Integrand size = 23, antiderivative size = 124 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {10 a e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {10 a e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \]
-2/9*a*(e*cos(d*x+c))^(9/2)/d/e+2/7*a*e*(e*cos(d*x+c))^(5/2)*sin(d*x+c)/d+ 10/21*a*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)+10/21*a*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 = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.52 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=-\frac {16 \sqrt [4]{2} a (e \cos (c+d x))^{9/2} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {9}{4},\frac {13}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{9 d e (1+\sin (c+d x))^{9/4}} \]
(-16*2^(1/4)*a*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[-9/4, 9/4, 13/4, ( 1 - Sin[c + d*x])/2])/(9*d*e*(1 + Sin[c + d*x])^(9/4))
Time = 0.55 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {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.
| \(\displaystyle \int (a \sin (c+d x)+a) (e \cos (c+d x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (c+d x)+a) (e \cos (c+d x))^{7/2}dx\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle a \int (e \cos (c+d x))^{7/2}dx-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}dx-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {5}{7} e^2 \int (e \cos (c+d x))^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5}{7} e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle a \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle a \left (\frac {5}{7} e^2 \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}\) |
(-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)
3.2.96.3.1 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])
Time = 5.84 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.86
| method | result | size |
| parts | \(-\frac {2 a \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{4} \left (48 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+128 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-72 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {2 a \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{9 d e}\) | \(231\) |
| default | \(-\frac {2 a \,e^{4} \left (-224 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+144 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+560 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-216 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-560 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+168 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+280 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-70 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(249\) |
-2/21*a*(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*(48* 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)^5-72* 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-2/9*a*(e*cos(d*x+c))^(9/2)/d/e
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\frac {-15 i \, \sqrt {2} a e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} a e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (7 \, a e^{3} \cos \left (d x + c\right )^{4} - 3 \, {\left (3 \, a e^{3} \cos \left (d x + c\right )^{2} + 5 \, a e^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{63 \, d} \]
1/63*(-15*I*sqrt(2)*a*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I* sin(d*x + c)) + 15*I*sqrt(2)*a*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*(7*a*e^3*cos(d*x + c)^4 - 3*(3*a*e^3*cos(d*x + c)^2 + 5*a*e^3)*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
\[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )} \,d x } \]
\[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+a\,\sin \left (c+d\,x\right )\right ) \,d x \]