\(\int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx\) [46]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 34.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.35 \[ \int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx=\int { \frac {{\left (b \coth \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.49 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {\left (a + b \coth {\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 375, normalized size of antiderivative = 18.75 \[ \int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx=\int { \frac {{\left (b \coth \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx=\int { \frac {{\left (b \coth \left (f x + e\right ) + a\right )}^{2}}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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