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

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

[Out]

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

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {\text {csch}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

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

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 0.66, size = 0, normalized size = 0.00 \[ \frac {{\left (i \, a d f^{2} x^{2} + 2 i \, a d e f x + i \, a d e^{2} + {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-i \, a d f^{2} x^{2} - 2 i \, a d e f x - i \, a d e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm integral}\left (\frac {{\left (-2 i \, d f x - 2 i \, d e - 4 i \, f\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (d f x + d e + 2 \, f\right )} e^{\left (d x + c\right )} + 8 i \, f}{i \, a d f^{3} x^{3} + 3 i \, a d e f^{2} x^{2} + 3 i \, a d e^{2} f x + i \, a d e^{3} + {\left (a d f^{3} x^{3} + 3 \, a d e f^{2} x^{2} + 3 \, a d e^{2} f x + a d e^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-i \, a d f^{3} x^{3} - 3 i \, a d e f^{2} x^{2} - 3 i \, a d e^{2} f x - i \, a d e^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (a d f^{3} x^{3} + 3 \, a d e f^{2} x^{2} + 3 \, a d e^{2} f x + a d e^{3}\right )} e^{\left (d x + c\right )}}, x\right ) - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} + 4 i}{i \, a d f^{2} x^{2} + 2 i \, a d e f x + i \, a d e^{2} + {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-i \, a d f^{2} x^{2} - 2 i \, a d e f x - i \, a d e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (d x + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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 - 2*I*d*e - 4*I*f)*e^(2*d*x + 2*c) - 2*(d*f*x + d*e + 2*f)*e^(d*x + c) + 8*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))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 1.67, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {csch}\left (d x +c \right )^{2}}{\left (f x +e \right )^{2} \left (a +i a \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^2/(f*x+e)^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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -4 i \, f \int \frac {1}{-i \, a d f^{3} x^{3} - 3 i \, a d e f^{2} x^{2} - 3 i \, a d e^{2} f x - i \, a d e^{3} + {\left (a d f^{3} x^{3} e^{c} + 3 \, a d e f^{2} x^{2} e^{c} + 3 \, a d e^{2} f x e^{c} + a d e^{3} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \frac {4 \, {\left (i \, e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (d x + c\right )} - 2 i\right )}}{2 i \, a d f^{2} x^{2} + 4 i \, a d e f x + 2 i \, a d e^{2} + 2 \, {\left (a d f^{2} x^{2} e^{\left (3 \, c\right )} + 2 \, a d e f x e^{\left (3 \, c\right )} + a d e^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (-2 i \, a d f^{2} x^{2} e^{\left (2 \, c\right )} - 4 i \, a d e f x e^{\left (2 \, c\right )} - 2 i \, a d e^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a d f^{2} x^{2} e^{c} + 2 \, a d e f x e^{c} + a d e^{2} e^{c}\right )} e^{\left (d x\right )}} - 4 \, \int -\frac {i \, d f x + i \, d e + 2 \, f}{4 \, {\left (a d f^{3} x^{3} + 3 \, a d e f^{2} x^{2} + 3 \, a d e^{2} f x + a d e^{3} - {\left (a d f^{3} x^{3} e^{c} + 3 \, a d e f^{2} x^{2} e^{c} + 3 \, a d e^{2} f x e^{c} + a d e^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} - 4 \, \int \frac {i \, d f x + i \, d e - 2 \, f}{4 \, {\left (a d f^{3} x^{3} + 3 \, a d e f^{2} x^{2} + 3 \, a d e^{2} f x + a d e^{3} + {\left (a d f^{3} x^{3} e^{c} + 3 \, a d e f^{2} x^{2} e^{c} + 3 \, a d e^{2} f x e^{c} + a d e^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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) - 4*I*a*d*e*f*x*e^(2*c) - 2*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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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