\(\int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx\) [1516]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx=\text {Int}\left (\frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2},x\right ) \] Output:

Defer(Int)(cos(f*x+e)^2*(c+d*sin(f*x+e))^(4/3)/(a+b*sin(f*x+e))^2,x)
 

Mathematica [N/A]

Not integrable

Time = 80.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx=\int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx \] Input:

Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x]) 
^2,x]
 

Output:

Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x]) 
^2, x]
 

Rubi [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {3042, 3404}

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.

\(\displaystyle \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (e+f x)^2 (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2}dx\)

\(\Big \downarrow \) 3404

\(\displaystyle \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2}dx\)

Input:

Int[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^(4/3))/(a + b*Sin[e + f*x])^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 3042

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

rule 3404

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Unin 
tegrable[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, 
x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[a^2 - b^2, 0]
 

Maple [N/A]

Not integrable

Time = 2.80 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94

\[\int \frac {\cos \left (f x +e \right )^{2} \left (c +d \sin \left (f x +e \right )\right )^{\frac {4}{3}}}{\left (a +b \sin \left (f x +e \right )\right )^{2}}d x\]

Input:

int(cos(f*x+e)^2*(c+d*sin(f*x+e))^(4/3)/(a+b*sin(f*x+e))^2,x)
 

Output:

int(cos(f*x+e)^2*(c+d*sin(f*x+e))^(4/3)/(a+b*sin(f*x+e))^2,x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)^2*(c+d*sin(f*x+e))^(4/3)/(a+b*sin(f*x+e))^2,x, algori 
thm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**2*(c+d*sin(f*x+e))**(4/3)/(a+b*sin(f*x+e))**2,x)
 

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 13.78 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(cos(f*x+e)^2*(c+d*sin(f*x+e))^(4/3)/(a+b*sin(f*x+e))^2,x, algori 
thm="maxima")
 

Output:

integrate((d*sin(f*x + e) + c)^(4/3)*cos(f*x + e)^2/(b*sin(f*x + e) + a)^2 
, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cos(f*x+e)^2*(c+d*sin(f*x+e))^(4/3)/(a+b*sin(f*x+e))^2,x, algori 
thm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[0,1,0]%%%} / %%%{1,[0,0,2]%%%} Error: Bad Argument Valu 
e
 

Mupad [N/A]

Not integrable

Time = 20.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2} \,d x \] Input:

int((cos(e + f*x)^2*(c + d*sin(e + f*x))^(4/3))/(a + b*sin(e + f*x))^2,x)
 

Output:

int((cos(e + f*x)^2*(c + d*sin(e + f*x))^(4/3))/(a + b*sin(e + f*x))^2, x)
 

Reduce [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.23 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^{4/3}}{(a+b \sin (e+f x))^2} \, dx=\left (\int \frac {\left (\sin \left (f x +e \right ) d +c \right )^{\frac {1}{3}} \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{2} b^{2}+2 \sin \left (f x +e \right ) a b +a^{2}}d x \right ) d +\left (\int \frac {\left (\sin \left (f x +e \right ) d +c \right )^{\frac {1}{3}} \cos \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{2} b^{2}+2 \sin \left (f x +e \right ) a b +a^{2}}d x \right ) c \] Input:

int(cos(f*x+e)^2*(c+d*sin(f*x+e))^(4/3)/(a+b*sin(f*x+e))^2,x)
 

Output:

int(((sin(e + f*x)*d + c)**(1/3)*cos(e + f*x)**2*sin(e + f*x))/(sin(e + f* 
x)**2*b**2 + 2*sin(e + f*x)*a*b + a**2),x)*d + int(((sin(e + f*x)*d + c)** 
(1/3)*cos(e + f*x)**2)/(sin(e + f*x)**2*b**2 + 2*sin(e + f*x)*a*b + a**2), 
x)*c