\(\int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [458]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 67.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.68 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\coth \left (d x + c\right )^{2}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.52 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\coth ^{2}{\left (c + d x \right )}}{\left (a + b \sinh {\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 311, normalized size of antiderivative = 11.11 \[ \int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\coth \left (d x + c\right )^{2}}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 1.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2}{\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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