\(\int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))^2} \, dx\) [43]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 41.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))^2} \, dx=\int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.80 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 7.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))^2} \, dx=\int \frac {1}{\left (a + b \sec {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 39.39 (sec) , antiderivative size = 2918, normalized size of antiderivative = 145.90 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

(2*a*b^3*d*sin(f*x + e) - (a^4 - a^2*b^2)*d*f*x - (a^4 - a^2*b^2)*c*f - ((a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^
2)*c*f)*cos(2*f*x + 2*e)^2 - 4*((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f)*cos(f*x + e)^2 - ((a^4 - a^2*b^2)
*d*f*x + (a^4 - a^2*b^2)*c*f)*sin(2*f*x + 2*e)^2 - 4*((a^2*b^2 - b^4)*d*f*x + (a^2*b^2 - b^4)*c*f)*sin(f*x + e
)^2 - 2*(a*b^3*d*sin(f*x + e) + (a^4 - a^2*b^2)*d*f*x + (a^4 - a^2*b^2)*c*f + 2*((a^3*b - a*b^3)*d*f*x + (a^3*
b - a*b^3)*c*f)*cos(f*x + e))*cos(2*f*x + 2*e) - 4*((a^3*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*cos(f*x + e)
- ((a^6 - a^4*b^2)*d^3*f*x^2 + 2*(a^6 - a^4*b^2)*c*d^2*f*x + (a^6 - a^4*b^2)*c^2*d*f + ((a^6 - a^4*b^2)*d^3*f*
x^2 + 2*(a^6 - a^4*b^2)*c*d^2*f*x + (a^6 - a^4*b^2)*c^2*d*f)*cos(2*f*x + 2*e)^2 + 4*((a^4*b^2 - a^2*b^4)*d^3*f
*x^2 + 2*(a^4*b^2 - a^2*b^4)*c*d^2*f*x + (a^4*b^2 - a^2*b^4)*c^2*d*f)*cos(f*x + e)^2 + ((a^6 - a^4*b^2)*d^3*f*
x^2 + 2*(a^6 - a^4*b^2)*c*d^2*f*x + (a^6 - a^4*b^2)*c^2*d*f)*sin(2*f*x + 2*e)^2 + 4*((a^5*b - a^3*b^3)*d^3*f*x
^2 + 2*(a^5*b - a^3*b^3)*c*d^2*f*x + (a^5*b - a^3*b^3)*c^2*d*f)*sin(2*f*x + 2*e)*sin(f*x + e) + 4*((a^4*b^2 -
a^2*b^4)*d^3*f*x^2 + 2*(a^4*b^2 - a^2*b^4)*c*d^2*f*x + (a^4*b^2 - a^2*b^4)*c^2*d*f)*sin(f*x + e)^2 + 2*((a^6 -
a^4*b^2)*d^3*f*x^2 + 2*(a^6 - a^4*b^2)*c*d^2*f*x + (a^6 - a^4*b^2)*c^2*d*f + 2*((a^5*b - a^3*b^3)*d^3*f*x^2 +
2*(a^5*b - a^3*b^3)*c*d^2*f*x + (a^5*b - a^3*b^3)*c^2*d*f)*cos(f*x + e))*cos(2*f*x + 2*e) + 4*((a^5*b - a^3*b
^3)*d^3*f*x^2 + 2*(a^5*b - a^3*b^3)*c*d^2*f*x + (a^5*b - a^3*b^3)*c^2*d*f)*cos(f*x + e))*integrate(-2*(2*a*b^3
*d*sin(f*x + e) - 2*((2*a^2*b^2 - b^4)*d*f*x + (2*a^2*b^2 - b^4)*c*f)*cos(f*x + e)^2 - 2*((2*a^2*b^2 - b^4)*d*
f*x + (2*a^2*b^2 - b^4)*c*f)*sin(f*x + e)^2 - (2*a*b^3*d*sin(f*x + e) + ((2*a^3*b - a*b^3)*d*f*x + (2*a^3*b -
a*b^3)*c*f)*cos(f*x + e))*cos(2*f*x + 2*e) - ((2*a^3*b - a*b^3)*d*f*x + (2*a^3*b - a*b^3)*c*f)*cos(f*x + e) +
(2*a*b^3*d*cos(f*x + e) + 2*a^2*b^2*d - ((2*a^3*b - a*b^3)*d*f*x + (2*a^3*b - a*b^3)*c*f)*sin(f*x + e))*sin(2*
