Integrand size = 25, antiderivative size = 118 \[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^3} \, dx=\frac {6 e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^3 d \sqrt {\cos (c+d x)}}-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {6 e (e \cos (c+d x))^{3/2}}{5 d \left (a^3+a^3 \sin (c+d x)\right )} \]
-4/5*e*(e*cos(d*x+c))^(3/2)/a/d/(a+a*sin(d*x+c))^2+6/5*e*(e*cos(d*x+c))^(3 /2)/d/(a^3+a^3*sin(d*x+c))+6/5*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d* x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/a^3/d/ cos(d*x+c)^(1/2)
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 = 0.56 \[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^3} \, dx=-\frac {(e \cos (c+d x))^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {7}{4},\frac {9}{4},\frac {11}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{7 \sqrt [4]{2} a^3 d e (1+\sin (c+d x))^{7/4}} \]
-1/7*((e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[7/4, 9/4, 11/4, (1 - Sin[c + d*x])/2])/(2^(1/4)*a^3*d*e*(1 + Sin[c + d*x])^(7/4))
Time = 0.52 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3159, 3042, 3162, 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.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{5/2}}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{5/2}}{(a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle -\frac {3 e^2 \int \frac {\sqrt {e \cos (c+d x)}}{\sin (c+d x) a+a}dx}{5 a^2}-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 e^2 \int \frac {\sqrt {e \cos (c+d x)}}{\sin (c+d x) a+a}dx}{5 a^2}-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3162 |
| \(\displaystyle -\frac {3 e^2 \left (-\frac {\int \sqrt {e \cos (c+d x)}dx}{a}-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}\right )}{5 a^2}-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 e^2 \left (-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}\right )}{5 a^2}-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle -\frac {3 e^2 \left (-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{a \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}\right )}{5 a^2}-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 e^2 \left (-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\cos (c+d x)}}-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}\right )}{5 a^2}-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {3 e^2 \left (-\frac {2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{a d \sqrt {\cos (c+d x)}}\right )}{5 a^2}-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
(-4*e*(e*Cos[c + d*x])^(3/2))/(5*a*d*(a + a*Sin[c + d*x])^2) - (3*e^2*((-2 *Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]]) - (2*(e*Cos[c + d*x])^(3/2))/(d*e*(a + a*Sin[c + d*x]))))/(5*a^2)
3.3.58.3.1 Defintions of rubi rules used
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 _)])^(m_), x_Symbol] :> Simp[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[b*((g*Cos[e + f*x])^(p + 1)/(a*f*g*(p - 1)*(a + b*S in[e + f*x]))), x] + Simp[p/(a*(p - 1)) Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && !GeQ[p, 1] && Intege rQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(331\) vs. \(2(130)=260\).
Time = 185.83 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.81
| method | result | size |
| default | \(-\frac {2 \left (24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+20 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e^{3}}{5 \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(332\) |
-2/5/(4*sin(1/2*d*x+1/2*c)^4-4*sin(1/2*d*x+1/2*c)^2+1)/a^3/sin(1/2*d*x+1/2 *c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d *x+1/2*c)-12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c) ,2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x +1/2*c)^4*cos(1/2*d*x+1/2*c)+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE (cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2* c)^2-20*sin(1/2*d*x+1/2*c)^5-2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*( sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos (1/2*d*x+1/2*c),2^(1/2))+20*sin(1/2*d*x+1/2*c)^3-sin(1/2*d*x+1/2*c))*e^3/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.69 \[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^3} \, dx=\frac {3 \, {\left (i \, \sqrt {2} e^{2} \cos \left (d x + c\right )^{2} - i \, \sqrt {2} e^{2} \cos \left (d x + c\right ) - 2 i \, \sqrt {2} e^{2} + {\left (-i \, \sqrt {2} e^{2} \cos \left (d x + c\right ) - 2 i \, \sqrt {2} e^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (-i \, \sqrt {2} e^{2} \cos \left (d x + c\right )^{2} + i \, \sqrt {2} e^{2} \cos \left (d x + c\right ) + 2 i \, \sqrt {2} e^{2} + {\left (i \, \sqrt {2} e^{2} \cos \left (d x + c\right ) + 2 i \, \sqrt {2} e^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, e^{2} \cos \left (d x + c\right )^{2} + e^{2} \cos \left (d x + c\right ) - 2 \, e^{2} + {\left (3 \, e^{2} \cos \left (d x + c\right ) + 2 \, e^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{5 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d - {\left (a^{3} d \cos \left (d x + c\right ) + 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
1/5*(3*(I*sqrt(2)*e^2*cos(d*x + c)^2 - I*sqrt(2)*e^2*cos(d*x + c) - 2*I*sq rt(2)*e^2 + (-I*sqrt(2)*e^2*cos(d*x + c) - 2*I*sqrt(2)*e^2)*sin(d*x + c))* sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I *sin(d*x + c))) + 3*(-I*sqrt(2)*e^2*cos(d*x + c)^2 + I*sqrt(2)*e^2*cos(d*x + c) + 2*I*sqrt(2)*e^2 + (I*sqrt(2)*e^2*cos(d*x + c) + 2*I*sqrt(2)*e^2)*s in(d*x + c))*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos (d*x + c) - I*sin(d*x + c))) - 2*(3*e^2*cos(d*x + c)^2 + e^2*cos(d*x + c) - 2*e^2 + (3*e^2*cos(d*x + c) + 2*e^2)*sin(d*x + c))*sqrt(e*cos(d*x + c))) /(a^3*d*cos(d*x + c)^2 - a^3*d*cos(d*x + c) - 2*a^3*d - (a^3*d*cos(d*x + c ) + 2*a^3*d)*sin(d*x + c))
Timed out. \[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^3} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
\[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^3} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^3} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]