\(\int \frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\) [216]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 31, 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 = {} \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int \frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 82.98 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int \frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.85 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 500, normalized size of antiderivative = 16.13 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

((I*a*d*f^2*x^2 + 2*I*a*d*e*f*x + I*a*d*e^2 + (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*e^(3*d*x + 3*c) + (-I*a*d*
f^2*x^2 - 2*I*a*d*e*f*x - I*a*d*e^2)*e^(2*d*x + 2*c) - (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*e^(d*x + c))*inte
gral(-2*((I*d*f*x + I*d*e + 2*I*f)*e^(2*d*x + 2*c) + (d*f*x + d*e + 2*f)*e^(d*x + c) - 4*I*f)/(I*a*d*f^3*x^3 +
3*I*a*d*e*f^2*x^2 + 3*I*a*d*e^2*f*x + I*a*d*e^3 + (a*d*f^3*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3)*e
^(3*d*x + 3*c) + (-I*a*d*f^3*x^3 - 3*I*a*d*e*f^2*x^2 - 3*I*a*d*e^2*f*x - I*a*d*e^3)*e^(2*d*x + 2*c) - (a*d*f^3
*x^3 + 3*a*d*e*f^2*x^2 + 3*a*d*e^2*f*x + a*d*e^3)*e^(d*x + c)), x) - 2*I*e^(2*d*x + 2*c) - 2*e^(d*x + c) + 4*I
)/(I*a*d*f^2*x^2 + 2*I*a*d*e*f*x + I*a*d*e^2 + (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*e^(3*d*x + 3*c) + (-I*a*d
*f^2*x^2 - 2*I*a*d*e*f*x - I*a*d*e^2)*e^(2*d*x + 2*c) - (a*d*f^2*x^2 + 2*a*d*e*f*x + a*d*e^2)*e^(d*x + c))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 477, normalized size of antiderivative = 15.39 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (f x + e\right )}^{2} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 1.47 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

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

[Out]

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