Integrand size = 25, antiderivative size = 237 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e} \] Output:
-2/1287*b*(177*a^2+44*b^2)*(e*cos(d*x+c))^(9/2)/d/e+10/231*a*(11*a^2+6*b^2 )*e^4*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d/(e*cos(d*x +c))^(1/2)+10/231*a*(11*a^2+6*b^2)*e^3*(e*cos(d*x+c))^(1/2)*sin(d*x+c)/d+2 /77*a*(11*a^2+6*b^2)*e*(e*cos(d*x+c))^(5/2)*sin(d*x+c)/d-34/143*a*b*(e*cos (d*x+c))^(9/2)*(a+b*sin(d*x+c))/d/e-2/13*b*(e*cos(d*x+c))^(9/2)*(a+b*sin(d *x+c))^2/d/e
Time = 3.05 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.86 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\frac {(e \cos (c+d x))^{7/2} \left (-154 b \left (78 a^2+11 b^2\right ) \sqrt {\cos (c+d x)}+2080 \left (11 a^3+6 a b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{3} \sqrt {\cos (c+d x)} \left (-77 b \left (624 a^2+73 b^2\right ) \cos (2 (c+d x))+154 b \left (-78 a^2+b^2\right ) \cos (4 (c+d x))+693 b^3 \cos (6 (c+d x))+156 a \left (506 a^2+213 b^2\right ) \sin (c+d x)+234 a \left (44 a^2-39 b^2\right ) \sin (3 (c+d x))-4914 a b^2 \sin (5 (c+d x))\right )\right )}{48048 d \cos ^{\frac {7}{2}}(c+d x)} \] Input:
Integrate[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3,x]
Output:
((e*Cos[c + d*x])^(7/2)*(-154*b*(78*a^2 + 11*b^2)*Sqrt[Cos[c + d*x]] + 208 0*(11*a^3 + 6*a*b^2)*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(-77* b*(624*a^2 + 73*b^2)*Cos[2*(c + d*x)] + 154*b*(-78*a^2 + b^2)*Cos[4*(c + d *x)] + 693*b^3*Cos[6*(c + d*x)] + 156*a*(506*a^2 + 213*b^2)*Sin[c + d*x] + 234*a*(44*a^2 - 39*b^2)*Sin[3*(c + d*x)] - 4914*a*b^2*Sin[5*(c + d*x)]))/ 3))/(48048*d*Cos[c + d*x]^(7/2))
Time = 1.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.98, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3171, 27, 3042, 3341, 27, 3042, 3148, 3042, 3115, 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.
| \(\displaystyle \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{13} \int \frac {1}{2} (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \left (13 a^2+17 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \left (13 a^2+17 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \left (13 a^2+17 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} (e \cos (c+d x))^{7/2} \left (13 a \left (11 a^2+6 b^2\right )+b \left (177 a^2+44 b^2\right ) \sin (c+d x)\right )dx-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int (e \cos (c+d x))^{7/2} \left (13 a \left (11 a^2+6 b^2\right )+b \left (177 a^2+44 b^2\right ) \sin (c+d x)\right )dx-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int (e \cos (c+d x))^{7/2} \left (13 a \left (11 a^2+6 b^2\right )+b \left (177 a^2+44 b^2\right ) \sin (c+d x)\right )dx-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \int (e \cos (c+d x))^{7/2}dx-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}dx-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \int (e \cos (c+d x))^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (13 a \left (11 a^2+6 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\) |
Input:
Int[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3,x]
Output:
(-2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^2)/(13*d*e) + ((-34*a*b* (e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x]))/(11*d*e) + ((-2*b*(177*a^2 + 44*b^2)*(e*Cos[c + d*x])^(9/2))/(9*d*e) + 13*a*(11*a^2 + 6*b^2)*((2*e*(e*C os[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + (5*e^2*((2*e^2*Sqrt[Cos[c + d*x]] *EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) + (2*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/7))/11)/13
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. \(513\) vs. \(2(212)=424\).
