Integrand size = 23, antiderivative size = 95 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx=-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac {2 a e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 a e \sqrt {e \cos (c+d x)} \sin (c+d x)}{3 d} \]
-2/5*a*(e*cos(d*x+c))^(5/2)/d/e+2/3*a*e^2*(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)+2/3*a*e*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 = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx=-\frac {8 \sqrt [4]{2} a (e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e (1+\sin (c+d x))^{5/4}} \]
(-8*2^(1/4)*a*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-5/4, 5/4, 9/4, (1 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(5/4))
Time = 0.41 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 3148, 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.
\(\displaystyle \int (a \sin (c+d x)+a) (e \cos (c+d x))^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \sin (c+d x)+a) (e \cos (c+d x))^{3/2}dx\) |
\(\Big \downarrow \) 3148 |
\(\displaystyle a \int (e \cos (c+d x))^{3/2}dx-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle a \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle a \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle a \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\) |
(-2*a*(e*Cos[c + d*x])^(5/2))/(5*d*e) + a*((2*e^2*Sqrt[Cos[c + d*x]]*Ellip ticF[(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))
3.2.98.3.1 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 = 2.95 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.88
method | result | size |
default | \(-\frac {2 a \,e^{2} \left (-24 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+36 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-18 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(179\) |
parts | \(-\frac {2 a \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{2} \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {2 a \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d e}\) | \(211\) |
-2/15/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a*e^2*(-24*si n(1/2*d*x+1/2*c)^7+20*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+36*sin(1/2*d *x+1/2*c)^5-10*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+5*(sin(1/2*d*x+1/2* c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c), 2^(1/2))-18*sin(1/2*d*x+1/2*c)^3+3*sin(1/2*d*x+1/2*c))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx=\frac {-5 i \, \sqrt {2} a e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} a e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (3 \, a e \cos \left (d x + c\right )^{2} - 5 \, a e \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{15 \, d} \]
1/15*(-5*I*sqrt(2)*a*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*s in(d*x + c)) + 5*I*sqrt(2)*a*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*(3*a*e*cos(d*x + c)^2 - 5*a*e*sin(d*x + c))*sqrt( e*cos(d*x + c)))/d
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )} \,d x } \]
\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )} \,d x } \]
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (a+a\,\sin \left (c+d\,x\right )\right ) \,d x \]