\(\int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx\) [20]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx=\int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 15.91 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx=\int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.95 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 7.34 (sec) , antiderivative size = 105, normalized size of antiderivative = 5.25 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx=\frac {\int \frac {1}{c^{2} \sec ^{2}{\left (e + f x \right )} + 2 c^{2} \sec {\left (e + f x \right )} + c^{2} + 2 c d x \sec ^{2}{\left (e + f x \right )} + 4 c d x \sec {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \sec ^{2}{\left (e + f x \right )} + 2 d^{2} x^{2} \sec {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{2}} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 38.23 (sec) , antiderivative size = 4471, normalized size of antiderivative = 223.55 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-1/3*(3*d^3*f^3*x^3 + 9*c*d^2*f^3*x^2 + 9*c^2*d*f^3*x + 3*c^3*f^3 + 3*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d
*f^3*x + c^3*f^3)*cos(3*f*x + 3*e)^2 + 3*(9*d^3*f^3*x^3 + 27*c*d^2*f^3*x^2 + 9*c^3*f^3 - 4*c*d^2*f + (27*c^2*d
*f^3 - 4*d^3*f)*x)*cos(2*f*x + 2*e)^2 + 3*(9*d^3*f^3*x^3 + 27*c*d^2*f^3*x^2 + 9*c^3*f^3 - 4*c*d^2*f + (27*c^2*
d*f^3 - 4*d^3*f)*x)*cos(f*x + e)^2 + 3*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + c^3*f^3)*sin(3*f*x + 3
*e)^2 + 3*(9*d^3*f^3*x^3 + 27*c*d^2*f^3*x^2 + 9*c^3*f^3 - 4*c*d^2*f + (27*c^2*d*f^3 - 4*d^3*f)*x)*sin(2*f*x +
2*e)^2 + 3*(9*d^3*f^3*x^3 + 27*c*d^2*f^3*x^2 + 9*c^3*f^3 - 4*c*d^2*f + (27*c^2*d*f^3 - 4*d^3*f)*x)*sin(f*x + e
)^2 + 2*(3*d^3*f^3*x^3 + 9*c*d^2*f^3*x^2 + 9*c^2*d*f^3*x + 3*c^3*f^3 + (9*d^3*f^3*x^3 + 27*c*d^2*f^3*x^2 + 9*c
^3*f^3 - 2*c*d^2*f + (27*c^2*d*f^3 - 2*d^3*f)*x)*cos(2*f*x + 2*e) + (9*d^3*f^3*x^3 + 27*c*d^2*f^3*x^2 + 9*c^3*
f^3 - 2*c*d^2*f + (27*c^2*d*f^3 - 2*d^3*f)*x)*cos(f*x + e) - 6*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - d^3)
*sin(2*f*x + 2*e) - 3*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - 4*d^3)*sin(f*x + e))*cos(3*f*x + 3*e) + 2
*(9*d^3*f^3*x^3 + 27*c*d^2*f^3*x^2 + 9*c^3*f^3 - 2*c*d^2*f + (27*c^2*d*f^3 - 2*d^3*f)*x + 3*(9*d^3*f^3*x^3 + 2
7*c*d^2*f^3*x^2 + 9*c^3*f^3 - 4*c*d^2*f + (27*c^2*d*f^3 - 4*d^3*f)*x)*cos(f*x + e) - 9*(d^3*f^2*x^2 + 2*c*d^2*
f^2*x + c^2*d*f^2 - 2*d^3)*sin(f*x + e))*cos(2*f*x + 2*e) + 2*(9*d^3*f^3*x^3 + 27*c*d^2*f^3*x^2 + 9*c^3*f^3 -
2*c*d^2*f + (27*c^2*d*f^3 - 2*d^3*f)*x)*cos(f*x + e) + 3*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^
3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3 + (a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x
^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(3*f*x + 3*e)^2 + 9*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^
2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(2*f*x + 2*e)^2 + 9*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4
*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(f*x + e)^2 + (a^2*d^5*f^3*x^4 + 4*
a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*sin(3*f*x + 3*e)^2 + 9*(a^2*d
^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*sin(2*f*x + 2*
e)^2 + 18*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3
)*sin(2*f*x + 2*e)*sin(f*x + e) + 9*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3
*d^2*f^3*x + a^2*c^4*d*f^3)*sin(f*x + e)^2 + 2*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2
+ 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3 + 3*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a
^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(2*f*x + 2*e) + 3*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*
f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(f*x + e))*cos(3*f*x + 3*e) + 6*(a^2*d^5*f^3*x^4 + 4*a^2*c*d
^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3 + 3*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^
3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(f*x + e))*cos(2*f*x + 2*e) + 6*(a^2*d
^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(f*x + e) +
6*((a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*sin(
2*f*x + 2*e) + (a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*
d*f^3)*sin(f*x + e))*sin(3*f*x + 3*e))*integrate(4/3*(5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2 - 12*d^3)*s
in(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*(5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2 - 6*d^3 + 6*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*
d*f^2 - d^3)*cos(2*f*x + 2*e) + 3*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - 4*d^3)*cos(f*x + e) + (9*d^3*
f^3*x^3 + 27*c*d^2*f^3*x^2 + 9*c^3*f^3 - 2*c*d^2*f + (27*c^2*d*f^3 - 2*d^3*f)*x)*sin(2*f*x + 2*e) + (9*d^3*f^3
*x^3 + 27*c*d^2*f^3*x^2 + 9*c^3*f^3 - 2*c*d^2*f + (27*c^2*d*f^3 - 2*d^3*f)*x)*sin(f*x + e))*sin(3*f*x + 3*e) +
6*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - 4*d^3 + 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 2*d^3)*
cos(f*x + e) + (9*d^3*f^3*x^3 + 27*c*d^2*f^3*x^2 + 9*c^3*f^3 - 4*c*d^2*f + (27*c^2*d*f^3 - 4*d^3*f)*x)*sin(f*x
+ e))*sin(2*f*x + 2*e) + 12*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - d^3)*sin(f*x + e))/(a^2*d^5*f^3*x^4 +
4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3 + (a^2*d^5*f^3*x^4 + 4*a^2*c
*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(3*f*x + 3*e)^2 + 9*(a^2*d^5*f^
3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(2*f*x + 2*e)^2
+ 9*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(
f*x + e)^2 + (a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*
f^3)*sin(3*f*x + 3*e)^2 + 9*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3
*x + a^2*c^4*d*f^3)*sin(2*f*x + 2*e)^2 + 18*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4
*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*sin(2*f*x + 2*e)*sin(f*x + e) + 9*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 +
6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*sin(f*x + e)^2 + 2*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^
4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3 + 3*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3
*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(2*f*x + 2*e) + 3*(a^2*d^5*f^3*x^4 + 4*
a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(f*x + e))*cos(3*f*x + 3*e
) + 6*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3 + 3
*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*cos(f*x
+ e))*cos(2*f*x + 2*e) + 6*(a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c^3*d^2*f^3
*x + a^2*c^4*d*f^3)*cos(f*x + e) + 6*((a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^2 + 4*a^2*c
^3*d^2*f^3*x + a^2*c^4*d*f^3)*sin(2*f*x + 2*e) + (a^2*d^5*f^3*x^4 + 4*a^2*c*d^4*f^3*x^3 + 6*a^2*c^2*d^3*f^3*x^
2 + 4*a^2*c^3*d^2*f^3*x + a^2*c^4*d*f^3)*sin(f*x + e))*sin(3*f*x + 3*e))

Giac [N/A]

Not integrable

Time = 1.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (a \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 14.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx=\int \frac {1}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[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)