\(\int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx\) [246]

Optimal result

Integrand size = 25, antiderivative size = 79 \[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx=-\frac {6 e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^2 d \sqrt {\cos (c+d x)}}-\frac {4 e (e \cos (c+d x))^{3/2}}{d \left (a^2+a^2 \sin (c+d x)\right )} \] Output:

-6*e^2*(e*cos(d*x+c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^2/d/co 
s(d*x+c)^(1/2)-4*e*(e*cos(d*x+c))^(3/2)/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 = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84 \[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx=-\frac {2^{3/4} (e \cos (c+d x))^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {7}{4},\frac {11}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{7 a^2 d e (1+\sin (c+d x))^{7/4}} \] Input:

Integrate[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^2,x]
 

Output:

-1/7*(2^(3/4)*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[5/4, 7/4, 11/4, (1 
- Sin[c + d*x])/2])/(a^2*d*e*(1 + Sin[c + d*x])^(7/4))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3159, 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.

\(\displaystyle \int \frac {(e \cos (c+d x))^{5/2}}{(a \sin (c+d x)+a)^2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3159

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3121

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

Input:

Int[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^2,x]
 

Output:

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

Defintions of rubi rules used

rule 3042

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

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 = 16.64 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.52

method	result	size
default	\(\frac {2 \left (4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e^{3}}{\sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} d}\)	\(120\)

Input:

int((e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

2/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/sin(1/2*d*x+1/2*c)/a^2*(4*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))-2*sin(1/2*d*x+1/ 
2*c))*e^3/d
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.33 \[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx=-\frac {2 \, {\left (3 \, \sqrt {\frac {1}{2}} {\left (i \, e^{2} \cos \left (d x + c\right ) + i \, e^{2} \sin \left (d x + c\right ) + i \, e^{2}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {\frac {1}{2}} {\left (-i \, e^{2} \cos \left (d x + c\right ) - i \, e^{2} \sin \left (d x + c\right ) - i \, e^{2}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (e^{2} \cos \left (d x + c\right ) - e^{2} \sin \left (d x + c\right ) + e^{2}\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} d} \] Input:

integrate((e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c))^2,x, algorithm="fricas")
 

Output:

-2*(3*sqrt(1/2)*(I*e^2*cos(d*x + c) + I*e^2*sin(d*x + c) + I*e^2)*sqrt(e)* 
weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x 
 + c))) + 3*sqrt(1/2)*(-I*e^2*cos(d*x + c) - I*e^2*sin(d*x + c) - I*e^2)*s 
qrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I* 
sin(d*x + c))) + 2*(e^2*cos(d*x + c) - e^2*sin(d*x + c) + e^2)*sqrt(e*cos( 
d*x + c)))/(a^2*d*cos(d*x + c) + a^2*d*sin(d*x + c) + a^2*d)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate((e*cos(d*x+c))**(5/2)/(a+a*sin(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c))^2,x, algorithm="maxima")
 

Output:

integrate((e*cos(d*x + c))^(5/2)/(a*sin(d*x + c) + a)^2, x)
 

Giac [F]

\[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c))^2,x, algorithm="giac")
 

Output:

integrate((e*cos(d*x + c))^(5/2)/(a*sin(d*x + c) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \] Input:

int((e*cos(c + d*x))^(5/2)/(a + a*sin(c + d*x))^2,x)
 

Output:

int((e*cos(c + d*x))^(5/2)/(a + a*sin(c + d*x))^2, x)
 

Reduce [F]

\[ \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1}d x \right ) e^{2}}{a^{2}} \] Input:

int((e*cos(d*x+c))^(5/2)/(a+a*sin(d*x+c))^2,x)
                                                                                    
                                                                                    
 

Output:

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