Integrand size = 25, antiderivative size = 197 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{315 d e}+\frac {2 a \left (7 a^2+6 b^2\right ) e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {2 a \left (7 a^2+6 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e} \]
-2/315*b*(89*a^2+28*b^2)*(e*cos(d*x+c))^(5/2)/d/e-26/63*a*b*(e*cos(d*x+c)) ^(5/2)*(a+b*sin(d*x+c))/d/e-2/9*b*(e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^2/ d/e+2/21*a*(7*a^2+6*b^2)*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2* c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/d/(e*cos(d*x+c)) ^(1/2)+2/21*a*(7*a^2+6*b^2)*e*sin(d*x+c)*(e*cos(d*x+c))^(1/2)/d
Time = 1.40 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.78 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\frac {(e \cos (c+d x))^{3/2} \left (80 \left (7 a^3+6 a b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {2}{3} \sqrt {\cos (c+d x)} \left (-756 a^2 b-147 b^3-28 \left (27 a^2 b+4 b^3\right ) \cos (2 (c+d x))+35 b^3 \cos (4 (c+d x))+840 a^3 \sin (c+d x)+450 a b^2 \sin (c+d x)-270 a b^2 \sin (3 (c+d x))\right )\right )}{840 d \cos ^{\frac {3}{2}}(c+d x)} \]
((e*Cos[c + d*x])^(3/2)*(80*(7*a^3 + 6*a*b^2)*EllipticF[(c + d*x)/2, 2] + (2*Sqrt[Cos[c + d*x]]*(-756*a^2*b - 147*b^3 - 28*(27*a^2*b + 4*b^3)*Cos[2* (c + d*x)] + 35*b^3*Cos[4*(c + d*x)] + 840*a^3*Sin[c + d*x] + 450*a*b^2*Si n[c + d*x] - 270*a*b^2*Sin[3*(c + d*x)]))/3))/(840*d*Cos[c + d*x]^(3/2))
Time = 0.92 (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, 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.
| \(\displaystyle \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{2} (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (9 a^2+13 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (9 a^2+13 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (9 a^2+13 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} (e \cos (c+d x))^{3/2} \left (9 a \left (7 a^2+6 b^2\right )+b \left (89 a^2+28 b^2\right ) \sin (c+d x)\right )dx-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int (e \cos (c+d x))^{3/2} \left (9 a \left (7 a^2+6 b^2\right )+b \left (89 a^2+28 b^2\right ) \sin (c+d x)\right )dx-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int (e \cos (c+d x))^{3/2} \left (9 a \left (7 a^2+6 b^2\right )+b \left (89 a^2+28 b^2\right ) \sin (c+d x)\right )dx-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \int (e \cos (c+d x))^{3/2}dx-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 b \left (89 a^2+28 b^2\right ) (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\right )-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}\) |
(-2*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2)/(9*d*e) + ((-26*a*b*( e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]))/(7*d*e) + ((-2*b*(89*a^2 + 28* b^2)*(e*Cos[c + d*x])^(5/2))/(5*d*e) + 9*a*(7*a^2 + 6*b^2)*((2*e^2*Sqrt[Co s[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) + (2*e*S qrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/7)/9
3.6.57.3.1 Defintions of rubi rules used
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[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[(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. \(449\) vs. \(2(201)=402\).
Time = 7.45 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.28
| method | result | size |
| default | \(\frac {2 e^{2} \left (-1120 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+2160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+2800 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-3240 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+1512 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -2296 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-420 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+1260 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-2268 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +644 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+210 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-90 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-105 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}-90 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}+1134 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +28 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-189 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -28 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}\right )}{315 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(450\) |
| parts | \(-\frac {2 a^{3} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{2} \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {2 b^{3} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} e^{2}}{5}\right )}{d \,e^{3}}-\frac {6 a^{2} b \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{5 e d}+\frac {4 a \,b^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{2} \left (24 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-60 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+50 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}\) | \(466\) |
2/315/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^2*(-1120*si n(1/2*d*x+1/2*c)^11*b^3+2160*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8*a*b^2 +2800*sin(1/2*d*x+1/2*c)^9*b^3-3240*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^ 6*a*b^2+1512*sin(1/2*d*x+1/2*c)^7*a^2*b-2296*sin(1/2*d*x+1/2*c)^7*b^3-420* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*a^3+1260*cos(1/2*d*x+1/2*c)*sin(1/ 2*d*x+1/2*c)^4*a*b^2-2268*sin(1/2*d*x+1/2*c)^5*a^2*b+644*sin(1/2*d*x+1/2*c )^5*b^3+210*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a^3-90*cos(1/2*d*x+1/2 *c)*sin(1/2*d*x+1/2*c)^2*a*b^2-105*(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))*a^3-90*(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))*a*b^2+1134*sin(1/2*d*x+1/2*c)^3*a^2*b+28*sin(1/2*d*x+1 /2*c)^3*b^3-189*sin(1/2*d*x+1/2*c)*a^2*b-28*sin(1/2*d*x+1/2*c)*b^3)/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.86 \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\frac {-15 i \, \sqrt {2} {\left (7 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} {\left (7 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (35 \, b^{3} e \cos \left (d x + c\right )^{4} - 63 \, {\left (3 \, a^{2} b + b^{3}\right )} e \cos \left (d x + c\right )^{2} - 15 \, {\left (9 \, a b^{2} e \cos \left (d x + c\right )^{2} - {\left (7 \, a^{3} + 6 \, a b^{2}\right )} e\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{315 \, d} \]
1/315*(-15*I*sqrt(2)*(7*a^3 + 6*a*b^2)*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*I*sqrt(2)*(7*a^3 + 6*a*b^2)*e^(3/2)*we ierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*(35*b^3*e*cos( d*x + c)^4 - 63*(3*a^2*b + b^3)*e*cos(d*x + c)^2 - 15*(9*a*b^2*e*cos(d*x + c)^2 - (7*a^3 + 6*a*b^2)*e)*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
\[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]