\(\int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx\) [63]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 15.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx=\int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.80 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx=\int \frac {1}{\left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 27.22 (sec) , antiderivative size = 1977, normalized size of antiderivative = 98.85 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((a^4 - b^4)*d*f*x + (a^4 - b^4)*c*f + ((a^4 - b^4)*d*f*x + (a^4 - b^4)*c*f)*cos(2*f*x + 2*e)^2 + ((a^4 - b^4
)*d*f*x + (a^4 - b^4)*c*f)*sin(2*f*x + 2*e)^2 + 2*(2*a*b^3*d + (a^4 - 2*a^2*b^2 + b^4)*d*f*x + (a^4 - 2*a^2*b^
2 + b^4)*c*f)*cos(2*f*x + 2*e) - ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^3*f*x^2 + 2*(a^6 + 3*a^4*b^2 + 3*a^2*b
^4 + b^6)*c*d^2*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d*f + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^3*f
*x^2 + 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d^2*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d*f)*cos(2*f*
x + 2*e)^2 + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^3*f*x^2 + 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d^2*f*x
+ (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d*f)*sin(2*f*x + 2*e)^2 + 2*((a^6 + a^4*b^2 - a^2*b^4 - b^6)*d^3*f*x
^2 + 2*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*c*d^2*f*x + (a^6 + a^4*b^2 - a^2*b^4 - b^6)*c^2*d*f)*cos(2*f*x + 2*e) +
4*((a^5*b + 2*a^3*b^3 + a*b^5)*d^3*f*x^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*c*d^2*f*x + (a^5*b + 2*a^3*b^3 + a*b
^5)*c^2*d*f)*sin(2*f*x + 2*e))*integrate(4*(2*(a^2*b^2*d*f*x + a^2*b^2*c*f - a*b^3*d)*cos(2*f*x + 2*e) - ((a^3
*b - a*b^3)*d*f*x + (a^3*b - a*b^3)*c*f - (a^2*b^2 - b^4)*d)*sin(2*f*x + 2*e))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^3*f*x^3 + 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d^2*f*x^2 + 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^
2*d*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^3*f + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^3*f*x^3 + 3*(a^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d^2*f*x^2 + 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d*f*x + (a^6 + 3*a^4*b^
2 + 3*a^2*b^4 + b^6)*c^3*f)*cos(2*f*x + 2*e)^2 + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^3*f*x^3 + 3*(a^6 + 3*a
^4*b^2 + 3*a^2*b^4 + b^6)*c*d^2*f*x^2 + 3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d*f*x + (a^6 + 3*a^4*b^2 + 3
*a^2*b^4 + b^6)*c^3*f)*sin(2*f*x + 2*e)^2 + 2*((a^6 + a^4*b^2 - a^2*b^4 - b^6)*d^3*f*x^3 + 3*(a^6 + a^4*b^2 -
a^2*b^4 - b^6)*c*d^2*f*x^2 + 3*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*c^2*d*f*x + (a^6 + a^4*b^2 - a^2*b^4 - b^6)*c^3
*f)*cos(2*f*x + 2*e) + 4*((a^5*b + 2*a^3*b^3 + a*b^5)*d^3*f*x^3 + 3*(a^5*b + 2*a^3*b^3 + a*b^5)*c*d^2*f*x^2 +
3*(a^5*b + 2*a^3*b^3 + a*b^5)*c^2*d*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c^3*f)*sin(2*f*x + 2*e)), x) + 2*(2*(a^3
*b - a*b^3)*d*f*x + 2*(a^3*b - a*b^3)*c*f - (a^2*b^2 - b^4)*d)*sin(2*f*x + 2*e))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*d^3*f*x^2 + 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d^2*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*
d*f + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^3*f*x^2 + 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d^2*f*x + (a^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d*f)*cos(2*f*x + 2*e)^2 + ((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^3*f*x^2 +
2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c*d^2*f*x + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d*f)*sin(2*f*x + 2*e
)^2 + 2*((a^6 + a^4*b^2 - a^2*b^4 - b^6)*d^3*f*x^2 + 2*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*c*d^2*f*x + (a^6 + a^4*
b^2 - a^2*b^4 - b^6)*c^2*d*f)*cos(2*f*x + 2*e) + 4*((a^5*b + 2*a^3*b^3 + a*b^5)*d^3*f*x^2 + 2*(a^5*b + 2*a^3*b
^3 + a*b^5)*c*d^2*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c^2*d*f)*sin(2*f*x + 2*e))

Giac [N/A]

Not integrable

Time = 22.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.99 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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