\(\int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx\) [38]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \sec \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx=\int \frac {1}{\left (a + b \sec {\left (e + f x \right )}\right ) \left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.33 (sec) , antiderivative size = 439, normalized size of antiderivative = 21.95 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \sec \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

-(2*(a*b*d^2*x + a*b*c*d)*integrate((a*cos(2*f*x + 2*e)*cos(f*x + e) + 2*b*cos(f*x + e)^2 + a*sin(2*f*x + 2*e)
*sin(f*x + e) + 2*b*sin(f*x + e)^2 + a*cos(f*x + e))/(a^3*d^2*x^2 + 2*a^3*c*d*x + a^3*c^2 + (a^3*d^2*x^2 + 2*a
^3*c*d*x + a^3*c^2)*cos(2*f*x + 2*e)^2 + 4*(a*b^2*d^2*x^2 + 2*a*b^2*c*d*x + a*b^2*c^2)*cos(f*x + e)^2 + (a^3*d
^2*x^2 + 2*a^3*c*d*x + a^3*c^2)*sin(2*f*x + 2*e)^2 + 4*(a^2*b*d^2*x^2 + 2*a^2*b*c*d*x + a^2*b*c^2)*sin(2*f*x +
2*e)*sin(f*x + e) + 4*(a*b^2*d^2*x^2 + 2*a*b^2*c*d*x + a*b^2*c^2)*sin(f*x + e)^2 + 2*(a^3*d^2*x^2 + 2*a^3*c*d
*x + a^3*c^2 + 2*(a^2*b*d^2*x^2 + 2*a^2*b*c*d*x + a^2*b*c^2)*cos(f*x + e))*cos(2*f*x + 2*e) + 4*(a^2*b*d^2*x^2
+ 2*a^2*b*c*d*x + a^2*b*c^2)*cos(f*x + e)), x) + 1)/(a*d^2*x + a*c*d)

Giac [N/A]

Not integrable

Time = 0.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \sec \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx=\int \frac {1}{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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