Integrand size = 25, antiderivative size = 169 \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\frac {26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac {26 e^8 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d \sqrt {e \cos (c+d x)}}+\frac {26 e^7 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 a^3 d}+\frac {26 e^5 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^3 d}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a+a \sin (c+d x))^2} \] Output:
26/45*e^3*(e*cos(d*x+c))^(9/2)/a^3/d+26/21*e^8*cos(d*x+c)^(1/2)*InverseJac obiAM(1/2*d*x+1/2*c,2^(1/2))/a^3/d/(e*cos(d*x+c))^(1/2)+26/21*e^7*(e*cos(d *x+c))^(1/2)*sin(d*x+c)/a^3/d+26/35*e^5*(e*cos(d*x+c))^(5/2)*sin(d*x+c)/a^ 3/d+4/5*e*(e*cos(d*x+c))^(13/2)/a/d/(a+a*sin(d*x+c))^2
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.43 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.39 \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=-\frac {4 \sqrt [4]{2} (e \cos (c+d x))^{17/2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {17}{4},\frac {21}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{17 a^3 d e (1+\sin (c+d x))^{17/4}} \] Input:
Integrate[(e*Cos[c + d*x])^(15/2)/(a + a*Sin[c + d*x])^3,x]
Output:
(-4*2^(1/4)*(e*Cos[c + d*x])^(17/2)*Hypergeometric2F1[-1/4, 17/4, 21/4, (1 - Sin[c + d*x])/2])/(17*a^3*d*e*(1 + Sin[c + d*x])^(17/4))
Time = 0.76 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3159, 3042, 3161, 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.
| \(\displaystyle \int \frac {(e \cos (c+d x))^{15/2}}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e \cos (c+d x))^{15/2}}{(a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle \frac {13 e^2 \int \frac {(e \cos (c+d x))^{11/2}}{\sin (c+d x) a+a}dx}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13 e^2 \int \frac {(e \cos (c+d x))^{11/2}}{\sin (c+d x) a+a}dx}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle \frac {13 e^2 \left (\frac {e^2 \int (e \cos (c+d x))^{7/2}dx}{a}+\frac {2 e (e \cos (c+d x))^{9/2}}{9 a d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13 e^2 \left (\frac {e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}dx}{a}+\frac {2 e (e \cos (c+d x))^{9/2}}{9 a d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {13 e^2 \left (\frac {e^2 \left (\frac {5}{7} e^2 \int (e \cos (c+d x))^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{a}+\frac {2 e (e \cos (c+d x))^{9/2}}{9 a d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13 e^2 \left (\frac {e^2 \left (\frac {5}{7} e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{a}+\frac {2 e (e \cos (c+d x))^{9/2}}{9 a d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {13 e^2 \left (\frac {e^2 \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{a}+\frac {2 e (e \cos (c+d x))^{9/2}}{9 a d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13 e^2 \left (\frac {e^2 \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{a}+\frac {2 e (e \cos (c+d x))^{9/2}}{9 a d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {13 e^2 \left (\frac {e^2 \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{a}+\frac {2 e (e \cos (c+d x))^{9/2}}{9 a d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13 e^2 \left (\frac {e^2 \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{a}+\frac {2 e (e \cos (c+d x))^{9/2}}{9 a d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {13 e^2 \left (\frac {e^2 \left (\frac {5}{7} e^2 \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )}{a}+\frac {2 e (e \cos (c+d x))^{9/2}}{9 a d}\right )}{5 a^2}+\frac {4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}\) |
Input:
Int[(e*Cos[c + d*x])^(15/2)/(a + a*Sin[c + d*x])^3,x]
Output:
(4*e*(e*Cos[c + d*x])^(13/2))/(5*a*d*(a + a*Sin[c + d*x])^2) + (13*e^2*((2 *e*(e*Cos[c + d*x])^(9/2))/(9*a*d) + (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))/a))/(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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[g*((g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1))), x] + Si mp[g^2/a Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g}, x ] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
Time = 140.32 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(\frac {2 e^{8} \left (1120 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+2160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-2800 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-3240 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+784 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+840 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1624 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+120 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-195 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1162 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+217 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\) | \(251\) |
Input:
int((e*cos(d*x+c))^(15/2)/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
2/315/a^3/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^8*(1120 *sin(1/2*d*x+1/2*c)^11+2160*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-2800*s in(1/2*d*x+1/2*c)^9-3240*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+784*sin(1 /2*d*x+1/2*c)^7+840*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+1624*sin(1/2*d *x+1/2*c)^5+120*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)*EllipticF(cos(1/2*d*x+1/2* c),2^(1/2))-1162*sin(1/2*d*x+1/2*c)^3+217*sin(1/2*d*x+1/2*c))/d
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.75 \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 \, {\left (195 i \, \sqrt {\frac {1}{2}} e^{\frac {15}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 195 i \, \sqrt {\frac {1}{2}} e^{\frac {15}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (35 \, e^{7} \cos \left (d x + c\right )^{4} - 252 \, e^{7} \cos \left (d x + c\right )^{2} + 15 \, {\left (9 \, e^{7} \cos \left (d x + c\right )^{2} - 13 \, e^{7}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{315 \, a^{3} d} \] Input:
integrate((e*cos(d*x+c))^(15/2)/(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-2/315*(195*I*sqrt(1/2)*e^(15/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 195*I*sqrt(1/2)*e^(15/2)*weierstrassPInverse(-4, 0, cos (d*x + c) - I*sin(d*x + c)) + (35*e^7*cos(d*x + c)^4 - 252*e^7*cos(d*x + c )^2 + 15*(9*e^7*cos(d*x + c)^2 - 13*e^7)*sin(d*x + c))*sqrt(e*cos(d*x + c) ))/(a^3*d)
Timed out. \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(15/2)/(a+a*sin(d*x+c))**3,x)
Output:
Timed out
\[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {15}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate((e*cos(d*x+c))^(15/2)/(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
integrate((e*cos(d*x + c))^(15/2)/(a*sin(d*x + c) + a)^3, x)
\[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {15}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate((e*cos(d*x+c))^(15/2)/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
integrate((e*cos(d*x + c))^(15/2)/(a*sin(d*x + c) + a)^3, x)
Timed out. \[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{15/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \] Input:
int((e*cos(c + d*x))^(15/2)/(a + a*sin(c + d*x))^3,x)
Output:
int((e*cos(c + d*x))^(15/2)/(a + a*sin(c + d*x))^3, x)
\[ \int \frac {(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{7}}{\sin \left (d x +c \right )^{3}+3 \sin \left (d x +c \right )^{2}+3 \sin \left (d x +c \right )+1}d x \right ) e^{7}}{a^{3}} \] Input:
int((e*cos(d*x+c))^(15/2)/(a+a*sin(d*x+c))^3,x)
Output:
(sqrt(e)*int((sqrt(cos(c + d*x))*cos(c + d*x)**7)/(sin(c + d*x)**3 + 3*sin (c + d*x)**2 + 3*sin(c + d*x) + 1),x)*e**7)/a**3