\(\int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\) [190]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\text {Int}\left (\frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))},x\right ) \] Output:

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

Mathematica [N/A]

Not integrable

Time = 12.50 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \] Input:

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

Output:

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

Rubi [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5048}

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.

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

\(\Big \downarrow \) 5048

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 5048

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ 
.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Unintegrable[((e + f*x)^m*F[c + d 
*x]^n)/(a + b*Sin[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && Tr 
igQ[F]
 

Maple [N/A]

Not integrable

Time = 1.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

Input:

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

Output:

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

Fricas [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \] Input:

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

Output:

integral(-(cos(d*x + c)^2 - 1)/(a*f^2*x^2 + 2*a*e*f*x + a*e^2 + (a*f^2*x^2 
 + 2*a*e*f*x + a*e^2)*sin(d*x + c)), x)
 

Sympy [N/A]

Not integrable

Time = 13.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\sin ^{2}{\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \] Input:

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

Output:

Integral(sin(c + d*x)**2/(e**2*sin(c + d*x) + e**2 + 2*e*f*x*sin(c + d*x) 
+ 2*e*f*x + f**2*x**2*sin(c + d*x) + f**2*x**2), x)/a
 

Maxima [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 1388, normalized size of antiderivative = 49.57 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \] Input:

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

Output:

-1/2*(d*e*(I*exp_integral_e(2, (I*d*f*x + I*d*e)/f) - I*exp_integral_e(2, 
-(I*d*f*x + I*d*e)/f))*cos(-(d*e - c*f)/f) + d*e*(exp_integral_e(2, (I*d*f 
*x + I*d*e)/f) + exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*sin(-(d*e - c*f) 
/f) + (d*e*(I*exp_integral_e(2, (I*d*f*x + I*d*e)/f) - I*exp_integral_e(2, 
 -(I*d*f*x + I*d*e)/f))*cos(-(d*e - c*f)/f) + d*e*(exp_integral_e(2, (I*d* 
f*x + I*d*e)/f) + exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*sin(-(d*e - c*f 
)/f) - 2*d*e + (d*f*(I*exp_integral_e(2, (I*d*f*x + I*d*e)/f) - I*exp_inte 
gral_e(2, -(I*d*f*x + I*d*e)/f))*cos(-(d*e - c*f)/f) + d*f*(exp_integral_e 
(2, (I*d*f*x + I*d*e)/f) + exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*sin(-( 
d*e - c*f)/f) - 2*d*f)*x)*cos(d*x + c)^2 + (d*e*(I*exp_integral_e(2, (I*d* 
f*x + I*d*e)/f) - I*exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*cos(-(d*e - c 
*f)/f) + d*e*(exp_integral_e(2, (I*d*f*x + I*d*e)/f) + exp_integral_e(2, - 
(I*d*f*x + I*d*e)/f))*sin(-(d*e - c*f)/f) - 2*d*e + (d*f*(I*exp_integral_e 
(2, (I*d*f*x + I*d*e)/f) - I*exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*cos( 
-(d*e - c*f)/f) + d*f*(exp_integral_e(2, (I*d*f*x + I*d*e)/f) + exp_integr 
al_e(2, -(I*d*f*x + I*d*e)/f))*sin(-(d*e - c*f)/f) - 2*d*f)*x)*sin(d*x + c 
)^2 - 2*d*e + (d*f*(I*exp_integral_e(2, (I*d*f*x + I*d*e)/f) - I*exp_integ 
ral_e(2, -(I*d*f*x + I*d*e)/f))*cos(-(d*e - c*f)/f) + d*f*(exp_integral_e( 
2, (I*d*f*x + I*d*e)/f) + exp_integral_e(2, -(I*d*f*x + I*d*e)/f))*sin(-(d 
*e - c*f)/f) - 2*d*f)*x + 4*f*cos(d*x + c) + 8*(a*d*f^4*x^2 + 2*a*d*e*f...
 

Giac [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 35.71 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^2}{{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\sin \left (d x +c \right )^{2}}{\sin \left (d x +c \right ) e^{2}+2 \sin \left (d x +c \right ) e f x +\sin \left (d x +c \right ) f^{2} x^{2}+e^{2}+2 e f x +f^{2} x^{2}}d x}{a} \] Input:

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

Output:

int(sin(c + d*x)**2/(sin(c + d*x)*e**2 + 2*sin(c + d*x)*e*f*x + sin(c + d* 
x)*f**2*x**2 + e**2 + 2*e*f*x + f**2*x**2),x)/a