3.137 \(\int \frac {1}{(c+d x)^2 (a+a \cos (e+f x))^2} \, dx\)
Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{(c+d x)^2 (a \cos (e+f x)+a)^2},x\right ) \]
[Out]
Unintegrable(1/(d*x+c)^2/(a+a*cos(f*x+e))^2,x)
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \[ \int \frac {1}{(c+d x)^2 (a+a \cos (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/((c + d*x)^2*(a + a*Cos[e + f*x])^2),x]
[Out]
Defer[Int][1/((c + d*x)^2*(a + a*Cos[e + f*x])^2), x]
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^2 (a+a \cos (e+f x))^2} \, dx &=\int \frac {1}{(c+d x)^2 (a+a \cos (e+f x))^2} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 12.78, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^2 (a+a \cos (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/((c + d*x)^2*(a + a*Cos[e + f*x])^2),x]
[Out]
Integrate[1/((c + d*x)^2*(a + a*Cos[e + f*x])^2), x]
________________________________________________________________________________________
fricas [A] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2} + {\left (a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2}\right )} \cos \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)^2/(a+a*cos(f*x+e))^2,x, algorithm="fricas")
[Out]
integral(1/(a^2*d^2*x^2 + 2*a^2*c*d*x + 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)*cos(f*x + e)), x)
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d x + c\right )}^{2} {\left (a \cos \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)^2/(a+a*cos(f*x+e))^2,x, algorithm="giac")
[Out]
integrate(1/((d*x + c)^2*(a*cos(f*x + e) + a)^2), x)
________________________________________________________________________________________
maple [A] time = 3.26, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right )^{2} \left (a +a \cos \left (f x +e \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(d*x+c)^2/(a+a*cos(f*x+e))^2,x)
[Out]
int(1/(d*x+c)^2/(a+a*cos(f*x+e))^2,x)
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)^2/(a+a*cos(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*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*sin(f*x + e) + 12*(d^2*f*x + c*d*f)*sin(f*x + e)^2 - 2*(6*d^2*sin(2*f*x + 2*e) - 2*(d^
2*f*x + c*d*f)*cos(2*f*x + 2*e) - 2*(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 +
4*d^2)*sin(f*x + e))*cos(3*f*x + 3*e) + 2*(2*d^2*f*x + 2*c*d*f + 12*(d^2*f*x + c*d*f)*cos(f*x + e) - 9*(d^2*f^
2*x^2 + 2*c*d*f^2*x + c^2*f^2 + 2*d^2)*sin(f*x + e))*cos(2*f*x + 2*e) + 4*(d^2*f*x + c*d*f)*cos(f*x + 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)*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)*co
s(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 + 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 + 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)*sin(2*f*x + 2*e)*sin(f*x + 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(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 + 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)*cos(f*x + 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)*cos(f*x + e))*cos(2*f*x + 2*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)*cos(f*x + 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(2*f*x +
2*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(f
*x + e))*sin(3*f*x + 3*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)*sin(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)*cos(f*x + e)), x) + 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 + 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(2*f*x + 2*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 + 3*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2 + 2*d^2)*cos(f*
x + e) + 4*(d^2*f*x + c*d*f)*sin(f*x + e))*sin(2*f*x + 2*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 + 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 + 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)*sin(2*f*x + 2*e)*sin(f
*x + 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(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 + 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)*co
s(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)*cos(f*x + 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)*cos(f*x + e))*cos(2*f*x + 2*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)*cos(f*x + 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(2*f*x + 2*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(f*x + e))*sin(3*f*x + 3*e))
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\left (a+a\,\cos \left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + a*cos(e + f*x))^2*(c + d*x)^2),x)
[Out]
int(1/((a + a*cos(e + f*x))^2*(c + d*x)^2), x)
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{c^{2} \cos ^{2}{\left (e + f x \right )} + 2 c^{2} \cos {\left (e + f x \right )} + c^{2} + 2 c d x \cos ^{2}{\left (e + f x \right )} + 4 c d x \cos {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \cos ^{2}{\left (e + f x \right )} + 2 d^{2} x^{2} \cos {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d*x+c)**2/(a+a*cos(f*x+e))**2,x)
[Out]
Integral(1/(c**2*cos(e + f*x)**2 + 2*c**2*cos(e + f*x) + c**2 + 2*c*d*x*cos(e + f*x)**2 + 4*c*d*x*cos(e + f*x)
+ 2*c*d*x + d**2*x**2*cos(e + f*x)**2 + 2*d**2*x**2*cos(e + f*x) + d**2*x**2), x)/a**2
________________________________________________________________________________________