\(\int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx\) [244]

Optimal result

Integrand size = 25, antiderivative size = 114 \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx=\frac {14 e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d \sqrt {\cos (c+d x)}}+\frac {14 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^2 d}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2+a^2 \sin (c+d x)\right )} \] Output:

14/5*e^4*(e*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^2/d/ 
cos(d*x+c)^(1/2)+14/15*e^3*(e*cos(d*x+c))^(3/2)*sin(d*x+c)/a^2/d+4/3*e*(e* 
cos(d*x+c))^(7/2)/d/(a^2+a^2*sin(d*x+c))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.58 \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx=-\frac {2\ 2^{3/4} (e \cos (c+d x))^{11/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{4},\frac {15}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{11 a^2 d e (1+\sin (c+d x))^{11/4}} \] Input:

Integrate[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^2,x]
 

Output:

(-2*2^(3/4)*(e*Cos[c + d*x])^(11/2)*Hypergeometric2F1[1/4, 11/4, 15/4, (1 
- Sin[c + d*x])/2])/(11*a^2*d*e*(1 + Sin[c + d*x])^(11/4))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.03, 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, 3115, 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.

\(\displaystyle \int \frac {(e \cos (c+d x))^{9/2}}{(a \sin (c+d x)+a)^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(e \cos (c+d x))^{9/2}}{(a \sin (c+d x)+a)^2}dx\)

\(\Big \downarrow \) 3159

\(\displaystyle \frac {7 e^2 \int (e \cos (c+d x))^{5/2}dx}{3 a^2}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2 \sin (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7 e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx}{3 a^2}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2 \sin (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {7 e^2 \left (\frac {3}{5} e^2 \int \sqrt {e \cos (c+d x)}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )}{3 a^2}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2 \sin (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7 e^2 \left (\frac {3}{5} e^2 \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )}{3 a^2}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2 \sin (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 3121

\(\displaystyle \frac {7 e^2 \left (\frac {3 e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{5 \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )}{3 a^2}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2 \sin (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7 e^2 \left (\frac {3 e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )}{3 a^2}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2 \sin (c+d x)+a^2\right )}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {7 e^2 \left (\frac {6 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )}{3 a^2}+\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2 \sin (c+d x)+a^2\right )}\)

Input:

Int[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^2,x]
 

Output:

(4*e*(e*Cos[c + d*x])^(7/2))/(3*d*(a^2 + a^2*Sin[c + d*x])) + (7*e^2*((6*e 
^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]] 
) + (2*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)))/(3*a^2)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

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]
 

rule 3119

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]
 

rule 3121

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]
 

rule 3159

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]
 

Maple [A] (verified)

Time = 15.52 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.67

method	result	size
default	\(-\frac {2 e^{5} \left (24 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-24 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-40 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+6 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+40 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-10 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\)	\(190\)

Input:

int((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

-2/15/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^5*(24*s 
in(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x 
+1/2*c)-40*sin(1/2*d*x+1/2*c)^5+6*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)- 
21*(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))+40*sin(1/2*d*x+1/2*c)^3-10*sin(1/2*d*x+1/2*c) 
)/d
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94 \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx=-\frac {2 \, {\left (-21 i \, \sqrt {\frac {1}{2}} e^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {\frac {1}{2}} e^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (3 \, e^{4} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 10 \, e^{4} \cos \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{15 \, a^{2} d} \] Input:

integrate((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c))^2,x, algorithm="fricas")
 

Output:

-2/15*(-21*I*sqrt(1/2)*e^(9/2)*weierstrassZeta(-4, 0, weierstrassPInverse( 
-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*I*sqrt(1/2)*e^(9/2)*weierstras 
sZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 
(3*e^4*cos(d*x + c)*sin(d*x + c) - 10*e^4*cos(d*x + c))*sqrt(e*cos(d*x + c 
)))/(a^2*d)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate((e*cos(d*x+c))**(9/2)/(a+a*sin(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c))^2,x, algorithm="maxima")
 

Output:

integrate((e*cos(d*x + c))^(9/2)/(a*sin(d*x + c) + a)^2, x)
 

Giac [F]

\[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c))^2,x, algorithm="giac")
 

Output:

integrate((e*cos(d*x + c))^(9/2)/(a*sin(d*x + c) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \] Input:

int((e*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x))^2,x)
 

Output:

int((e*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x))^2, x)
 

Reduce [F]

\[ \int \frac {(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1}d x \right ) e^{4}}{a^{2}} \] Input:

int((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c))^2,x)
 

Output:

(sqrt(e)*int((sqrt(cos(c + d*x))*cos(c + d*x)**4)/(sin(c + d*x)**2 + 2*sin 
(c + d*x) + 1),x)*e**4)/a**2