Integrand size = 25, antiderivative size = 193 \[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx=\frac {10 \left (11 a^2+2 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{231 d \sqrt {e \sin (c+d x)}}-\frac {10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e} \] Output:
10/231*(11*a^2+2*b^2)*e^4*InverseJacobiAM(1/2*c-1/4*Pi+1/2*d*x,2^(1/2))*si n(d*x+c)^(1/2)/d/(e*sin(d*x+c))^(1/2)-10/231*(11*a^2+2*b^2)*e^3*cos(d*x+c) *(e*sin(d*x+c))^(1/2)/d-2/77*(11*a^2+2*b^2)*e*cos(d*x+c)*(e*sin(d*x+c))^(5 /2)/d+26/99*a*b*(e*sin(d*x+c))^(9/2)/d/e+2/11*b*(a+b*cos(d*x+c))*(e*sin(d* x+c))^(9/2)/d/e
Time = 2.78 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.81 \[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx=\frac {\left (\frac {1}{6} \left (924 a b-6 \left (506 a^2+71 b^2\right ) \cos (c+d x)-1232 a b \cos (2 (c+d x))+396 a^2 \cos (3 (c+d x))-117 b^2 \cos (3 (c+d x))+308 a b \cos (4 (c+d x))+63 b^2 \cos (5 (c+d x))\right ) \csc ^3(c+d x)-\frac {40 \left (11 a^2+2 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )}{\sin ^{\frac {7}{2}}(c+d x)}\right ) (e \sin (c+d x))^{7/2}}{924 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2),x]
Output:
((((924*a*b - 6*(506*a^2 + 71*b^2)*Cos[c + d*x] - 1232*a*b*Cos[2*(c + d*x) ] + 396*a^2*Cos[3*(c + d*x)] - 117*b^2*Cos[3*(c + d*x)] + 308*a*b*Cos[4*(c + d*x)] + 63*b^2*Cos[5*(c + d*x)])*Csc[c + d*x]^3)/6 - (40*(11*a^2 + 2*b^ 2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2])/Sin[c + d*x]^(7/2))*(e*Sin[c + d*x ])^(7/2))/(924*d)
Time = 0.80 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.95, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3171, 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.
| \(\displaystyle \int (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} \left (11 a^2+13 b \cos (c+d x) a+2 b^2\right ) (e \sin (c+d x))^{7/2}dx+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \left (11 a^2+13 b \cos (c+d x) a+2 b^2\right ) (e \sin (c+d x))^{7/2}dx+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2} \left (11 a^2-13 b \sin \left (c+d x-\frac {\pi }{2}\right ) a+2 b^2\right )dx+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{11} \left (\left (11 a^2+2 b^2\right ) \int (e \sin (c+d x))^{7/2}dx+\frac {26 a b (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\left (11 a^2+2 b^2\right ) \int (e \sin (c+d x))^{7/2}dx+\frac {26 a b (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\left (11 a^2+2 b^2\right ) \left (\frac {5}{7} e^2 \int (e \sin (c+d x))^{3/2}dx-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {26 a b (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\left (11 a^2+2 b^2\right ) \left (\frac {5}{7} e^2 \int (e \sin (c+d x))^{3/2}dx-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {26 a b (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\left (11 a^2+2 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin (c+d x)}}dx-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {26 a b (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\left (11 a^2+2 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin (c+d x)}}dx-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {26 a b (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{11} \left (\left (11 a^2+2 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{3 \sqrt {e \sin (c+d x)}}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {26 a b (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\left (11 a^2+2 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{3 \sqrt {e \sin (c+d x)}}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {26 a b (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\left (11 a^2+2 b^2\right ) \left (\frac {5}{7} e^2 \left (\frac {2 e^2 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}\right )+\frac {26 a b (e \sin (c+d x))^{9/2}}{9 d e}\right )+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}\) |
Input:
Int[(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2),x]
Output:
(2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(9/2))/(11*d*e) + ((26*a*b*(e*S in[c + d*x])^(9/2))/(9*d*e) + (11*a^2 + 2*b^2)*((-2*e*Cos[c + d*x]*(e*Sin[ c + d*x])^(5/2))/(7*d) + (5*e^2*((2*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*S qrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) - (2*e*Cos[c + d*x]*Sqrt[e*S in[c + d*x]])/(3*d)))/7))/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[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])
