\(\int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))^2} \, dx\) [116]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 10.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))^2} \, dx=\int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 102, normalized size of antiderivative = 5.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(1/(2*a^2*d^2*x^2 + 4*a^2*c*d*x + 2*a^2*c^2 - (a^2*d^2*x^2 + 2*a^2*c*d*x + a^2*c^2)*cos(f*x + e)^2 + 2
*(a^2*d^2*x^2 + 2*a^2*c*d*x + a^2*c^2)*sin(f*x + e)), x)

Sympy [N/A]

Not integrable

Time = 6.43 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.25 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))^2} \, dx=\frac {\int \frac {1}{c^{2} \sin ^{2}{\left (e + f x \right )} + 2 c^{2} \sin {\left (e + f x \right )} + c^{2} + 2 c d x \sin ^{2}{\left (e + f x \right )} + 4 c d x \sin {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \sin ^{2}{\left (e + f x \right )} + 2 d^{2} x^{2} \sin {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{2}} \]

[In]

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

[Out]

Integral(1/(c**2*sin(e + f*x)**2 + 2*c**2*sin(e + f*x) + c**2 + 2*c*d*x*sin(e + f*x)**2 + 4*c*d*x*sin(e + f*x)
+ 2*c*d*x + d**2*x**2*sin(e + f*x)**2 + 2*d**2*x**2*sin(e + f*x) + d**2*x**2), x)/a**2

Maxima [N/A]

Not integrable

Time = 16.29 (sec) , antiderivative size = 3521, normalized size of antiderivative = 176.05 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

1/3*(12*(d^2*f*x + c*d*f)*cos(2*f*x + 2*e)^2 - 12*d^2*cos(f*x + e) + 12*(d^2*f*x + c*d*f)*cos(f*x + e)^2 + 12*
(d^2*f*x + c*d*f)*sin(2*f*x + 2*e)^2 + 12*(d^2*f*x + c*d*f)*sin(f*x + e)^2 + 2*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^
2*f^2 - 6*d^2*cos(2*f*x + 2*e) + 6*d^2 - 2*(d^2*f*x + c*d*f)*cos(f*x + e) - 2*(d^2*f*x + c*d*f)*sin(2*f*x + 2*
e) + 3*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2 + 4*d^2)*sin(f*x + e))*cos(3*f*x + 3*e) - 2*(2*d^2*f*x + 2*c*d*f +
9*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2 + 2*d^2)*cos(f*x + e) + 12*(d^2*f*x + c*d*f)*sin(f*x + e))*cos(2*f*x +
2*e) - 3*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3 + (
a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*cos(3*f*x + 3
*e)^2 + 9*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*co
s(2*f*x + 2*e)^2 + 9*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*
c^4*f^3)*cos(f*x + e)^2 + (a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x +
a^2*c^4*f^3)*sin(3*f*x + 3*e)^2 + 18*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c
^3*d*f^3*x + a^2*c^4*f^3)*cos(f*x + e)*sin(2*f*x + 2*e) + 9*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2
*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*sin(2*f*x + 2*e)^2 + 9*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3
+ 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*sin(f*x + e)^2 - 6*((a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*
f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*cos(f*x + e) + (a^2*d^4*f^3*x^4 + 4*a^2*c*d
^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - 6*(
a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3 + 3*(a^2*d^4*f
^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*sin(f*x + e))*cos(2*f*
x + 2*e) - 2*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3
- 3*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*cos(2*f*
x + 2*e) + 3*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)
*sin(f*x + e))*sin(3*f*x + 3*e) + 6*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3
*d*f^3*x + a^2*c^4*f^3)*sin(f*x + e))*integrate(4/3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + 12*d^3)*cos(f*x
+ e)/(a^2*d^5*f^3*x^5 + 5*a^2*c*d^4*f^3*x^4 + 10*a^2*c^2*d^3*f^3*x^3 + 10*a^2*c^3*d^2*f^3*x^2 + 5*a^2*c^4*d*f
^3*x + a^2*c^5*f^3 + (a^2*d^5*f^3*x^5 + 5*a^2*c*d^4*f^3*x^4 + 10*a^2*c^2*d^3*f^3*x^3 + 10*a^2*c^3*d^2*f^3*x^2
+ 5*a^2*c^4*d*f^3*x + a^2*c^5*f^3)*cos(f*x + e)^2 + (a^2*d^5*f^3*x^5 + 5*a^2*c*d^4*f^3*x^4 + 10*a^2*c^2*d^3*f^
3*x^3 + 10*a^2*c^3*d^2*f^3*x^2 + 5*a^2*c^4*d*f^3*x + a^2*c^5*f^3)*sin(f*x + e)^2 + 2*(a^2*d^5*f^3*x^5 + 5*a^2*
c*d^4*f^3*x^4 + 10*a^2*c^2*d^3*f^3*x^3 + 10*a^2*c^3*d^2*f^3*x^2 + 5*a^2*c^4*d*f^3*x + a^2*c^5*f^3)*sin(f*x + e
)), x) - 2*(6*d^2*sin(2*f*x + 2*e) - 2*(d^2*f*x + c*d*f)*cos(2*f*x + 2*e) + 3*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2
*f^2 + 4*d^2)*cos(f*x + e) + 2*(d^2*f*x + c*d*f)*sin(f*x + e))*sin(3*f*x + 3*e) - 6*(d^2*f^2*x^2 + 2*c*d*f^2*x
+ c^2*f^2 + 4*d^2 - 4*(d^2*f*x + c*d*f)*cos(f*x + e) + 3*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2 + 2*d^2)*sin(f*
x + e))*sin(2*f*x + 2*e) + 4*(d^2*f*x + c*d*f)*sin(f*x + e))/(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^
2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3 + (a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x
^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*cos(3*f*x + 3*e)^2 + 9*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^
2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*cos(2*f*x + 2*e)^2 + 9*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3
+ 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*cos(f*x + e)^2 + (a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^
3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*sin(3*f*x + 3*e)^2 + 18*(a^2*d^4*f^3*x^4 + 4*
a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*cos(f*x + e)*sin(2*f*x + 2*e) + 9
*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*sin(2*f*x +
2*e)^2 + 9*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*
sin(f*x + e)^2 - 6*((a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c
^4*f^3)*cos(f*x + e) + (a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^
2*c^4*f^3)*sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - 6*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x
^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3 + 3*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^
2*c^3*d*f^3*x + a^2*c^4*f^3)*sin(f*x + e))*cos(2*f*x + 2*e) - 2*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2
*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3 - 3*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*
f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*cos(2*f*x + 2*e) + 3*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2
*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*sin(f*x + e))*sin(3*f*x + 3*e) + 6*(a^2*d^4*f^3*x^4 + 4*a^
2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*sin(f*x + e))

Giac [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sin (e+f x))^2} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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