Integrand size = 25, antiderivative size = 258 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx=-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{3003 d e}+\frac {2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 d \sqrt {\cos (c+d x)}}+\frac {2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{195 d}-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e} \] Output:
-10/3003*a*b*(115*a^2+94*b^2)*(e*cos(d*x+c))^(7/2)/d/e+2/65*(39*a^4+52*a^2 *b^2+4*b^4)*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)+2/195*(39*a^4+52*a^2*b^2+4*b^4)*e*(e*cos(d*x+c))^(3/2) *sin(d*x+c)/d-2/429*b*(73*a^2+22*b^2)*(e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c) )/d/e-38/143*a*b*(e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^2/d/e-2/13*b*(e*cos (d*x+c))^(7/2)*(a+b*sin(d*x+c))^3/d/e
Time = 3.67 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.81 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx=\frac {(e \cos (c+d x))^{5/2} \left (2 \left (39 a^4+52 a^2 b^2+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+65 \sqrt {\cos (c+d x)} \left (-\frac {1}{77} a b \left (66 a^2+31 b^2\right ) \cos (c+d x)-\frac {1}{154} a b \left (44 a^2+9 b^2\right ) \cos (3 (c+d x))+\frac {1}{22} a b^3 \cos (5 (c+d x))+\frac {\left (624 a^4-208 a^2 b^2-61 b^4\right ) \sin (2 (c+d x))}{3120}-\frac {1}{78} b^2 \left (13 a^2+b^2\right ) \sin (4 (c+d x))+\frac {1}{208} b^4 \sin (6 (c+d x))\right )\right )}{65 d \cos ^{\frac {5}{2}}(c+d x)} \] Input:
Integrate[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^4,x]
Output:
((e*Cos[c + d*x])^(5/2)*(2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*EllipticE[(c + d* x)/2, 2] + 65*Sqrt[Cos[c + d*x]]*(-1/77*(a*b*(66*a^2 + 31*b^2)*Cos[c + d*x ]) - (a*b*(44*a^2 + 9*b^2)*Cos[3*(c + d*x)])/154 + (a*b^3*Cos[5*(c + d*x)] )/22 + ((624*a^4 - 208*a^2*b^2 - 61*b^4)*Sin[2*(c + d*x)])/3120 - (b^2*(13 *a^2 + b^2)*Sin[4*(c + d*x)])/78 + (b^4*Sin[6*(c + d*x)])/208)))/(65*d*Cos [c + d*x]^(5/2))
Time = 1.37 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3042, 3171, 27, 3042, 3341, 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.
| \(\displaystyle \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{13} \int \frac {1}{2} (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \left (13 a^2+19 b \sin (c+d x) a+6 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \left (13 a^2+19 b \sin (c+d x) a+6 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \left (13 a^2+19 b \sin (c+d x) a+6 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (a \left (143 a^2+142 b^2\right )+3 b \left (73 a^2+22 b^2\right ) \sin (c+d x)\right )dx-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (a \left (143 a^2+142 b^2\right )+3 b \left (73 a^2+22 b^2\right ) \sin (c+d x)\right )dx-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (a \left (143 a^2+142 b^2\right )+3 b \left (73 a^2+22 b^2\right ) \sin (c+d x)\right )dx-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {3}{2} (e \cos (c+d x))^{5/2} \left (11 \left (39 a^4+52 b^2 a^2+4 b^4\right )+5 a b \left (115 a^2+94 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{3} \int (e \cos (c+d x))^{5/2} \left (11 \left (39 a^4+52 b^2 a^2+4 b^4\right )+5 a b \left (115 a^2+94 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{3} \int (e \cos (c+d x))^{5/2} \left (11 \left (39 a^4+52 b^2 a^2+4 b^4\right )+5 a b \left (115 a^2+94 b^2\right ) \sin (c+d x)\right )dx-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{3} \left (11 \left (39 a^4+52 a^2 b^2+4 b^4\right ) \int (e \cos (c+d x))^{5/2}dx-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{3} \left (11 \left (39 a^4+52 a^2 b^2+4 b^4\right ) \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{3} \left (11 \left (39 a^4+52 a^2 b^2+4 b^4\right ) \left (\frac {3}{5} e^2 \int \sqrt {e \cos (c+d x)}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{3} \left (11 \left (39 a^4+52 a^2 b^2+4 b^4\right ) \left (\frac {3}{5} e^2 \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{3} \left (11 \left (39 a^4+52 a^2 b^2+4 b^4\right ) \left (\frac {3 e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{5 \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{3} \left (11 \left (39 a^4+52 a^2 b^2+4 b^4\right ) \left (\frac {3 e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{3} \left (11 \left (39 a^4+52 a^2 b^2+4 b^4\right ) \left (\frac {6 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )-\frac {10 a b \left (115 a^2+94 b^2\right ) (e \cos (c+d x))^{7/2}}{7 d e}\right )-\frac {2 b \left (73 a^2+22 b^2\right ) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{3 d e}\right )-\frac {38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\right )-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}\) |
Input:
Int[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^4,x]
Output:
(-2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3)/(13*d*e) + ((-38*a*b* (e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2)/(11*d*e) + ((-2*b*(73*a^2 + 22*b^2)*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(3*d*e) + ((-10*a*b* (115*a^2 + 94*b^2)*(e*Cos[c + d*x])^(7/2))/(7*d*e) + 11*(39*a^4 + 52*a^2*b ^2 + 4*b^4)*((6*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*S qrt[Cos[c + d*x]]) + (2*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)))/3)/ 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[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. \(757\) vs. \(2(234)=468\).
