\(\int \frac {\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\) [189]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int \frac {\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int \frac {\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (f x + e\right )} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.67 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\frac {\int \frac {\sin ^{2}{\left (c + d x \right )}}{e \sin {\left (c + d x \right )} + e + f x \sin {\left (c + d x \right )} + f x}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 1266, normalized size of antiderivative = 45.21 \[ \int \frac {\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (f x + e\right )} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

-1/2*(d*e*(I*exp_integral_e(1, (I*d*f*x + I*d*e)/f) - I*exp_integral_e(1, -(I*d*f*x + I*d*e)/f))*cos(-(d*e - c
*f)/f) + d*e*(exp_integral_e(1, (I*d*f*x + I*d*e)/f) + exp_integral_e(1, -(I*d*f*x + I*d*e)/f))*sin(-(d*e - c*
f)/f) + (d*e*(I*exp_integral_e(1, (I*d*f*x + I*d*e)/f) - I*exp_integral_e(1, -(I*d*f*x + I*d*e)/f))*cos(-(d*e
- c*f)/f) + d*e*(exp_integral_e(1, (I*d*f*x + I*d*e)/f) + exp_integral_e(1, -(I*d*f*x + I*d*e)/f))*sin(-(d*e -
c*f)/f) + (d*f*(I*exp_integral_e(1, (I*d*f*x + I*d*e)/f) - I*exp_integral_e(1, -(I*d*f*x + I*d*e)/f))*cos(-(d
*e - c*f)/f) + d*f*(exp_integral_e(1, (I*d*f*x + I*d*e)/f) + exp_integral_e(1, -(I*d*f*x + I*d*e)/f))*sin(-(d*
e - c*f)/f))*x)*cos(d*x + c)^2 + (d*e*(I*exp_integral_e(1, (I*d*f*x + I*d*e)/f) - I*exp_integral_e(1, -(I*d*f*
x + I*d*e)/f))*cos(-(d*e - c*f)/f) + d*e*(exp_integral_e(1, (I*d*f*x + I*d*e)/f) + exp_integral_e(1, -(I*d*f*x
+ I*d*e)/f))*sin(-(d*e - c*f)/f) + (d*f*(I*exp_integral_e(1, (I*d*f*x + I*d*e)/f) - I*exp_integral_e(1, -(I*d
*f*x + I*d*e)/f))*cos(-(d*e - c*f)/f) + d*f*(exp_integral_e(1, (I*d*f*x + I*d*e)/f) + exp_integral_e(1, -(I*d*
f*x + I*d*e)/f))*sin(-(d*e - c*f)/f))*x)*sin(d*x + c)^2 + (d*f*(I*exp_integral_e(1, (I*d*f*x + I*d*e)/f) - I*e
xp_integral_e(1, -(I*d*f*x + I*d*e)/f))*cos(-(d*e - c*f)/f) + d*f*(exp_integral_e(1, (I*d*f*x + I*d*e)/f) + ex
p_integral_e(1, -(I*d*f*x + I*d*e)/f))*sin(-(d*e - c*f)/f))*x + 4*f*cos(d*x + c) + 4*(a*d*f^3*x + a*d*e*f^2 +
(a*d*f^3*x + a*d*e*f^2)*cos(d*x + c)^2 + (a*d*f^3*x + a*d*e*f^2)*sin(d*x + c)^2 + 2*(a*d*f^3*x + a*d*e*f^2)*si
n(d*x + c))*integrate(cos(d*x + c)/(a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2 + (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2
)*cos(d*x + c)^2 + (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*sin(d*x + c)^2 + 2*(a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e
^2)*sin(d*x + c)), x) + 2*(d*f*x + (d*f*x + d*e)*cos(d*x + c)^2 + (d*f*x + d*e)*sin(d*x + c)^2 + d*e + 2*(d*f*
x + d*e)*sin(d*x + c))*log(f*x + e) - 2*(d*e*(-I*exp_integral_e(1, (I*d*f*x + I*d*e)/f) + I*exp_integral_e(1,
-(I*d*f*x + I*d*e)/f))*cos(-(d*e - c*f)/f) - d*e*(exp_integral_e(1, (I*d*f*x + I*d*e)/f) + exp_integral_e(1, -
(I*d*f*x + I*d*e)/f))*sin(-(d*e - c*f)/f) + (d*f*(-I*exp_integral_e(1, (I*d*f*x + I*d*e)/f) + I*exp_integral_e
(1, -(I*d*f*x + I*d*e)/f))*cos(-(d*e - c*f)/f) - d*f*(exp_integral_e(1, (I*d*f*x + I*d*e)/f) + exp_integral_e(
1, -(I*d*f*x + I*d*e)/f))*sin(-(d*e - c*f)/f))*x)*sin(d*x + c))/(a*d*f^2*x + a*d*e*f + (a*d*f^2*x + a*d*e*f)*c
os(d*x + c)^2 + (a*d*f^2*x + a*d*e*f)*sin(d*x + c)^2 + 2*(a*d*f^2*x + a*d*e*f)*sin(d*x + c))

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (f x + e\right )} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^2}{\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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