized size of antiderivative = 2.92
| 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^{3}+\left (720 A \,b^{3}+2160 B a \,b^{2}+2240 b^{3} B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1512 A a \,b^{2}-1080 A \,b^{3}-1512 B \,a^{2} b -3240 B a \,b^{2}-2072 b^{3} B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (1260 A \,a^{2} b +1512 A a \,b^{2}+840 A \,b^{3}+420 a^{3} B +1512 B \,a^{2} b +2520 B a \,b^{2}+952 b^{3} B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-630 A \,a^{2} b -378 A a \,b^{2}-240 A \,b^{3}-210 a^{3} B -378 B \,a^{2} b -720 B a \,b^{2}-168 b^{3} B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+315 A \,a^{2} 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 )+75 A \,b^{3} \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 )-315 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^{3}-567 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^{2}+105 a^{3} 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 )+225 B 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 )-567 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} b -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^{3}\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}\) | \(745\) |
| parts | \(\text {Expression too large to display}\) | \(971\) |
Input:
int(cos(d*x+c)^(1/2)*(a+cos(d*x+c)*b)^3*(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^3+(720*A*b^3+2160*B*a*b^2+2240*B* b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-1512*A*a*b^2-1080*A*b^3-151 2*B*a^2*b-3240*B*a*b^2-2072*B*b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c) +(1260*A*a^2*b+1512*A*a*b^2+840*A*b^3+420*B*a^3+1512*B*a^2*b+2520*B*a*b^2+ 952*B*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-630*A*a^2*b-378*A*a*b ^2-240*A*b^3-210*B*a^3-378*B*a^2*b-720*B*a*b^2-168*B*b^3)*sin(1/2*d*x+1/2* c)^2*cos(1/2*d*x+1/2*c)+315*A*a^2*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))+75*A*b^3*(si n(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))-315*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^3-567*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^2+105*a^3*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))+225*B*a*b^2*(si n(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))-567*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*b-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^3)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.26 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {2 \, {\left (35 \, B b^{3} \cos \left (d x + c\right )^{3} + 105 \, B a^{3} + 315 \, A a^{2} b + 225 \, B a b^{2} + 75 \, A b^{3} + 45 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{2} + 7 \, {\left (27 \, B a^{2} b + 27 \, A a b^{2} + 7 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (7 i \, B a^{3} + 21 i \, A a^{2} b + 15 i \, B a b^{2} + 5 i \, A b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-7 i \, B a^{3} - 21 i \, A a^{2} b - 15 i \, B a b^{2} - 5 i \, A b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-15 i \, A a^{3} - 27 i \, B a^{2} b - 27 i \, A a b^{2} - 7 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (15 i \, A a^{3} + 27 i \, B a^{2} b + 27 i \, A a b^{2} + 7 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{315 \, d} \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)),x, algorith m="fricas")
Output:
1/315*(2*(35*B*b^3*cos(d*x + c)^3 + 105*B*a^3 + 315*A*a^2*b + 225*B*a*b^2 + 75*A*b^3 + 45*(3*B*a*b^2 + A*b^3)*cos(d*x + c)^2 + 7*(27*B*a^2*b + 27*A* a*b^2 + 7*B*b^3)*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - 15*sqrt(2 )*(7*I*B*a^3 + 21*I*A*a^2*b + 15*I*B*a*b^2 + 5*I*A*b^3)*weierstrassPInvers e(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*sqrt(2)*(-7*I*B*a^3 - 21*I*A* a^2*b - 15*I*B*a*b^2 - 5*I*A*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*sqrt(2)*(-15*I*A*a^3 - 27*I*B*a^2*b - 27*I*A*a*b^2 - 7*I*B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c ) + I*sin(d*x + c))) - 21*sqrt(2)*(15*I*A*a^3 + 27*I*B*a^2*b + 27*I*A*a*b^ 2 + 7*I*B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/d
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(1/2)*(a+b*cos(d*x+c))**3*(A+B*cos(d*x+c)),x)
Output:
Timed out
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)),x, algorith m="maxima")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3*sqrt(cos(d*x + c)), x)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)),x, algorith m="giac")
Output:
integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3*sqrt(cos(d*x + c)), x)
Time = 24.77 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.29 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\frac {2\,\left (A\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a^2\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a^2\,b\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {B\,a^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,A\,b^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,b^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,A\,a\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,B\,a^2\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3,x)
Output:
(2*(A*a^3*ellipticE(c/2 + (d*x)/2, 2) + A*a^2*b*ellipticF(c/2 + (d*x)/2, 2 ) + A*a^2*b*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (B*a^3*((2*cos(c + d*x)^ (1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (2*A*b^3*c os(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^3*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 )) - (6*A*a*b^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)) - (6*B*a^2*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)) - (2*B*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))/(3*d*(sin(c + d*x)^2)^(1/2))
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx=\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{4}+4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{3} b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) b^{4}+4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a \,b^{3}+6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} b^{2} \] Input:
int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)),x)
Output:
int(sqrt(cos(c + d*x)),x)*a**4 + 4*int(sqrt(cos(c + d*x))*cos(c + d*x),x)* a**3*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*b**4 + 4*int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a*b**3 + 6*int(sqrt(cos(c + d*x))*cos(c + d*x) **2,x)*a**2*b**2