Integrand size = 25, antiderivative size = 145 \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx=\frac {6 e^6 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 a^2 d \sqrt {e \cos (c+d x)}}+\frac {6 e^5 \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 a^2 d}+\frac {18 e^3 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^2 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2+a^2 \sin (c+d x)\right )} \] Output:
6/7*e^6*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/a^2/d/(e*c os(d*x+c))^(1/2)+6/7*e^5*(e*cos(d*x+c))^(1/2)*sin(d*x+c)/a^2/d+18/35*e^3*( e*cos(d*x+c))^(5/2)*sin(d*x+c)/a^2/d+4/5*e*(e*cos(d*x+c))^(9/2)/d/(a^2+a^2 *sin(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.46 \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx=-\frac {4 \sqrt [4]{2} (e \cos (c+d x))^{13/2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {13}{4},\frac {17}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{13 a^2 d e (1+\sin (c+d x))^{13/4}} \] Input:
Integrate[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^2,x]
Output:
(-4*2^(1/4)*(e*Cos[c + d*x])^(13/2)*Hypergeometric2F1[-1/4, 13/4, 17/4, (1 - Sin[c + d*x])/2])/(13*a^2*d*e*(1 + Sin[c + d*x])^(13/4))
Time = 0.60 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3159, 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.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{11/2}}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{11/2}}{(a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle \frac {9 e^2 \int (e \cos (c+d x))^{7/2}dx}{5 a^2}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}dx}{5 a^2}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \int (e \cos (c+d x))^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {9 e^2 \left (\frac {5}{7} e^2 \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )}\) |
Input:
Int[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^2,x]
Output:
(4*e*(e*Cos[c + d*x])^(9/2))/(5*d*(a^2 + a^2*Sin[c + d*x])) + (9*e^2*((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))/(5*a^2)
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 _)])^(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]
Time = 19.49 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.40
| method | result | size |
| default | \(\frac {2 e^{6} \left (80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-120 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-112 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+168 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+20 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-84 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+14 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\) | \(203\) |
Input:
int((e*cos(d*x+c))^(11/2)/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
2/35/a^2/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^6*(80*co s(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-120*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x +1/2*c)-112*sin(1/2*d*x+1/2*c)^7+168*sin(1/2*d*x+1/2*c)^5+20*sin(1/2*d*x+1 /2*c)^2*cos(1/2*d*x+1/2*c)-15*(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))-84*sin(1/2*d*x+1/2 *c)^3+14*sin(1/2*d*x+1/2*c))/d
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.78 \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {\frac {1}{2}} e^{\frac {11}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {\frac {1}{2}} e^{\frac {11}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - {\left (14 \, e^{5} \cos \left (d x + c\right )^{2} - 5 \, {\left (e^{5} \cos \left (d x + c\right )^{2} - 3 \, e^{5}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{35 \, a^{2} d} \] Input:
integrate((e*cos(d*x+c))^(11/2)/(a+a*sin(d*x+c))^2,x, algorithm="fricas")
Output:
-2/35*(15*I*sqrt(1/2)*e^(11/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I *sin(d*x + c)) - 15*I*sqrt(1/2)*e^(11/2)*weierstrassPInverse(-4, 0, cos(d* x + c) - I*sin(d*x + c)) - (14*e^5*cos(d*x + c)^2 - 5*(e^5*cos(d*x + c)^2 - 3*e^5)*sin(d*x + c))*sqrt(e*cos(d*x + c)))/(a^2*d)
Timed out. \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(11/2)/(a+a*sin(d*x+c))**2,x)
Output:
Timed out
\[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*cos(d*x+c))^(11/2)/(a+a*sin(d*x+c))^2,x, algorithm="maxima")
Output:
integrate((e*cos(d*x + c))^(11/2)/(a*sin(d*x + c) + a)^2, x)
\[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*cos(d*x+c))^(11/2)/(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
integrate((e*cos(d*x + c))^(11/2)/(a*sin(d*x + c) + a)^2, x)
Timed out. \[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{11/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \] Input:
int((e*cos(c + d*x))^(11/2)/(a + a*sin(c + d*x))^2,x)
Output:
int((e*cos(c + d*x))^(11/2)/(a + a*sin(c + d*x))^2, x)
\[ \int \frac {(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}}{\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1}d x \right ) e^{5}}{a^{2}} \] Input:
int((e*cos(d*x+c))^(11/2)/(a+a*sin(d*x+c))^2,x)
Output:
(sqrt(e)*int((sqrt(cos(c + d*x))*cos(c + d*x)**5)/(sin(c + d*x)**2 + 2*sin (c + d*x) + 1),x)*e**5)/a**2