\(\int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [485]

Optimal result

Integrand size = 34, antiderivative size = 34 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 34, 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 = {} \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\coth ^2(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 = 70.68 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.60 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\coth \left (d x + c\right )^{2} \operatorname {csch}\left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integral(coth(d*x + c)^2*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 = 15.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\coth ^{2}{\left (c + d x \right )} \operatorname {csch}{\left (c + d x \right )}}{\left (a + b \sinh {\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 762, normalized size of antiderivative = 22.41 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\coth \left (d x + c\right )^{2} \operatorname {csch}\left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

-2*(a^2*b*e^c + b^3*e^c)*integrate(-e^(d*x)/(a^3*b*f*x + a^3*b*e - (a^3*b*f*x*e^(2*c) + a^3*b*e*e^(2*c))*e^(2*
d*x) - 2*(a^4*f*x*e^c + a^4*e*e^c)*e^(d*x)), x) - (2*b*d*f*x + 2*b*d*e + (a*d*f*x*e^(3*c) + (d*e - f)*a*e^(3*c
))*e^(3*d*x) - 2*(b*d*f*x*e^(2*c) + b*d*e*e^(2*c))*e^(2*d*x) + (a*d*f*x*e^c + (d*e + f)*a*e^c)*e^(d*x))/(a^2*d
^2*f^2*x^2 + 2*a^2*d^2*e*f*x + a^2*d^2*e^2 + (a^2*d^2*f^2*x^2*e^(4*c) + 2*a^2*d^2*e*f*x*e^(4*c) + a^2*d^2*e^2*
e^(4*c))*e^(4*d*x) - 2*(a^2*d^2*f^2*x^2*e^(2*c) + 2*a^2*d^2*e*f*x*e^(2*c) + a^2*d^2*e^2*e^(2*c))*e^(2*d*x)) +
2*integrate(-1/4*(2*b^2*d^2*e^2 + 2*a*b*d*e*f + (d^2*e^2 + 2*f^2)*a^2 + (a^2*d^2*f^2 + 2*b^2*d^2*f^2)*x^2 + 2*
(a^2*d^2*e*f + 2*b^2*d^2*e*f + a*b*d*f^2)*x)/(a^3*d^2*f^3*x^3 + 3*a^3*d^2*e*f^2*x^2 + 3*a^3*d^2*e^2*f*x + a^3*
d^2*e^3 - (a^3*d^2*f^3*x^3*e^c + 3*a^3*d^2*e*f^2*x^2*e^c + 3*a^3*d^2*e^2*f*x*e^c + a^3*d^2*e^3*e^c)*e^(d*x)),
x) + 2*integrate(1/4*(2*b^2*d^2*e^2 - 2*a*b*d*e*f + (d^2*e^2 + 2*f^2)*a^2 + (a^2*d^2*f^2 + 2*b^2*d^2*f^2)*x^2
+ 2*(a^2*d^2*e*f + 2*b^2*d^2*e*f - a*b*d*f^2)*x)/(a^3*d^2*f^3*x^3 + 3*a^3*d^2*e*f^2*x^2 + 3*a^3*d^2*e^2*f*x +
a^3*d^2*e^3 + (a^3*d^2*f^3*x^3*e^c + 3*a^3*d^2*e*f^2*x^2*e^c + 3*a^3*d^2*e^2*f*x*e^c + a^3*d^2*e^3*e^c)*e^(d*x
)), x)

Giac [F(-1)]

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

[In]

integrate(coth(d*x+c)^2*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 = 1.60 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^2}{\mathrm {sinh}\left (c+d\,x\right )\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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