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

Optimal. Leaf size=33 \[ \text{Unintegrable}\left (\frac{\text{csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.075928, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Csch[c + d*x]^2/((e + f*x)*(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 = 131.951, size = 0, normalized size = 0. \[ \int \frac{\text{csch}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

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]

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Maple [A]  time = 0.571, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{ \left ( fx+e \right ) \left ( a+ia\sinh \left ( dx+c \right ) \right ) }}\, dx \end{align*}

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)

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

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*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) - 2*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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (i \, a d f x + i \, a d e +{\left (a d f x + a d e\right )} e^{\left (3 \, d x + 3 \, c\right )} +{\left (-i \, a d f x - i \, a d e\right )} e^{\left (2 \, d x + 2 \, c\right )} -{\left (a d f x + a d e\right )} e^{\left (d x + c\right )}\right )}{\rm integral}\left (\frac{{\left (-2 i \, d f x - 2 i \, d e - 2 i \, f\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \,{\left (d f x + d e + f\right )} e^{\left (d x + c\right )} + 4 i \, f}{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 )}}, 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 x + i \, a d e +{\left (a d f x + a d e\right )} e^{\left (3 \, d x + 3 \, c\right )} +{\left (-i \, a d f x - i \, a d e\right )} e^{\left (2 \, d x + 2 \, c\right )} -{\left (a d f x + a d e\right )} e^{\left (d x + c\right )}} \end{align*}

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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

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