Integrand size = 25, antiderivative size = 168 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=-\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {130 a^2 e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {130 a^2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {26 a^2 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e} \]
-26/99*a^2*(e*cos(d*x+c))^(9/2)/d/e+26/77*a^2*e*(e*cos(d*x+c))^(5/2)*sin(d *x+c)/d-2/11*(e*cos(d*x+c))^(9/2)*(a^2+a^2*sin(d*x+c))/d/e+130/231*a^2*e^4 *(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)+130/231*a^2*e^3*sin(d *x+c)*(e*cos(d*x+c))^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=-\frac {32 \sqrt [4]{2} a^2 (e \cos (c+d x))^{9/2} \operatorname {Hypergeometric2F1}\left (-\frac {13}{4},\frac {9}{4},\frac {13}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{9 d e (1+\sin (c+d x))^{9/4}} \]
(-32*2^(1/4)*a^2*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[-13/4, 9/4, 13/4 , (1 - Sin[c + d*x])/2])/(9*d*e*(1 + Sin[c + d*x])^(9/4))
Time = 0.74 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3157, 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.
| \(\displaystyle \int (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{7/2}dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {13}{11} a \int (e \cos (c+d x))^{7/2} (\sin (c+d x) a+a)dx-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13}{11} a \int (e \cos (c+d x))^{7/2} (\sin (c+d x) a+a)dx-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {13}{11} a \left (a \int (e \cos (c+d x))^{7/2}dx-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13}{11} a \left (a \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}dx-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {13}{11} a \left (a \left (\frac {5}{7} e^2 \int (e \cos (c+d x))^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13}{11} a \left (a \left (\frac {5}{7} e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {13}{11} a \left (a \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13}{11} a \left (a \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {13}{11} a \left (a \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13}{11} a \left (a \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {13}{11} a \left (a \left (\frac {5}{7} e^2 \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}\) |
(-2*(e*Cos[c + d*x])^(9/2)*(a^2 + a^2*Sin[c + d*x]))/(11*d*e) + (13*a*((-2 *a*(e*Cos[c + d*x])^(9/2))/(9*d*e) + a*((2*e*(e*Cos[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
3.3.4.3.1 Defintions of rubi rules used
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[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Time = 18.39 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.76
| method | result | size |
| default | \(\frac {2 a^{2} e^{4} \left (4032 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10080 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+4928 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8208 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12320 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2232 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12320 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-924 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-6160 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+498 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-195 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1540 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-154 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{693 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(295\) |
| parts | \(-\frac {2 a^{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^{4} \left (48 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-72 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \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 )+16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \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 {4 a^{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^{4} \left (672 \left (\cos ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2352 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3312 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2400 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+922 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-159 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \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 )+5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{231 \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 {4 a^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{9 d e}\) | \(473\) |
2/693/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a^2*e^4*(4032 *cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12-10080*sin(1/2*d*x+1/2*c)^10*cos( 1/2*d*x+1/2*c)+4928*sin(1/2*d*x+1/2*c)^11+8208*cos(1/2*d*x+1/2*c)*sin(1/2* d*x+1/2*c)^8-12320*sin(1/2*d*x+1/2*c)^9-2232*sin(1/2*d*x+1/2*c)^6*cos(1/2* d*x+1/2*c)+12320*sin(1/2*d*x+1/2*c)^7-924*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x +1/2*c)-6160*sin(1/2*d*x+1/2*c)^5+498*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2 *c)-195*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Elli pticF(cos(1/2*d*x+1/2*c),2^(1/2))+1540*sin(1/2*d*x+1/2*c)^3-154*sin(1/2*d* x+1/2*c))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.85 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\frac {-195 i \, \sqrt {2} a^{2} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 i \, \sqrt {2} a^{2} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (154 \, a^{2} e^{3} \cos \left (d x + c\right )^{4} + 3 \, {\left (21 \, a^{2} e^{3} \cos \left (d x + c\right )^{4} - 39 \, a^{2} e^{3} \cos \left (d x + c\right )^{2} - 65 \, a^{2} e^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{693 \, d} \]
1/693*(-195*I*sqrt(2)*a^2*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 195*I*sqrt(2)*a^2*e^(7/2)*weierstrassPInverse(-4, 0, c os(d*x + c) - I*sin(d*x + c)) - 2*(154*a^2*e^3*cos(d*x + c)^4 + 3*(21*a^2* e^3*cos(d*x + c)^4 - 39*a^2*e^3*cos(d*x + c)^2 - 65*a^2*e^3)*sin(d*x + c)) *sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
\[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
\[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]