Time = 146.19 (sec) , antiderivative size = 758, normalized size of antiderivative = 2.94
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(758\) |
| default | \(\text {Expression too large to display}\) | \(776\) |
Input:
int((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
-2/5*a^4*(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-8/195*b^4*(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*(480*cos(1/2*d*x+1/2*c)^15-19 20*cos(1/2*d*x+1/2*c)^13+3040*cos(1/2*d*x+1/2*c)^11-2400*cos(1/2*d*x+1/2*c )^9+958*cos(1/2*d*x+1/2*c)^7-156*cos(1/2*d*x+1/2*c)^5-5*cos(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)*Ellipt icE(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+8*a*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))+8/15*b^2*a^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*cos(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*cos(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(co s(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-8/7*a^3*b/d*(e*cos(d*x+c))^(7/2)/e
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.92 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx=-\frac {2 \, {\left (-231 i \, \sqrt {\frac {1}{2}} {\left (39 \, a^{4} + 52 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {\frac {1}{2}} {\left (39 \, a^{4} + 52 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (5460 \, a b^{3} e^{2} \cos \left (d x + c\right )^{5} - 8580 \, {\left (a^{3} b + a b^{3}\right )} e^{2} \cos \left (d x + c\right )^{3} + 77 \, {\left (15 \, b^{4} e^{2} \cos \left (d x + c\right )^{5} - 5 \, {\left (26 \, a^{2} b^{2} + 5 \, b^{4}\right )} e^{2} \cos \left (d x + c\right )^{3} + {\left (39 \, a^{4} + 52 \, a^{2} b^{2} + 4 \, b^{4}\right )} e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{15015 \, d} \] Input:
integrate((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^4,x, algorithm="fricas")
Output:
-2/15015*(-231*I*sqrt(1/2)*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e^(5/2)*weierstra ssZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*I*sqrt(1/2)*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (5460*a*b ^3*e^2*cos(d*x + c)^5 - 8580*(a^3*b + a*b^3)*e^2*cos(d*x + c)^3 + 77*(15*b ^4*e^2*cos(d*x + c)^5 - 5*(26*a^2*b^2 + 5*b^4)*e^2*cos(d*x + c)^3 + (39*a^ 4 + 52*a^2*b^2 + 4*b^4)*e^2*cos(d*x + c))*sin(d*x + c))*sqrt(e*cos(d*x + c )))/d
Timed out. \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(5/2)*(a+b*sin(d*x+c))**4,x)
Output:
Timed out
\[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \] Input:
integrate((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^4,x, algorithm="maxima")
Output:
integrate((e*cos(d*x + c))^(5/2)*(b*sin(d*x + c) + a)^4, x)
\[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \] Input:
integrate((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^4,x, algorithm="giac")
Output:
integrate((e*cos(d*x + c))^(5/2)*(b*sin(d*x + c) + a)^4, x)
Timed out. \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4 \,d x \] Input:
int((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^4,x)
Output:
int((e*cos(c + d*x))^(5/2)*(a + b*sin(c + d*x))^4, x)
\[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx=\frac {\sqrt {e}\, e^{2} \left (-8 \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} a^{3} b +7 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{4}d x \right ) b^{4} d +28 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{3}d x \right ) a \,b^{3} d +42 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}d x \right ) a^{2} b^{2} d +7 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{4} d \right )}{7 d} \] Input:
int((e*cos(d*x+c))^(5/2)*(a+b*sin(d*x+c))^4,x)
Output:
(sqrt(e)*e**2*( - 8*sqrt(cos(c + d*x))*cos(c + d*x)**3*a**3*b + 7*int(sqrt (cos(c + d*x))*cos(c + d*x)**2*sin(c + d*x)**4,x)*b**4*d + 28*int(sqrt(cos (c + d*x))*cos(c + d*x)**2*sin(c + d*x)**3,x)*a*b**3*d + 42*int(sqrt(cos(c + d*x))*cos(c + d*x)**2*sin(c + d*x)**2,x)*a**2*b**2*d + 7*int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a**4*d))/(7*d)