\(\int \frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [453]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 32, 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 = {} \[ \int \frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 76.91 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.69 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\coth \left (d x + c\right ) \operatorname {csch}\left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.56 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\coth {\left (c + d x \right )} \operatorname {csch}{\left (c + d x \right )}}{\left (a + b \sinh {\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 316, normalized size of antiderivative = 9.88 \[ \int \frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\coth \left (d x + c\right ) \operatorname {csch}\left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 1.82 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {\coth (c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\mathrm {coth}\left (c+d\,x\right )}{\mathrm {sinh}\left (c+d\,x\right )\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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