\(\int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx\) [58]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.76 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx=\int \frac {1}{\left (a + b \tan {\left (e + f x \right )}\right ) \left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.73 (sec) , antiderivative size = 424, normalized size of antiderivative = 21.20 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 4.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c+d x)^2 (a+b \tan (e+f x))} \, dx=\int \frac {1}{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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