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

Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 91.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[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 [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {csch}\left (d x +c \right )^{2}}{\left (f x +e \right ) \left (a +i a \sinh \left (d x +c \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*f*x*e + I*a*d*e^2 + (a*d*f^2*x^2 + 2*a*d*f*x*e + a*d*e^2)*e^(3*d*x + 3*c) + (-I
*a*d*f^2*x^2 - 2*I*a*d*f*x*e - I*a*d*e^2)*e^(2*d*x + 2*c) - (a*d*f^2*x^2 + 2*a*d*f*x*e + 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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________