Integrand size = 41, antiderivative size = 211 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a (9 A+7 (B+C)) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {10 a (11 A+11 B+9 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {10 a (11 A+11 B+9 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a (9 A+7 (B+C)) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 a (11 A+11 B+9 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 a (B+C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 a C \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d} \] Output:
2/15*a*(9*A+7*B+7*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+10/231*a*(11* A+11*B+9*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+10/231*a*(11*A+11*B+9 *C)*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/45*a*(9*A+7*B+7*C)*cos(d*x+c)^(3/2)*si n(d*x+c)/d+2/77*a*(11*A+11*B+9*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/d+2/9*a*(B+C )*cos(d*x+c)^(7/2)*sin(d*x+c)/d+2/11*a*C*cos(d*x+c)^(9/2)*sin(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.20 (sec) , antiderivative size = 1344, normalized size of antiderivative = 6.37 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C* Cos[c + d*x]^2),x]
Output:
a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/15*((9*A + 7*B + 7*C)*Cot[c])/d + ((506*A + 506*B + 435*C)*Cos[d*x]*Sin[c])/(1848* d) + ((18*A + 19*B + 19*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((44*A + 44*B + 57*C)*Cos[3*d*x]*Sin[3*c])/(1232*d) + ((B + C)*Cos[4*d*x]*Sin[4*c])/(72*d) + (C*Cos[5*d*x]*Sin[5*c])/(176*d) + ((506*A + 506*B + 435*C)*Cos[c]*Sin[d *x])/(1848*d) + ((18*A + 19*B + 19*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + ((44* A + 44*B + 57*C)*Cos[3*c]*Sin[3*d*x])/(1232*d) + ((B + C)*Cos[4*c]*Sin[4*d *x])/(72*d) + (C*Cos[5*c]*Sin[5*d*x])/(176*d)) - (5*A*(1 + Cos[c + d*x])*C sc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Se c[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot [c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (5*B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[ c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]] ])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (15*C *(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - Arc Tan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(77*d*Sqrt[1 + Cot[...
Time = 0.97 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.95, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.366, Rules used = {3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3115, 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.
| \(\displaystyle \int \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right ) \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} \cos ^{\frac {5}{2}}(c+d x) \left (11 a (B+C) \cos ^2(c+d x)+a (11 A+11 B+9 C) \cos (c+d x)+11 a A\right )dx+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \cos ^{\frac {5}{2}}(c+d x) \left (11 a (B+C) \cos ^2(c+d x)+a (11 A+11 B+9 C) \cos (c+d x)+11 a A\right )dx+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (11 a (B+C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (11 A+11 B+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+11 a A\right )dx+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \cos ^{\frac {5}{2}}(c+d x) (11 a (9 A+7 (B+C))+9 a (11 A+11 B+9 C) \cos (c+d x))dx+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \cos ^{\frac {5}{2}}(c+d x) (11 a (9 A+7 (B+C))+9 a (11 A+11 B+9 C) \cos (c+d x))dx+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (11 a (9 A+7 (B+C))+9 a (11 A+11 B+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 a (9 A+7 (B+C)) \int \cos ^{\frac {5}{2}}(c+d x)dx+9 a (11 A+11 B+9 C) \int \cos ^{\frac {7}{2}}(c+d x)dx\right )+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 a (9 A+7 (B+C)) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx+9 a (11 A+11 B+9 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}dx\right )+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 a (9 A+7 (B+C)) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+9 a (11 A+11 B+9 C) \left (\frac {5}{7} \int \cos ^{\frac {3}{2}}(c+d x)dx+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )\right )+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 a (9 A+7 (B+C)) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+9 a (11 A+11 B+9 C) \left (\frac {5}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )\right )+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (9 a (11 A+11 B+9 C) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+11 a (9 A+7 (B+C)) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (9 a (11 A+11 B+9 C) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+11 a (9 A+7 (B+C)) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (9 a (11 A+11 B+9 C) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+11 a (9 A+7 (B+C)) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (9 a (11 A+11 B+9 C) \left (\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {5}{7} \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+11 a (9 A+7 (B+C)) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {22 a (B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
Input:
Int[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
(2*a*C*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(11*d) + ((22*a*(B + C)*Cos[c + d* x]^(7/2)*Sin[c + d*x])/(9*d) + (11*a*(9*A + 7*(B + C))*((6*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)) + 9*a*(11*A + 11*B + 9*C)*((2*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (5*((2*Ellipti cF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/7) )/9)/11
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_.)*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. \(542\) vs. \(2(192)=384\).
