\(\int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [215]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

Defer[Int][Csch[c + d*x]^2/((e + f*x)*(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) (a+i a \sinh (c+d x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 70.54 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 342, normalized size of antiderivative = 11.03 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 15.97 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=- \frac {i \int \frac {\operatorname {csch}^{2}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 330, normalized size of antiderivative = 10.65 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

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

[Out]

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