Time = 40.54 (sec) , antiderivative size = 514, normalized size of antiderivative = 2.17
| 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^{4} \left (48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+128 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-72 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+5 \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 {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \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 {13}{2}}}{13}-\frac {e^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{9}\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^{4} \left (672 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}-2352 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+3312 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-2400 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+922 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-159 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-5 \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 {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{77 \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 a^{2} b \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{3 d e}\) | \(514\) |
| default | \(\frac {2 e^{4} \left (381888 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a \,b^{2}-179712 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a \,b^{2}+36036 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a \,b^{2}-1170 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{2}+157248 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-393120 a \,b^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2145 \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 ) a^{3}-88704 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{15} b^{3}+308000 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} b^{3}-113960 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b^{3}+18172 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{3}+308 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{3}-308 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}+310464 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}-433664 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-1170 \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 ) a \,b^{2}-120120 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{2} b +30030 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2} b -3003 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b -20592 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a^{3}+30888 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{3}-24024 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a^{3}+6864 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{3}+96096 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} a^{2} b -240240 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} a^{2} b +240240 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a^{2} b \right )}{9009 \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}\) | \(618\) |
Input:
int((e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-2/21*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^4*(4 8*cos(1/2*d*x+1/2*c)^9-120*cos(1/2*d*x+1/2*c)^7+128*cos(1/2*d*x+1/2*c)^5-7 2*cos(1/2*d*x+1/2*c)^3+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2* c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+16*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+2*b^3/d/e^3*(1/13*(e*cos(d*x+c)) ^(13/2)-1/9*e^2*(e*cos(d*x+c))^(9/2))+4/77*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^4*(672*cos(1/2*d*x+1/2*c)^13-2352*cos(1 /2*d*x+1/2*c)^11+3312*cos(1/2*d*x+1/2*c)^9-2400*cos(1/2*d*x+1/2*c)^7+922*c os(1/2*d*x+1/2*c)^5-159*cos(1/2*d*x+1/2*c)^3-5*(sin(1/2*d*x+1/2*c)^2)^(1/2 )*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+ 5*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-2/3*a^2*b/d *(e*cos(d*x+c))^(9/2)/e
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.85 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=-\frac {2 \, {\left (195 i \, \sqrt {\frac {1}{2}} {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 195 i \, \sqrt {\frac {1}{2}} {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - {\left (693 \, b^{3} e^{3} \cos \left (d x + c\right )^{6} - 1001 \, {\left (3 \, a^{2} b + b^{3}\right )} e^{3} \cos \left (d x + c\right )^{4} - 39 \, {\left (63 \, a b^{2} e^{3} \cos \left (d x + c\right )^{4} - 3 \, {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{3} \cos \left (d x + c\right )^{2} - 5 \, {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{9009 \, d} \] Input:
integrate((e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-2/9009*(195*I*sqrt(1/2)*(11*a^3 + 6*a*b^2)*e^(7/2)*weierstrassPInverse(-4 , 0, cos(d*x + c) + I*sin(d*x + c)) - 195*I*sqrt(1/2)*(11*a^3 + 6*a*b^2)*e ^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - (693*b^ 3*e^3*cos(d*x + c)^6 - 1001*(3*a^2*b + b^3)*e^3*cos(d*x + c)^4 - 39*(63*a* b^2*e^3*cos(d*x + c)^4 - 3*(11*a^3 + 6*a*b^2)*e^3*cos(d*x + c)^2 - 5*(11*a ^3 + 6*a*b^2)*e^3)*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(7/2)*(a+b*sin(d*x+c))**3,x)
Output:
Timed out
\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^3,x, algorithm="maxima")
Output:
integrate((e*cos(d*x + c))^(7/2)*(b*sin(d*x + c) + a)^3, x)
\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
integrate((e*cos(d*x + c))^(7/2)*(b*sin(d*x + c) + a)^3, x)
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \] Input:
int((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^3,x)
Output:
int((e*cos(c + d*x))^(7/2)*(a + b*sin(c + d*x))^3, x)
\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx=\frac {\sqrt {e}\, e^{3} \left (-2 \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} a^{2} b +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{3}d x \right ) b^{3} d +9 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}d x \right ) a \,b^{2} d +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a^{3} d \right )}{3 d} \] Input:
int((e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^3,x)
Output:
(sqrt(e)*e**3*( - 2*sqrt(cos(c + d*x))*cos(c + d*x)**4*a**2*b + 3*int(sqrt (cos(c + d*x))*cos(c + d*x)**3*sin(c + d*x)**3,x)*b**3*d + 9*int(sqrt(cos( c + d*x))*cos(c + d*x)**3*sin(c + d*x)**2,x)*a*b**2*d + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a**3*d))/(3*d)