\(\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [221]
Optimal result
Integrand size = 31, antiderivative size = 31 \[
\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right )
\]
[Out]
Unintegrable(csch(d*x+c)^3/(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 = {} \[
\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx
\]
[In]
Int[Csch[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])),x]
[Out]
Defer[Int][Csch[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \\
\end{align*}
Mathematica [F(-1)]
Timed out. \[
\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\text {\$Aborted}
\]
[In]
Integrate[Csch[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])),x]
[Out]
$Aborted
Maple [N/A] (verified)
Not integrable
Time = 0.91 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
\[\int \frac {\operatorname {csch}\left (d x +c \right )^{3}}{\left (f x +e \right ) \left (a +i a \sinh \left (d x +c \right )\right )}d x\]
[In]
int(csch(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
[Out]
int(csch(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
Fricas [N/A]
Not integrable
Time = 0.29 (sec) , antiderivative size = 903, normalized size of antiderivative = 29.13
\[
\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x }
\]
[In]
integrate(csch(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")
[Out]
-(4*d*f*x + 4*d*e + (3*d*f*x + 3*d*e - f)*e^(4*d*x + 4*c) - (3*I*d*f*x + 3*I*d*e - I*f)*e^(3*d*x + 3*c) - (5*d
*f*x + 5*d*e - f)*e^(2*d*x + 2*c) - (-I*d*f*x - I*d*e + I*f)*e^(d*x + c) - (-I*a*d^2*f^2*x^2 - 2*I*a*d^2*e*f*x
- I*a*d^2*e^2 + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*e^(5*d*x + 5*c) + (-I*a*d^2*f^2*x^2 - 2*I*a*d^2*e
*f*x - I*a*d^2*e^2)*e^(4*d*x + 4*c) - 2*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*e^(3*d*x + 3*c) - 2*(-I*a*
d^2*f^2*x^2 - 2*I*a*d^2*e*f*x - I*a*d^2*e^2)*e^(2*d*x + 2*c) + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*e^(
d*x + c))*integral((4*d*f^2*x + 4*d*e*f - (3*d^2*f^2*x^2 + 3*d^2*e^2 + 2*d*e*f - 2*f^2 + 2*(3*d^2*e*f + d*f^2)
*x)*e^(2*d*x + 2*c) + (3*I*d^2*f^2*x^2 + 3*I*d^2*e^2 + 2*I*d*e*f - 2*I*f^2 - 2*(-3*I*d^2*e*f - I*d*f^2)*x)*e^(
d*x + c))/(I*a*d^2*f^3*x^3 + 3*I*a*d^2*e*f^2*x^2 + 3*I*a*d^2*e^2*f*x + I*a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*
e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*e^(3*d*x + 3*c) + (-I*a*d^2*f^3*x^3 - 3*I*a*d^2*e*f^2*x^2 - 3*I*a*d^2
*e^2*f*x - I*a*d^2*e^3)*e^(2*d*x + 2*c) - (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*e^
(d*x + c)), x))/(-I*a*d^2*f^2*x^2 - 2*I*a*d^2*e*f*x - I*a*d^2*e^2 + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2
)*e^(5*d*x + 5*c) + (-I*a*d^2*f^2*x^2 - 2*I*a*d^2*e*f*x - I*a*d^2*e^2)*e^(4*d*x + 4*c) - 2*(a*d^2*f^2*x^2 + 2*
a*d^2*e*f*x + a*d^2*e^2)*e^(3*d*x + 3*c) - 2*(-I*a*d^2*f^2*x^2 - 2*I*a*d^2*e*f*x - I*a*d^2*e^2)*e^(2*d*x + 2*c
) + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*e^(d*x + c))
Sympy [N/A]
Not integrable
Time = 42.90 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32
\[
\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=- \frac {i \int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a}
\]
[In]
integrate(csch(d*x+c)**3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)
[Out]
-I*Integral(csch(c + d*x)**3/(e*sinh(c + d*x) - I*e + f*x*sinh(c + d*x) - I*f*x), x)/a
Maxima [N/A]
Not integrable
Time = 0.82 (sec) , antiderivative size = 775, normalized size of antiderivative = 25.00
\[
\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x }
\]
[In]
integrate(csch(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")
[Out]
-8*f*integrate(1/(-4*I*a*d*f^2*x^2 - 8*I*a*d*e*f*x - 4*I*a*d*e^2 + 4*(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) - 8*(4*d*f*x + 4*d*e + (3*d*f*x*e^(4*c) + (3*d*e - f)*e^(4*c))*e^(4*d*x) + (-3*I*d*f*x*e
^(3*c) + (-3*I*d*e + I*f)*e^(3*c))*e^(3*d*x) - (5*d*f*x*e^(2*c) + (5*d*e - f)*e^(2*c))*e^(2*d*x) + (I*d*f*x*e^
c + (I*d*e - I*f)*e^c)*e^(d*x))/(-8*I*a*d^2*f^2*x^2 - 16*I*a*d^2*e*f*x - 8*I*a*d^2*e^2 + 8*(a*d^2*f^2*x^2*e^(5
*c) + 2*a*d^2*e*f*x*e^(5*c) + a*d^2*e^2*e^(5*c))*e^(5*d*x) - 8*(I*a*d^2*f^2*x^2*e^(4*c) + 2*I*a*d^2*e*f*x*e^(4
*c) + I*a*d^2*e^2*e^(4*c))*e^(4*d*x) - 16*(a*d^2*f^2*x^2*e^(3*c) + 2*a*d^2*e*f*x*e^(3*c) + a*d^2*e^2*e^(3*c))*
e^(3*d*x) - 16*(-I*a*d^2*f^2*x^2*e^(2*c) - 2*I*a*d^2*e*f*x*e^(2*c) - I*a*d^2*e^2*e^(2*c))*e^(2*d*x) + 8*(a*d^2
*f^2*x^2*e^c + 2*a*d^2*e*f*x*e^c + a*d^2*e^2*e^c)*e^(d*x)) - 8*integrate(1/16*(3*d^2*f^2*x^2 + 3*d^2*e^2 + 2*I
*d*e*f - 2*f^2 + 2*(3*d^2*e*f + I*d*f^2)*x)/(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 +
(a*d^2*f^3*x^3*e^c + 3*a*d^2*e*f^2*x^2*e^c + 3*a*d^2*e^2*f*x*e^c + a*d^2*e^3*e^c)*e^(d*x)), x) - 8*integrate(
-1/16*(3*d^2*f^2*x^2 + 3*d^2*e^2 - 2*I*d*e*f - 2*f^2 + 2*(3*d^2*e*f - I*d*f^2)*x)/(a*d^2*f^3*x^3 + 3*a*d^2*e*f
^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 - (a*d^2*f^3*x^3*e^c + 3*a*d^2*e*f^2*x^2*e^c + 3*a*d^2*e^2*f*x*e^c + a*d^
2*e^3*e^c)*e^(d*x)), x)
Giac [F(-1)]
Timed out. \[
\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\text {Timed out}
\]
[In]
integrate(csch(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
[Out]
Timed out
Mupad [N/A]
Not integrable
Time = 4.64 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
\[
\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x
\]
[In]
int(1/(sinh(c + d*x)^3*(e + f*x)*(a + a*sinh(c + d*x)*1i)),x)
[Out]
int(1/(sinh(c + d*x)^3*(e + f*x)*(a + a*sinh(c + d*x)*1i)), x)