\(\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx\) [115]
Optimal result
Integrand size = 20, antiderivative size = 20 \[
\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx=\text {Int}\left (\frac {1}{(c+d x) (a+a \sin (e+f x))^2},x\right )
\]
[Out]
Unintegrable(1/(d*x+c)/(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 = {} \[
\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx=\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx
\]
[In]
Int[1/((c + d*x)*(a + a*Sin[e + f*x])^2),x]
[Out]
Defer[Int][1/((c + d*x)*(a + a*Sin[e + f*x])^2), x]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx \\
\end{align*}
Mathematica [N/A]
Not integrable
Time = 9.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10
\[
\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx=\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx
\]
[In]
Integrate[1/((c + d*x)*(a + a*Sin[e + f*x])^2),x]
[Out]
Integrate[1/((c + d*x)*(a + a*Sin[e + f*x])^2), x]
Maple [N/A] (verified)
Not integrable
Time = 0.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \frac {1}{\left (d x +c \right ) \left (a +a \sin \left (f x +e \right )\right )^{2}}d x\]
[In]
int(1/(d*x+c)/(a+a*sin(f*x+e))^2,x)
[Out]
int(1/(d*x+c)/(a+a*sin(f*x+e))^2,x)
Fricas [N/A]
Not integrable
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.00
\[
\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x }
\]
[In]
integrate(1/(d*x+c)/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
[Out]
integral(1/(2*a^2*d*x + 2*a^2*c - (a^2*d*x + a^2*c)*cos(f*x + e)^2 + 2*(a^2*d*x + a^2*c)*sin(f*x + e)), x)
Sympy [N/A]
Not integrable
Time = 2.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.70
\[
\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx=\frac {\int \frac {1}{c \sin ^{2}{\left (e + f x \right )} + 2 c \sin {\left (e + f x \right )} + c + d x \sin ^{2}{\left (e + f x \right )} + 2 d x \sin {\left (e + f x \right )} + d x}\, dx}{a^{2}}
\]
[In]
integrate(1/(d*x+c)/(a+a*sin(f*x+e))**2,x)
[Out]
Integral(1/(c*sin(e + f*x)**2 + 2*c*sin(e + f*x) + c + d*x*sin(e + f*x)**2 + 2*d*x*sin(e + f*x) + d*x), x)/a**
2
Maxima [N/A]
Not integrable
Time = 7.98 (sec) , antiderivative size = 2914, normalized size of antiderivative = 145.70
\[
\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x }
\]
[In]
integrate(1/(d*x+c)/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
[Out]
1/3*(6*(d^2*f*x + c*d*f)*cos(2*f*x + 2*e)^2 - 4*d^2*cos(f*x + e) + 6*(d^2*f*x + c*d*f)*cos(f*x + e)^2 + 6*(d^2
*f*x + c*d*f)*sin(2*f*x + 2*e)^2 + 6*(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
- 2*d^2*cos(2*f*x + 2*e) + 2*d^2 - (d^2*f*x + c*d*f)*cos(f*x + e) - (d^2*f*x + c*d*f)*sin(2*f*x + 2*e) + (3*d
^2*f^2*x^2 + 6*c*d*f^2*x + 3*c^2*f^2 + 4*d^2)*sin(f*x + e))*cos(3*f*x + 3*e) - 2*(d^2*f*x + c*d*f + 3*(3*d^2*f
^2*x^2 + 6*c*d*f^2*x + 3*c^2*f^2 + 2*d^2)*cos(f*x + e) + 6*(d^2*f*x + c*d*f)*sin(f*x + e))*cos(2*f*x + 2*e) -
3*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3 + (a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^
3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*cos(3*f*x + 3*e)^2 + 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2
*c^2*d*f^3*x + a^2*c^3*f^3)*cos(2*f*x + 2*e)^2 + 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x
+ a^2*c^3*f^3)*cos(f*x + e)^2 + (a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(
3*f*x + 3*e)^2 + 18*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*cos(f*x + e)*sin
(2*f*x + 2*e) + 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(2*f*x + 2*e)^2
+ 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(f*x + e)^2 - 6*((a^2*d^3*f^
3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*cos(f*x + e) + (a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f
^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - 6*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^
2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3 + 3*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a
^2*c^3*f^3)*sin(f*x + e))*cos(2*f*x + 2*e) - 2*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^
2*c^3*f^3 - 3*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*cos(2*f*x + 2*e) + 3*(
a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(f*x + e))*sin(3*f*x + 3*e) + 6*(a
^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(f*x + e))*integrate(2/3*(d^3*f^2*x
^2 + 2*c*d^2*f^2*x + c^2*d*f^2 + 6*d^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 + (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(f*x + e)^2 + 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(f*x + e)), x) - 2*(2*d^2*sin(2*f*x + 2*e) - (d^2*f*x + c*d*f)*co
s(2*f*x + 2*e) + (3*d^2*f^2*x^2 + 6*c*d*f^2*x + 3*c^2*f^2 + 4*d^2)*cos(f*x + e) + (d^2*f*x + c*d*f)*sin(f*x +
e))*sin(3*f*x + 3*e) - 2*(3*d^2*f^2*x^2 + 6*c*d*f^2*x + 3*c^2*f^2 + 4*d^2 - 6*(d^2*f*x + c*d*f)*cos(f*x + e) +
3*(3*d^2*f^2*x^2 + 6*c*d*f^2*x + 3*c^2*f^2 + 2*d^2)*sin(f*x + e))*sin(2*f*x + 2*e) + 2*(d^2*f*x + c*d*f)*sin(
f*x + e))/(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3 + (a^2*d^3*f^3*x^3 + 3*a^2*
c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*cos(3*f*x + 3*e)^2 + 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2
+ 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*cos(2*f*x + 2*e)^2 + 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*
d*f^3*x + a^2*c^3*f^3)*cos(f*x + e)^2 + (a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f
^3)*sin(3*f*x + 3*e)^2 + 18*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*cos(f*x
+ e)*sin(2*f*x + 2*e) + 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(2*f*x
+ 2*e)^2 + 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(f*x + e)^2 - 6*((a^
2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*cos(f*x + e) + (a^2*d^3*f^3*x^3 + 3*a^2
*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - 6*(a^2*d^3*f^3*x^3 + 3*
a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3 + 3*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f
^3*x + a^2*c^3*f^3)*sin(f*x + e))*cos(2*f*x + 2*e) - 2*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^
3*x + a^2*c^3*f^3 - 3*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*cos(2*f*x + 2*
e) + 3*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(f*x + e))*sin(3*f*x + 3*e
) + 6*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*sin(f*x + e))
Giac [N/A]
Not integrable
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10
\[
\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x }
\]
[In]
integrate(1/(d*x+c)/(a+a*sin(f*x+e))^2,x, algorithm="giac")
[Out]
integrate(1/((d*x + c)*(a*sin(f*x + e) + a)^2), x)
Mupad [N/A]
Not integrable
Time = 0.66 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10
\[
\int \frac {1}{(c+d x) (a+a \sin (e+f x))^2} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,\left (c+d\,x\right )} \,d x
\]
[In]
int(1/((a + a*sin(e + f*x))^2*(c + d*x)),x)
[Out]
int(1/((a + a*sin(e + f*x))^2*(c + d*x)), x)