Integrand size = 33, antiderivative size = 196 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a (9 A+7 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {10 a (11 A+9 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {10 a (11 A+9 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a (9 A+7 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 a (11 A+9 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 a C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 a C \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d} \] Output:
2/15*a*(9*A+7*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+10/231*a*(11*A+9* C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+10/231*a*(11*A+9*C)*cos(d*x+c) ^(1/2)*sin(d*x+c)/d+2/45*a*(9*A+7*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/77*a* (11*A+9*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/d+2/9*a*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 = 6.99 (sec) , antiderivative size = 964, normalized size of antiderivative = 4.92 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + 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*C)*Cot[c])/d + ((506*A + 435*C)*Cos[d*x]*Sin[c])/(1848*d) + ((18*A + 19*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((44*A + 57*C)*Cos[3*d*x]*Sin[3*c])/( 1232*d) + (C*Cos[4*d*x]*Sin[4*c])/(72*d) + (C*Cos[5*d*x]*Sin[5*c])/(176*d) + ((506*A + 435*C)*Cos[c]*Sin[d*x])/(1848*d) + ((18*A + 19*C)*Cos[2*c]*Si n[2*d*x])/(180*d) + ((44*A + 57*C)*Cos[3*c]*Sin[3*d*x])/(1232*d) + (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])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Co t[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 - A rcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(77*d*Sqrt[1 + Cot[c ]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((Hypergeometr icPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[T an[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + Ar cTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*S qrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c...
Time = 0.93 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 3513, 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.
| \(\displaystyle \int \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a) \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right ) \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3513 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} \cos ^{\frac {5}{2}}(c+d x) \left (11 a C \cos ^2(c+d x)+a (11 A+9 C) \cos (c+d x)+11 a A\right )dx+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \cos ^{\frac {5}{2}}(c+d x) \left (11 a C \cos ^2(c+d x)+a (11 A+9 C) \cos (c+d x)+11 a A\right )dx+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (11 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (11 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+11 a A\right )dx+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \cos ^{\frac {5}{2}}(c+d x) (11 a (9 A+7 C)+9 a (11 A+9 C) \cos (c+d x))dx+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \cos ^{\frac {5}{2}}(c+d x) (11 a (9 A+7 C)+9 a (11 A+9 C) \cos (c+d x))dx+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (11 a (9 A+7 C)+9 a (11 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 a (9 A+7 C) \int \cos ^{\frac {5}{2}}(c+d x)dx+9 a (11 A+9 C) \int \cos ^{\frac {7}{2}}(c+d x)dx\right )+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 a (9 A+7 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx+9 a (11 A+9 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}dx\right )+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 a (9 A+7 C) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+9 a (11 A+9 C) \left (\frac {5}{7} \int \cos ^{\frac {3}{2}}(c+d x)dx+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )\right )+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (11 a (9 A+7 C) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+9 a (11 A+9 C) \left (\frac {5}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )\right )+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (9 a (11 A+9 C) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+11 a (9 A+7 C) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (9 a (11 A+9 C) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+11 a (9 A+7 C) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (9 a (11 A+9 C) \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+11 a (9 A+7 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (9 a (11 A+9 C) \left (\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {5}{7} \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+11 a (9 A+7 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {22 a C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {2 a C \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}\) |
Input:
Int[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + 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*C*Cos[c + d*x]^(7/ 2)*Sin[c + d*x])/(9*d) + (11*a*(9*A + 7*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 + 9*C)*((2* Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (5*((2*EllipticF[(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_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[ (-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3) )), x] + Simp[1/(b*(m + 3)) Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c *(m + 3) + b*d*(C*(m + 2) + A*(m + 3))*Sin[e + f*x] - (2*a*C*d - b*c*C*(m + 3))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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. \(433\) vs. \(2(175)=350\).
Time = 33.08 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.21
| 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}-62720 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (7920 A +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 -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 +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 -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 )+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}\) | \(434\) |
| parts | \(\text {Expression too large to display}\) | \(844\) |
Input:
int(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+C*cos(d*x+c)^2),x,method=_RETURNV ERBOSE)
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)^12-62720*C*cos(1/2*d*x+1/2*c)*sin( 1/2*d*x+1/2*c)^10+(7920*A+81520*C)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c) +(-17424*A-57712*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(14784*A+24332 *C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-4026*A-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(cos(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))+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 = 221, normalized size of antiderivative = 1.13 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {-75 i \, \sqrt {2} {\left (11 \, A + 9 \, C\right )} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 75 i \, \sqrt {2} {\left (11 \, A + 9 \, C\right )} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (9 \, A + 7 \, C\right )} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (9 \, A + 7 \, C\right )} a {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (315 \, C a \cos \left (d x + c\right )^{4} + 385 \, C a \cos \left (d x + c\right )^{3} + 45 \, {\left (11 \, A + 9 \, C\right )} a \cos \left (d x + c\right )^{2} + 77 \, {\left (9 \, A + 7 \, C\right )} a \cos \left (d x + c\right ) + 75 \, {\left (11 \, A + 9 \, C\right )} a\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{3465 \, d} \] Input:
integrate(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+C*cos(d*x+c)^2),x, algorith m="fricas")
Output:
1/3465*(-75*I*sqrt(2)*(11*A + 9*C)*a*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 75*I*sqrt(2)*(11*A + 9*C)*a*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*I*sqrt(2)*(9*A + 7*C)*a*weierstra ssZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*I*sqrt(2)*(9*A + 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*C*a *cos(d*x + c)^3 + 45*(11*A + 9*C)*a*cos(d*x + c)^2 + 77*(9*A + 7*C)*a*cos( d*x + c) + 75*(11*A + 9*C)*a)*sqrt(cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(5/2)*(a+a*cos(d*x+c))*(A+C*cos(d*x+c)**2),x)
Output:
Timed out
\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+C*cos(d*x+c)^2),x, algorith m="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)*cos(d*x + c)^(5/2), x)
\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+C*cos(d*x+c)^2),x, algorith m="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(a*cos(d*x + c) + a)*cos(d*x + c)^(5/2), x)
Time = 0.67 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.90 \[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {2\,A\,a\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(cos(c + d*x)^(5/2)*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x)),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*C*a*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 1 5/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (2*C*a*cos(c + d*x)^ (13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*d*(s in(c + d*x)^2)^(1/2))
\[ \int \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx=a \left (\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a \right ) \] Input:
int(cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))*(A+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)*c + int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a + int(sqr t(cos(c + d*x))*cos(c + d*x)**2,x)*a)