\(\int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [209]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 29, 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 = {} \[ \int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 35.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.78 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 235, normalized size of antiderivative = 8.10 \[ \int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(csch(d*x+c)/(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^(d*x + c))*integral(2*((d*f*x + d*e + f)*e^(2*d*x + 2*c) - (I*d*f
*x + I*d*e)*e^(d*x + c) - 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*a*d*f*x - I*a*d*e + (a*d*f*x + a*d*e)*e^(d*x + c))

Sympy [N/A]

Not integrable

Time = 14.90 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=- \frac {i \int \frac {\operatorname {csch}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 169, normalized size of antiderivative = 5.83 \[ \int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 17.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {\text {csch}(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {1}{\mathrm {sinh}\left (c+d\,x\right )\,\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

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

[Out]

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