Integrand size = 25, antiderivative size = 197 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e} \] Output:
-2/231*b*(43*a^2+12*b^2)*(e*cos(d*x+c))^(7/2)/d/e+2/5*a*(3*a^2+2*b^2)*e^2* (e*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d/cos(d*x+c)^(1 /2)+2/15*a*(3*a^2+2*b^2)*e*(e*cos(d*x+c))^(3/2)*sin(d*x+c)/d-10/33*a*b*(e* cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))/d/e-2/11*b*(e*cos(d*x+c))^(7/2)*(a+b*si n(d*x+c))^2/d/e
Time = 2.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.76 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=\frac {(e \cos (c+d x))^{5/2} \left (1848 \left (3 a^3+2 a b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^{\frac {3}{2}}(c+d x) \left (-1980 a^2 b-345 b^3-60 \left (33 a^2 b+4 b^3\right ) \cos (2 (c+d x))+105 b^3 \cos (4 (c+d x))+1848 a^3 \sin (c+d x)+462 a b^2 \sin (c+d x)-770 a b^2 \sin (3 (c+d x))\right )\right )}{4620 d \cos ^{\frac {5}{2}}(c+d x)} \] Input:
Integrate[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3,x]
Output:
((e*Cos[c + d*x])^(5/2)*(1848*(3*a^3 + 2*a*b^2)*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^(3/2)*(-1980*a^2*b - 345*b^3 - 60*(33*a^2*b + 4*b^3)*Cos[2* (c + d*x)] + 105*b^3*Cos[4*(c + d*x)] + 1848*a^3*Sin[c + d*x] + 462*a*b^2* Sin[c + d*x] - 770*a*b^2*Sin[3*(c + d*x)])))/(4620*d*Cos[c + d*x]^(5/2))
Time = 0.95 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3171, 27, 3042, 3341, 27, 3042, 3148, 3042, 3115, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (11 a^2+15 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (11 a^2+15 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (11 a^2+15 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {3}{2} (e \cos (c+d x))^{5/2} \left (11 a \left (3 a^2+2 b^2\right )+b \left (43 a^2+12 b^2\right ) \sin (c+d x)\right )dx-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \int (e \cos (c+d x))^{5/2} \left (11 a \left (3 a^2+2 b^2\right )+b \left (43 a^2+12 b^2\right ) \sin (c+d x)\right )dx-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \int (e \cos (c+d x))^{5/2} \left (11 a \left (3 a^2+2 b^2\right )+b \left (43 a^2+12 b^2\right ) \sin (c+d x)\right )dx-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \int (e \cos (c+d x))^{5/2}dx-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \left (\frac {3}{5} e^2 \int \sqrt {e \cos (c+d x)}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \left (\frac {3}{5} e^2 \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \left (\frac {3 e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{5 \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \left (\frac {3 e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \left (\frac {6 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\) |
Input:
Int[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3,x]
Output:
(-2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2)/(11*d*e) + ((-10*a*b* (e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(3*d*e) + ((-2*b*(43*a^2 + 12 *b^2)*(e*Cos[c + d*x])^(7/2))/(7*d*e) + 11*a*(3*a^2 + 2*b^2)*((6*e^2*Sqrt[ e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*e *(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)))/3)/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[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[1/(m + p) Int[(g*Cos[e + f*x])^p* (a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1) *Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S implerQ[c + d*x, a + b*x])
Leaf count of result is larger than twice the leaf count of optimal. \(503\) vs. \(2(177)=354\).
