\(\int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx\) [45]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 42.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx=\int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx=\int { \frac {{\left (b \coth \left (f x + e\right ) + a\right )}^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.49 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx=\int \frac {\left (a + b \coth {\left (e + f x \right )}\right )^{2}}{c + d x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 226, normalized size of antiderivative = 11.30 \[ \int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx=\int { \frac {{\left (b \coth \left (f x + e\right ) + a\right )}^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx=\int { \frac {{\left (b \coth \left (f x + e\right ) + a\right )}^{2}}{d x + c} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \coth (e+f x))^2}{c+d x} \, dx=\int \frac {{\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2}{c+d\,x} \,d x \]

[In]

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

[Out]

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