Time = 45.45 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.57
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (-1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a \left (20160 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-12320 B -62720 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7920 A +32560 B +81520 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-17424 A -34672 B -57712 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (14784 A +19712 B +24332 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-4026 A -4488 B -4638 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+825 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 )-2079 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 )+825 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 )-1617 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 )+675 C \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 )-1617 C \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 )\right )}{3465 \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 {-1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}\) | \(543\) |
| parts | \(\text {Expression too large to display}\) | \(854\) |
Input:
int(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,me thod=_RETURNVERBOSE)
Output:
-2/3465*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a*(20160* C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-12320*B-62720*C)*sin(1/2*d*x+ 1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A+32560*B+81520*C)*sin(1/2*d*x+1/2*c)^8 *cos(1/2*d*x+1/2*c)+(-17424*A-34672*B-57712*C)*sin(1/2*d*x+1/2*c)^6*cos(1/ 2*d*x+1/2*c)+(14784*A+19712*B+24332*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/ 2*c)+(-4026*A-4488*B-4638*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+825*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(c os(1/2*d*x+1/2*c),2^(1/2))-2079*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))+825*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))-1617*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))+675*C*(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))-1617*C*(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)))/(-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)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2) /d
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.16 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {-75 i \, \sqrt {2} {\left (11 \, A + 11 \, B + 9 \, C\right )} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 75 i \, \sqrt {2} {\left (11 \, A + 11 \, B + 9 \, C\right )} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (9 \, A + 7 \, B + 7 \, C\right )} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (9 \, A + 7 \, B + 7 \, C\right )} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (315 \, C a \cos \left (d x + c\right )^{4} + 385 \, {\left (B + C\right )} a \cos \left (d x + c\right )^{3} + 45 \, {\left (11 \, A + 11 \, B + 9 \, C\right )} a \cos \left (d x + c\right )^{2} + 77 \, {\left (9 \, A + 7 \, B + 7 \, C\right )} a \cos \left (d x + c\right ) + 75 \, {\left (11 \, A + 11 \, B + 9 \, C\right )} a\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{3465 \, d} \] Input:
integrate(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x, algorithm="fricas")
Output:
1/3465*(-75*I*sqrt(2)*(11*A + 11*B + 9*C)*a*weierstrassPInverse(-4, 0, cos (d*x + c) + I*sin(d*x + c)) + 75*I*sqrt(2)*(11*A + 11*B + 9*C)*a*weierstra ssPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*I*sqrt(2)*(9*A + 7* B + 7*C)*a*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*I*sqrt(2)*(9*A + 7*B + 7*C)*a*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(315*C* a*cos(d*x + c)^4 + 385*(B + C)*a*cos(d*x + c)^3 + 45*(11*A + 11*B + 9*C)*a *cos(d*x + c)^2 + 77*(9*A + 7*B + 7*C)*a*cos(d*x + c) + 75*(11*A + 11*B + 9*C)*a)*sqrt(cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(5/2)*(a+a*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)* *2),x)
Output:
Timed out
\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)*cos (d*x + c)^(5/2), x)
\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)*cos (d*x + c)^(5/2), x)
Time = 0.94 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.26 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {2\,A\,a\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
- (2*A*a*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*a*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*a*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*a*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)) - (2*C*a*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)) - (2*C*a*co s(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2 ))/(13*d*(sin(c + d*x)^2)^(1/2))
\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=a \left (\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a \right ) \] Input:
int(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
a*(int(sqrt(cos(c + d*x))*cos(c + d*x)**5,x)*c + int(sqrt(cos(c + d*x))*co s(c + d*x)**4,x)*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*c + int(sqr t(cos(c + d*x))*cos(c + d*x)**3,x)*a + int(sqrt(cos(c + d*x))*cos(c + d*x) **3,x)*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a)