Time = 38.85 (sec) , antiderivative size = 504, normalized size of antiderivative = 2.56
| method | result | size |
| parts | \(-\frac {2 a^{3} \sqrt {e \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}}\, e^{3} \left (-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-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 {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{5 \sqrt {-e \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, d}+\frac {2 b^{3} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {11}{2}}}{11}-\frac {e^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7}\right )}{d \,e^{3}}+\frac {4 a \,b^{2} \sqrt {e \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}}\, e^{3} \left (80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+272 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-144 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+35 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \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 )-3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 \sqrt {-e \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, d}-\frac {6 a^{2} b \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d e}\) | \(504\) |
| default | \(-\frac {2 e^{3} \left (-6720 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+12320 a \,b^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+20160 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-24640 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a \,b^{2}+7920 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} a^{2} b -22560 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} b^{3}-1848 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{3}+17248 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a \,b^{2}-15840 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a^{2} b +11520 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b^{3}+1848 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}-4928 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a \,b^{2}+11880 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{2} b -2340 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{3}-462 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}+462 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{2}-693 \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}-462 \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}-3960 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} b -60 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{3}+495 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b +60 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}\right )}{1155 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\) | \(534\) |
Input:
int((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-2/5*a^3*(e*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e^3*(-8 *sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+8*sin(1/2*d*x+1/2*c)^4*cos(1/2*d* x+1/2*c)-2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*(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)))/(-e*(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)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d+2*b^3/d/e^3*(1/11*(e*cos(d* x+c))^(11/2)-1/7*e^2*(e*cos(d*x+c))^(7/2))+4/15*a*b^2*(e*(2*cos(1/2*d*x+1/ 2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e^3*(80*cos(1/2*d*x+1/2*c)^11-240*co s(1/2*d*x+1/2*c)^9+272*cos(1/2*d*x+1/2*c)^7-144*cos(1/2*d*x+1/2*c)^5+35*co s(1/2*d*x+1/2*c)^3+3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2 +1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-3*cos(1/2*d*x+1/2*c))/(-e* (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)/(e *(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d-6/7*a^2*b/d*(e*cos(d*x+c))^(7/2)/e
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.96 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=-\frac {2 \, {\left (-231 i \, \sqrt {\frac {1}{2}} {\left (3 \, a^{3} + 2 \, a b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {\frac {1}{2}} {\left (3 \, a^{3} + 2 \, a b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (105 \, b^{3} e^{2} \cos \left (d x + c\right )^{5} - 165 \, {\left (3 \, a^{2} b + b^{3}\right )} e^{2} \cos \left (d x + c\right )^{3} - 77 \, {\left (5 \, a b^{2} e^{2} \cos \left (d x + c\right )^{3} - {\left (3 \, a^{3} + 2 \, a b^{2}\right )} e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{1155 \, d} \] Input:
integrate((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-2/1155*(-231*I*sqrt(1/2)*(3*a^3 + 2*a*b^2)*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*I*sqrt(1 /2)*(3*a^3 + 2*a*b^2)*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(- 4, 0, cos(d*x + c) - I*sin(d*x + c))) - (105*b^3*e^2*cos(d*x + c)^5 - 165* (3*a^2*b + b^3)*e^2*cos(d*x + c)^3 - 77*(5*a*b^2*e^2*cos(d*x + c)^3 - (3*a ^3 + 2*a*b^2)*e^2*cos(d*x + c))*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(5/2)*(a+b*sin(d*x+c))**3,x)
Output:
Timed out
\[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^3,x, algorithm="maxima")
Output:
integrate((e*cos(d*x + c))^(5/2)*(b*sin(d*x + c) + a)^3, x)
\[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
integrate((e*cos(d*x + c))^(5/2)*(b*sin(d*x + c) + a)^3, x)
Timed out. \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \] Input:
int((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^3,x)
Output:
int((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^3, x)
\[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=\frac {\sqrt {e}\, e^{2} \left (-6 \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} a^{2} b +7 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{3}d x \right ) b^{3} d +21 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}d x \right ) a \,b^{2} d +7 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{3} d \right )}{7 d} \] Input:
int((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^3,x)
Output:
(sqrt(e)*e**2*( - 6*sqrt(cos(c + d*x))*cos(c + d*x)**3*a**2*b + 7*int(sqrt (cos(c + d*x))*cos(c + d*x)**2*sin(c + d*x)**3,x)*b**3*d + 21*int(sqrt(cos (c + d*x))*cos(c + d*x)**2*sin(c + d*x)**2,x)*a*b**2*d + 7*int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a**3*d))/(7*d)