\(\int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx\) [111]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx=\text {Int}\left (\frac {1}{(c+d x)^2 (a+a \sin (e+f x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx \]

[In]

Int[1/((c + d*x)^2*(a + a*Sin[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)^2*(a + a*Sin[e + f*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.92 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx \]

[In]

Integrate[1/((c + d*x)^2*(a + a*Sin[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)^2*(a + a*Sin[e + f*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d x +c \right )^{2} \left (a +a \sin \left (f x +e \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)^2/(a+a*sin(f*x+e)),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.90 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx=\frac {\int \frac {1}{c^{2} \sin {\left (e + f x \right )} + c^{2} + 2 c d x \sin {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \sin {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a} \]

[In]

integrate(1/(d*x+c)**2/(a+a*sin(f*x+e)),x)

[Out]

Integral(1/(c**2*sin(e + f*x) + c**2 + 2*c*d*x*sin(e + f*x) + 2*c*d*x + d**2*x**2*sin(e + f*x) + d**2*x**2), x
)/a

Maxima [N/A]

Not integrable

Time = 0.74 (sec) , antiderivative size = 442, normalized size of antiderivative = 22.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)^2/(a+a*sin(f*x+e)),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)^2/(a+a*sin(f*x+e)),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))} \, dx=\int \frac {1}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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