Integrand size = 25, antiderivative size = 154 \[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\frac {2 \left (9 a^2+2 b^2\right ) e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 d \sqrt {\sin (c+d x)}}-\frac {2 \left (9 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 d}+\frac {22 a b (e \sin (c+d x))^{7/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{9 d e} \] Output:
-2/15*(9*a^2+2*b^2)*e^2*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*si n(d*x+c))^(1/2)/d/sin(d*x+c)^(1/2)-2/45*(9*a^2+2*b^2)*e*cos(d*x+c)*(e*sin( d*x+c))^(3/2)/d+22/63*a*b*(e*sin(d*x+c))^(7/2)/d/e+2/9*b*(a+b*cos(d*x+c))* (e*sin(d*x+c))^(7/2)/d/e
Time = 1.43 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.75 \[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=-\frac {(e \sin (c+d x))^{5/2} \left (84 \left (9 a^2+2 b^2\right ) E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )+\left (21 \left (12 a^2+b^2\right ) \cos (c+d x)+5 b (-36 a+36 a \cos (2 (c+d x))+7 b \cos (3 (c+d x)))\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{630 d \sin ^{\frac {5}{2}}(c+d x)} \] Input:
Integrate[(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2),x]
Output:
-1/630*((e*Sin[c + d*x])^(5/2)*(84*(9*a^2 + 2*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + (21*(12*a^2 + b^2)*Cos[c + d*x] + 5*b*(-36*a + 36*a*Cos[2*( c + d*x)] + 7*b*Cos[3*(c + d*x)]))*Sin[c + d*x]^(3/2)))/(d*Sin[c + d*x]^(5 /2))
Time = 0.67 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3171, 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.
| \(\displaystyle \int (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{2} \left (9 a^2+11 b \cos (c+d x) a+2 b^2\right ) (e \sin (c+d x))^{5/2}dx+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \left (9 a^2+11 b \cos (c+d x) a+2 b^2\right ) (e \sin (c+d x))^{5/2}dx+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2} \left (9 a^2-11 b \sin \left (c+d x-\frac {\pi }{2}\right ) a+2 b^2\right )dx+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \int (e \sin (c+d x))^{5/2}dx+\frac {22 a b (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \int (e \sin (c+d x))^{5/2}dx+\frac {22 a b (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \left (\frac {3}{5} e^2 \int \sqrt {e \sin (c+d x)}dx-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}\right )+\frac {22 a b (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \left (\frac {3}{5} e^2 \int \sqrt {e \sin (c+d x)}dx-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}\right )+\frac {22 a b (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \left (\frac {3 e^2 \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{5 \sqrt {\sin (c+d x)}}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}\right )+\frac {22 a b (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \left (\frac {3 e^2 \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{5 \sqrt {\sin (c+d x)}}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}\right )+\frac {22 a b (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\left (9 a^2+2 b^2\right ) \left (\frac {6 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}\right )+\frac {22 a b (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}\) |
Input:
Int[(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2),x]
Output:
(2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2))/(9*d*e) + ((22*a*b*(e*Si n[c + d*x])^(7/2))/(7*d*e) + (9*a^2 + 2*b^2)*((6*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*d)))/9
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])
Leaf count of result is larger than twice the leaf count of optimal. \(331\) vs. \(2(136)=272\).
Time = 10.17 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.16
| method | result | size |
| default | \(\frac {\frac {4 a b \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7 e}-\frac {e^{3} \left (10 \sin \left (d x +c \right )^{6} b^{2}+54 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}+12 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-27 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}-6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-18 a^{2} \sin \left (d x +c \right )^{4}-14 \sin \left (d x +c \right )^{4} b^{2}+18 a^{2} \sin \left (d x +c \right )^{2}+4 b^{2} \sin \left (d x +c \right )^{2}\right )}{45 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(332\) |
| parts | \(-\frac {a^{2} e^{3} \left (6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{2}\right )}{5 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b^{2} e^{3} \left (5 \sin \left (d x +c \right )^{6}+6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-7 \sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{2}\right )}{45 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {4 a b \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d e}\) | \(340\) |
Input:
int((a+cos(d*x+c)*b)^2*(e*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
(4/7/e*a*b*(e*sin(d*x+c))^(7/2)-1/45*e^3*(10*sin(d*x+c)^6*b^2+54*(1-sin(d* x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticE((1-sin(d*x+c ))^(1/2),1/2*2^(1/2))*a^2+12*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*s in(d*x+c)^(1/2)*EllipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*b^2-27*(1-sin( d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x +c))^(1/2),1/2*2^(1/2))*a^2-6*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)* sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*b^2-18*a^2*si n(d*x+c)^4-14*sin(d*x+c)^4*b^2+18*a^2*sin(d*x+c)^2+4*b^2*sin(d*x+c)^2)/cos (d*x+c)/(e*sin(d*x+c))^(1/2))/d
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=-\frac {2 \, {\left (-21 i \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} \sqrt {-\frac {1}{2} i \, e} e^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} \sqrt {\frac {1}{2} i \, e} e^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (35 \, b^{2} e^{2} \cos \left (d x + c\right )^{3} + 90 \, a b e^{2} \cos \left (d x + c\right )^{2} - 90 \, a b e^{2} + 21 \, {\left (3 \, a^{2} - b^{2}\right )} e^{2} \cos \left (d x + c\right )\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d} \] Input:
integrate((a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(5/2),x, algorithm="fricas")
Output:
-2/315*(-21*I*(9*a^2 + 2*b^2)*sqrt(-1/2*I*e)*e^2*weierstrassZeta(4, 0, wei erstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*I*(9*a^2 + 2*b ^2)*sqrt(1/2*I*e)*e^2*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos( d*x + c) - I*sin(d*x + c))) + (35*b^2*e^2*cos(d*x + c)^3 + 90*a*b*e^2*cos( d*x + c)^2 - 90*a*b*e^2 + 21*(3*a^2 - b^2)*e^2*cos(d*x + c))*sqrt(e*sin(d* x + c))*sin(d*x + c))/d
Timed out. \[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**2*(e*sin(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(5/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^2*(e*sin(d*x + c))^(5/2), x)
\[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(5/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^2*(e*sin(d*x + c))^(5/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2 \,d x \] Input:
int((e*sin(c + d*x))^(5/2)*(a + b*cos(c + d*x))^2,x)
Output:
int((e*sin(c + d*x))^(5/2)*(a + b*cos(c + d*x))^2, x)
\[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\frac {\sqrt {e}\, e^{2} \left (4 \sqrt {\sin \left (d x +c \right )}\, \sin \left (d x +c \right )^{3} a b +7 \left (\int \sqrt {\sin \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}d x \right ) b^{2} d +7 \left (\int \sqrt {\sin \left (d x +c \right )}\, \sin \left (d x +c \right )^{2}d x \right ) a^{2} d \right )}{7 d} \] Input:
int((a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(5/2),x)
Output:
(sqrt(e)*e**2*(4*sqrt(sin(c + d*x))*sin(c + d*x)**3*a*b + 7*int(sqrt(sin(c + d*x))*cos(c + d*x)**2*sin(c + d*x)**2,x)*b**2*d + 7*int(sqrt(sin(c + d* x))*sin(c + d*x)**2,x)*a**2*d))/(7*d)