\(\int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx\) [10]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 24.75 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.40 \[ \int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.80 \[ \int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx=a^{2} \left (\int \frac {2 \sec {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.38 (sec) , antiderivative size = 624, normalized size of antiderivative = 31.20 \[ \int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 29.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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