\(\int \frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [468]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 41.56 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.87 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 6.70 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2} \operatorname {sech}\left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 472, normalized size of antiderivative = 13.88 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2} \operatorname {sech}\left (d x + c\right )}{{\left (f x + e\right )} {\left (b \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(csch(d*x+c)^2*sech(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)) - 8*integrate(-1/4*(a*b^3*e^(d*x
+ c) - b^4)/(a^4*b*e + a^2*b^3*e + (a^4*b*f + a^2*b^3*f)*x - (a^4*b*e*e^(2*c) + a^2*b^3*e*e^(2*c) + (a^4*b*f*
e^(2*c) + a^2*b^3*f*e^(2*c))*x)*e^(2*d*x) - 2*(a^5*e*e^c + a^3*b^2*e*e^c + (a^5*f*e^c + a^3*b^2*f*e^c)*x)*e^(d
*x)), x) - 8*integrate(-1/8*(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) + 8*integrate(1/8*(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) - 8*inte
grate(1/4*(a*e^(d*x + c) + b)/(a^2*e + b^2*e + (a^2*f + b^2*f)*x + (a^2*e*e^(2*c) + b^2*e*e^(2*c) + (a^2*f*e^(
2*c) + b^2*f*e^(2*c))*x)*e^(2*d*x)), x)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 3.66 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx=\int \frac {1}{\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

int(1/(cosh(c + d*x)*sinh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(1/(cosh(c + d*x)*sinh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))), x)