\(\int \frac {(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx\) [48]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 14.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.35 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.68 (sec) , antiderivative size = 609, normalized size of antiderivative = 30.45 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((a^2 - b^2)*d*f*x - 2*b^2*d*sin(2*f*x + 2*e) + (a^2 - b^2)*c*f + ((a^2 - b^2)*d*f*x + (a^2 - b^2)*c*f)*cos(2
*f*x + 2*e)^2 + ((a^2 - b^2)*d*f*x + (a^2 - b^2)*c*f)*sin(2*f*x + 2*e)^2 + 2*((a^2 - b^2)*d*f*x + (a^2 - b^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*b*d*f*x + a*b*c*f + b^2*d)*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 = 15.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.64 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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