Time = 11.24 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\frac {\frac {4 a b \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9 e}-\frac {e^{4} \left (-42 b^{2} \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )-66 a^{2} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+72 b^{2} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+55 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}+10 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}+176 a^{2} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )-10 b^{2} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )\right )}{231 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(252\) |
| parts | \(-\frac {a^{2} e^{4} \left (-6 \sin \left (d x +c \right )^{5}+5 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-4 \sin \left (d x +c \right )^{3}+10 \sin \left (d x +c \right )\right )}{21 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b^{2} e^{4} \left (21 \sin \left (d x +c \right )^{7}-27 \sin \left (d x +c \right )^{5}+5 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-4 \sin \left (d x +c \right )^{3}+10 \sin \left (d x +c \right )\right )}{231 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {4 a b \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9 d e}\) | \(252\) |
Input:
int((a+cos(d*x+c)*b)^2*(e*sin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
Output:
(4/9/e*a*b*(e*sin(d*x+c))^(9/2)-1/231*e^4*(-42*b^2*cos(d*x+c)^6*sin(d*x+c) -66*a^2*cos(d*x+c)^4*sin(d*x+c)+72*b^2*cos(d*x+c)^4*sin(d*x+c)+55*(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+10*(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+176*a^2*c os(d*x+c)^2*sin(d*x+c)-10*b^2*cos(d*x+c)^2*sin(d*x+c))/cos(d*x+c)/(e*sin(d *x+c))^(1/2))/d
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.01 \[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx=\frac {2 \, {\left (15 \, {\left (11 \, a^{2} + 2 \, b^{2}\right )} \sqrt {-\frac {1}{2} i \, e} e^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, {\left (11 \, a^{2} + 2 \, b^{2}\right )} \sqrt {\frac {1}{2} i \, e} e^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (63 \, b^{2} e^{3} \cos \left (d x + c\right )^{5} + 154 \, a b e^{3} \cos \left (d x + c\right )^{4} - 308 \, a b e^{3} \cos \left (d x + c\right )^{2} + 9 \, {\left (11 \, a^{2} - 12 \, b^{2}\right )} e^{3} \cos \left (d x + c\right )^{3} + 154 \, a b e^{3} - 3 \, {\left (88 \, a^{2} - 5 \, b^{2}\right )} e^{3} \cos \left (d x + c\right )\right )} \sqrt {e \sin \left (d x + c\right )}\right )}}{693 \, d} \] Input:
integrate((a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(7/2),x, algorithm="fricas")
Output:
2/693*(15*(11*a^2 + 2*b^2)*sqrt(-1/2*I*e)*e^3*weierstrassPInverse(4, 0, co s(d*x + c) + I*sin(d*x + c)) + 15*(11*a^2 + 2*b^2)*sqrt(1/2*I*e)*e^3*weier strassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + (63*b^2*e^3*cos(d*x + c)^5 + 154*a*b*e^3*cos(d*x + c)^4 - 308*a*b*e^3*cos(d*x + c)^2 + 9*(11*a ^2 - 12*b^2)*e^3*cos(d*x + c)^3 + 154*a*b*e^3 - 3*(88*a^2 - 5*b^2)*e^3*cos (d*x + c))*sqrt(e*sin(d*x + c)))/d
Timed out. \[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**2*(e*sin(d*x+c))**(7/2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(7/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^2*(e*sin(d*x + c))^(7/2), x)
\[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(7/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^2*(e*sin(d*x + c))^(7/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2 \,d x \] Input:
int((e*sin(c + d*x))^(7/2)*(a + b*cos(c + d*x))^2,x)
Output:
int((e*sin(c + d*x))^(7/2)*(a + b*cos(c + d*x))^2, x)
\[ \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx=\frac {\sqrt {e}\, e^{3} \left (4 \sqrt {\sin \left (d x +c \right )}\, \sin \left (d x +c \right )^{4} a b +9 \left (\int \sqrt {\sin \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{3}d x \right ) b^{2} d +9 \left (\int \sqrt {\sin \left (d x +c \right )}\, \sin \left (d x +c \right )^{3}d x \right ) a^{2} d \right )}{9 d} \] Input:
int((a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(7/2),x)
Output:
(sqrt(e)*e**3*(4*sqrt(sin(c + d*x))*sin(c + d*x)**4*a*b + 9*int(sqrt(sin(c + d*x))*cos(c + d*x)**2*sin(c + d*x)**3,x)*b**2*d + 9*int(sqrt(sin(c + d* x))*sin(c + d*x)**3,x)*a**2*d))/(9*d)