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

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 7.90 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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 = 12.75 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\sin ^{2}{\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \]

[In]

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

[Out]

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 = 1.19 (sec) , antiderivative size = 1388, normalized size of antiderivative = 49.57 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

-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_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_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_integr
al_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_integra
l_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_integral_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_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_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^3*x + a*d*e^2*f^2 + (a*d*f^4*x^2 + 2*a*d*e*f^3*x + a*d*e^2*f^2
)*cos(d*x + c)^2 + (a*d*f^4*x^2 + 2*a*d*e*f^3*x + a*d*e^2*f^2)*sin(d*x + c)^2 + 2*(a*d*f^4*x^2 + 2*a*d*e*f^3*x
+ a*d*e^2*f^2)*sin(d*x + c))*integrate(cos(d*x + c)/(a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3
+ (a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3)*cos(d*x + c)^2 + (a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 +
3*a*d*e^2*f*x + a*d*e^3)*sin(d*x + c)^2 + 2*(a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3)*sin(d*x
+ c)), x) - 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_integral_e(2, -(I*d*f*x + I*d
*e)/f))*sin(-(d*e - c*f)/f) + 2*d*f)*x)*sin(d*x + c))/(a*d*f^3*x^2 + 2*a*d*e*f^2*x + a*d*e^2*f + (a*d*f^3*x^2
+ 2*a*d*e*f^2*x + a*d*e^2*f)*cos(d*x + c)^2 + (a*d*f^3*x^2 + 2*a*d*e*f^2*x + a*d*e^2*f)*sin(d*x + c)^2 + 2*(a*
d*f^3*x^2 + 2*a*d*e*f^2*x + a*d*e^2*f)*sin(d*x + c))

Giac [N/A]

Not integrable

Time = 0.79 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^2(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^2}{{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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