\(\int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx\) [14]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx=\int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 7.80 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx=\int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sec \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx=\frac {\int \frac {1}{c \sec {\left (e + f x \right )} + c + d x \sec {\left (e + f x \right )} + d x}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 391, normalized size of antiderivative = 19.55 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sec \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sec \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))} \, dx=\int \frac {1}{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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