\(\int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx\) [28]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx=\text {Int}\left (\frac {a+b \sec (e+f x)}{(c+d x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx=\int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx=\int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.53 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx=\int { \frac {b \sec \left (f x + e\right ) + a}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.62 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx=\int \frac {a + b \sec {\left (e + f x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 172, normalized size of antiderivative = 9.56 \[ \int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx=\int { \frac {b \sec \left (f x + e\right ) + a}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 2.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx=\int { \frac {b \sec \left (f x + e\right ) + a}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {a+b \sec (e+f x)}{(c+d x)^2} \, dx=\int \frac {a+\frac {b}{\cos \left (e+f\,x\right )}}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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