f*x + 2*e))/((a^6 - a^4*b^2)*d^3*f*x^3 + 3*(a^6 - a^4*b^2)*c*d^2*f*x^2 + 3*(a^6 - a^4*b^2)*c^2*d*f*x + (a^6 -
a^4*b^2)*c^3*f + ((a^6 - a^4*b^2)*d^3*f*x^3 + 3*(a^6 - a^4*b^2)*c*d^2*f*x^2 + 3*(a^6 - a^4*b^2)*c^2*d*f*x + (a
^6 - a^4*b^2)*c^3*f)*cos(2*f*x + 2*e)^2 + 4*((a^4*b^2 - a^2*b^4)*d^3*f*x^3 + 3*(a^4*b^2 - a^2*b^4)*c*d^2*f*x^2
+ 3*(a^4*b^2 - a^2*b^4)*c^2*d*f*x + (a^4*b^2 - a^2*b^4)*c^3*f)*cos(f*x + e)^2 + ((a^6 - a^4*b^2)*d^3*f*x^3 +
3*(a^6 - a^4*b^2)*c*d^2*f*x^2 + 3*(a^6 - a^4*b^2)*c^2*d*f*x + (a^6 - a^4*b^2)*c^3*f)*sin(2*f*x + 2*e)^2 + 4*((
a^5*b - a^3*b^3)*d^3*f*x^3 + 3*(a^5*b - a^3*b^3)*c*d^2*f*x^2 + 3*(a^5*b - a^3*b^3)*c^2*d*f*x + (a^5*b - a^3*b^
3)*c^3*f)*sin(2*f*x + 2*e)*sin(f*x + e) + 4*((a^4*b^2 - a^2*b^4)*d^3*f*x^3 + 3*(a^4*b^2 - a^2*b^4)*c*d^2*f*x^2
+ 3*(a^4*b^2 - a^2*b^4)*c^2*d*f*x + (a^4*b^2 - a^2*b^4)*c^3*f)*sin(f*x + e)^2 + 2*((a^6 - a^4*b^2)*d^3*f*x^3
+ 3*(a^6 - a^4*b^2)*c*d^2*f*x^2 + 3*(a^6 - a^4*b^2)*c^2*d*f*x + (a^6 - a^4*b^2)*c^3*f + 2*((a^5*b - a^3*b^3)*d
^3*f*x^3 + 3*(a^5*b - a^3*b^3)*c*d^2*f*x^2 + 3*(a^5*b - a^3*b^3)*c^2*d*f*x + (a^5*b - a^3*b^3)*c^3*f)*cos(f*x
+ e))*cos(2*f*x + 2*e) + 4*((a^5*b - a^3*b^3)*d^3*f*x^3 + 3*(a^5*b - a^3*b^3)*c*d^2*f*x^2 + 3*(a^5*b - a^3*b^3
)*c^2*d*f*x + (a^5*b - a^3*b^3)*c^3*f)*cos(f*x + e)), x) + 2*(a*b^3*d*cos(f*x + e) + a^2*b^2*d - 2*((a^3*b - a
*b^3)*d*f*x + (a^3*b - a*b^3)*c*f)*sin(f*x + e))*sin(2*f*x + 2*e))/((a^6 - a^4*b^2)*d^3*f*x^2 + 2*(a^6 - a^4*b
^2)*c*d^2*f*x + (a^6 - a^4*b^2)*c^2*d*f + ((a^6 - a^4*b^2)*d^3*f*x^2 + 2*(a^6 - a^4*b^2)*c*d^2*f*x + (a^6 - a^
4*b^2)*c^2*d*f)*cos(2*f*x + 2*e)^2 + 4*((a^4*b^2 - a^2*b^4)*d^3*f*x^2 + 2*(a^4*b^2 - a^2*b^4)*c*d^2*f*x + (a^4
*b^2 - a^2*b^4)*c^2*d*f)*cos(f*x + e)^2 + ((a^6 - a^4*b^2)*d^3*f*x^2 + 2*(a^6 - a^4*b^2)*c*d^2*f*x + (a^6 - a^
4*b^2)*c^2*d*f)*sin(2*f*x + 2*e)^2 + 4*((a^5*b - a^3*b^3)*d^3*f*x^2 + 2*(a^5*b - a^3*b^3)*c*d^2*f*x + (a^5*b -
a^3*b^3)*c^2*d*f)*sin(2*f*x + 2*e)*sin(f*x + e) + 4*((a^4*b^2 - a^2*b^4)*d^3*f*x^2 + 2*(a^4*b^2 - a^2*b^4)*c*
d^2*f*x + (a^4*b^2 - a^2*b^4)*c^2*d*f)*sin(f*x + e)^2 + 2*((a^6 - a^4*b^2)*d^3*f*x^2 + 2*(a^6 - a^4*b^2)*c*d^2
*f*x + (a^6 - a^4*b^2)*c^2*d*f + 2*((a^5*b - a^3*b^3)*d^3*f*x^2 + 2*(a^5*b - a^3*b^3)*c*d^2*f*x + (a^5*b - a^3
*b^3)*c^2*d*f)*cos(f*x + e))*cos(2*f*x + 2*e) + 4*((a^5*b - a^3*b^3)*d^3*f*x^2 + 2*(a^5*b - a^3*b^3)*c*d^2*f*x
+ (a^5*b - a^3*b^3)*c^2*d*f)*cos(f*x + e))

Giac [N/A]

Not integrable

Time = 3.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 17.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))^2} \, dx=\int \frac {1}{{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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