\(\int \frac {1}{(c+d x) (a+a \sec (e+f x))^2} \, dx\) [19]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 13.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))^2} \, dx=\int \frac {1}{(c+d x) (a+a \sec (e+f x))^2} \, dx \]

[In]

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

[Out]

Integrate[1/((c + d*x)*(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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.85 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.97 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))^2} \, dx=\frac {\int \frac {1}{c \sec ^{2}{\left (e + f x \right )} + 2 c \sec {\left (e + f x \right )} + c + d x \sec ^{2}{\left (e + f x \right )} + 2 d x \sec {\left (e + f x \right )} + d x}\, dx}{a^{2}} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 10.62 (sec) , antiderivative size = 3956, normalized size of antiderivative = 197.80 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

1/3*(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*log(d*x + c) + 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)*log(d*x + c)*sin(3*f*x + 3*e)^2 + 3*(2*d^3*f*x + 2*c*d^2*f +
9*(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)*log(d*x + c))*cos(2*f*x + 2*e)^2 + 3*(2*d^3*f*x +
2*c*d^2*f + 9*(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)*log(d*x + c))*cos(f*x + e)^2 + 3*(2*d^
3*f*x + 2*c*d^2*f + 9*(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)*log(d*x + c))*sin(2*f*x + 2*e)
^2 + 3*(2*d^3*f*x + 2*c*d^2*f + 9*(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)*log(d*x + c))*sin(
f*x + e)^2 + 2*((d^3*f*x + c*d^2*f + 9*(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)*log(d*x + c))
*cos(2*f*x + 2*e) + (d^3*f*x + c*d^2*f + 9*(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)*log(d*x +
c))*cos(f*x + e) + 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)*log(d*x + c) + 2*(3*d^3*f^2*x^
2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - d^3)*sin(2*f*x + 2*e) + (9*d^3*f^2*x^2 + 18*c*d^2*f^2*x + 9*c^2*d*f^2 - 4*d^
3)*sin(f*x + e))*cos(3*f*x + 3*e) + 2*(d^3*f*x + c*d^2*f + 3*(2*d^3*f*x + 2*c*d^2*f + 9*(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)*log(d*x + c))*cos(f*x + e) + 9*(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)*log(d*x + c) + 3*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - 2*d^3)*sin(f*x + e))*cos(2*f*x
+ 2*e) + 2*(d^3*f*x + c*d^2*f + 9*(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)*log(d*x + c))*cos(
f*x + e) - 3*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 + (a^2*d^4*f^3*x^3 +
3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(3*f*x + 3*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*
d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(2*f*x + 2*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x
^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e)^2 + (a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*
d^2*f^3*x + a^2*c^3*d*f^3)*sin(3*f*x + 3*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x
+ a^2*c^3*d*f^3)*sin(2*f*x + 2*e)^2 + 18*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c
^3*d*f^3)*sin(2*f*x + 2*e)*sin(f*x + e) + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2
*c^3*d*f^3)*sin(f*x + e)^2 + 2*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 +
3*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(2*f*x + 2*e) + 3*(a^2*d^4*
f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e))*cos(3*f*x + 3*e) + 6*(a^2*d
^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 + 3*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*
x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e))*cos(2*f*x + 2*e) + 6*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f
^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e) + 6*((a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2
*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*sin(2*f*x + 2*e) + (a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*
x + a^2*c^3*d*f^3)*sin(f*x + e))*sin(3*f*x + 3*e))*integrate(2/3*(5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2
- 6*d^3)*sin(f*x + e)/(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^
2*c^4*f^3 + (a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*
cos(f*x + e)^2 + (a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*
f^3)*sin(f*x + e)^2 + 2*(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a
^2*c^4*f^3)*cos(f*x + e)), x) + 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)*log(d*x + c) - 2*(
5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2 - 2*d^3 + 2*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - d^3)*c
os(2*f*x + 2*e) + (9*d^3*f^2*x^2 + 18*c*d^2*f^2*x + 9*c^2*d*f^2 - 4*d^3)*cos(f*x + e) - (d^3*f*x + c*d^2*f + 9
*(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)*log(d*x + c))*sin(2*f*x + 2*e) - (d^3*f*x + c*d^2*f
+ 9*(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)*log(d*x + c))*sin(f*x + e))*sin(3*f*x + 3*e) -
2*(9*d^3*f^2*x^2 + 18*c*d^2*f^2*x + 9*c^2*d*f^2 - 4*d^3 + 3*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - 2*d
^3)*cos(f*x + e) - 3*(2*d^3*f*x + 2*c*d^2*f + 9*(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)*log(
d*x + c))*sin(f*x + e))*sin(2*f*x + 2*e) - 4*(3*d^3*f^2*x^2 + 6*c*d^2*f^2*x + 3*c^2*d*f^2 - d^3)*sin(f*x + e))
/(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 + (a^2*d^4*f^3*x^3 + 3*a^2*c*d^3
*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(3*f*x + 3*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2
+ 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(2*f*x + 2*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c
^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e)^2 + (a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x +
a^2*c^3*d*f^3)*sin(3*f*x + 3*e)^2 + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d
*f^3)*sin(2*f*x + 2*e)^2 + 18*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*si
n(2*f*x + 2*e)*sin(f*x + e) + 9*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*
sin(f*x + e)^2 + 2*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 + 3*(a^2*d^4*f
^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(2*f*x + 2*e) + 3*(a^2*d^4*f^3*x^3 + 3*
a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e))*cos(3*f*x + 3*e) + 6*(a^2*d^4*f^3*x^3 +
3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3 + 3*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*
c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e))*cos(2*f*x + 2*e) + 6*(a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a
^2*c^2*d^2*f^3*x + a^2*c^3*d*f^3)*cos(f*x + e) + 6*((a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3
*x + a^2*c^3*d*f^3)*sin(2*f*x + 2*e) + (a^2*d^4*f^3*x^3 + 3*a^2*c*d^3*f^3*x^2 + 3*a^2*c^2*d^2*f^3*x + a^2*c^3*
d*f^3)*sin(f*x + e))*sin(3*f*x + 3*e))

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (a \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 14.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(c+d x) (a+a \sec (e+f x))^2} \, dx=\int \frac {1}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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