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

Optimal. Leaf size=36 \[ \text{Unintegrable}\left (\frac{\coth (c+d x) \text{csch}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0920873, 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 = {} \[ \int \frac{\coth (c+d x) \text{csch}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Coth[c + d*x]*Csch[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Defer[Int][(Coth[c + d*x]*Csch[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Rubi steps

\begin{align*} \int \frac{\coth (c+d x) \text{csch}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx &=\int \frac{\coth (c+d x) \text{csch}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\\ \end{align*}

Mathematica [F]  time = 180.001, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [A]  time = 1.971, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm coth} \left (dx+c\right ) \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{ \left ( fx+e \right ) \left ( a+b\sinh \left ( dx+c \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)*csch(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(coth(d*x+c)*csch(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a f - 2 \,{\left (b d f x e^{\left (3 \, c\right )} + b d e e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (2 \, a d f x e^{\left (2 \, c\right )} +{\left (2 \, d e - f\right )} a e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 2 \,{\left (b d f x e^{c} + b d e e^{c}\right )} e^{\left (d x\right )}}{a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} d^{2} e f x + a^{2} d^{2} e^{2} +{\left (a^{2} d^{2} f^{2} x^{2} e^{\left (4 \, c\right )} + 2 \, a^{2} d^{2} e f x e^{\left (4 \, c\right )} + a^{2} d^{2} e^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 2 \,{\left (a^{2} d^{2} f^{2} x^{2} e^{\left (2 \, c\right )} + 2 \, a^{2} d^{2} e f x e^{\left (2 \, c\right )} + a^{2} d^{2} e^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + 4 \, \int -\frac{b^{2} d^{2} f^{2} x^{2} + b^{2} d^{2} e^{2} + a b d e f + a^{2} f^{2} +{\left (2 \, b^{2} d^{2} e f + a b d f^{2}\right )} x}{4 \,{\left (a^{3} d^{2} f^{3} x^{3} + 3 \, a^{3} d^{2} e f^{2} x^{2} + 3 \, a^{3} d^{2} e^{2} f x + a^{3} d^{2} e^{3} -{\left (a^{3} d^{2} f^{3} x^{3} e^{c} + 3 \, a^{3} d^{2} e f^{2} x^{2} e^{c} + 3 \, a^{3} d^{2} e^{2} f x e^{c} + a^{3} d^{2} e^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} - 4 \, \int \frac{b^{2} d^{2} f^{2} x^{2} + b^{2} d^{2} e^{2} - a b d e f + a^{2} f^{2} +{\left (2 \, b^{2} d^{2} e f - a b d f^{2}\right )} x}{4 \,{\left (a^{3} d^{2} f^{3} x^{3} + 3 \, a^{3} d^{2} e f^{2} x^{2} + 3 \, a^{3} d^{2} e^{2} f x + a^{3} d^{2} e^{3} +{\left (a^{3} d^{2} f^{3} x^{3} e^{c} + 3 \, a^{3} d^{2} e f^{2} x^{2} e^{c} + 3 \, a^{3} d^{2} e^{2} f x e^{c} + a^{3} d^{2} e^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} + 4 \, \int -\frac{a b^{2} e^{\left (d x + c\right )} - b^{3}}{2 \,{\left (a^{3} b f x + a^{3} b e -{\left (a^{3} b f x e^{\left (2 \, c\right )} + a^{3} b e e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \,{\left (a^{4} f x e^{c} + a^{4} e 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(coth(d*x+c)*csch(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\coth \left (d x + c\right ) \operatorname{csch}\left (d x + c\right )^{2}}{a f x + a e +{\left (b f x + b e\right )} \sinh \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(coth(d*x + c)*csch(d*x + c)^2/(a*f*x + a*e + (b*f*x + b*e)*sinh(d*x + c)), x)

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

[Out]

Timed out