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

Optimal. Leaf size=34 \[ \text{Unintegrable}\left (\frac{\coth (c+d x) \text{csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0663352, 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 = {} \[ \int \frac{\coth (c+d x) \text{csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\coth (c+d x) \text{csch}(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx &=\int \frac{\coth (c+d x) \text{csch}(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])/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

$Aborted

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Maple [A]  time = 0.916, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )}{ \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)/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(coth(d*x+c)*csch(d*x+c)/(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{2 \, e^{\left (d x + c\right )}}{a d f x + a d e -{\left (a d f x e^{\left (2 \, c\right )} + a d e e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} - 2 \, \int -\frac{b d f x + b d e + a f}{2 \,{\left (a^{2} d f^{2} x^{2} + 2 \, a^{2} d e f x + a^{2} d e^{2} -{\left (a^{2} d f^{2} x^{2} e^{c} + 2 \, a^{2} d e f x e^{c} + a^{2} d e^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} + 2 \, \int \frac{b d f x + b d e - a f}{2 \,{\left (a^{2} d f^{2} x^{2} + 2 \, a^{2} d e f x + a^{2} d e^{2} +{\left (a^{2} d f^{2} x^{2} e^{c} + 2 \, a^{2} d e f x e^{c} + a^{2} d e^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} - 2 \, \int -\frac{a b e^{\left (d x + c\right )} - b^{2}}{a^{2} b f x + a^{2} b e -{\left (a^{2} b f x e^{\left (2 \, c\right )} + a^{2} b e e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \,{\left (a^{3} f x e^{c} + a^{3} e e^{c}\right )} e^{\left (d x\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)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

2*e^(d*x + c)/(a*d*f*x + a*d*e - (a*d*f*x*e^(2*c) + a*d*e*e^(2*c))*e^(2*d*x)) - 2*integrate(-1/2*(b*d*f*x + b*
d*e + a*f)/(a^2*d*f^2*x^2 + 2*a^2*d*e*f*x + a^2*d*e^2 - (a^2*d*f^2*x^2*e^c + 2*a^2*d*e*f*x*e^c + a^2*d*e^2*e^c
)*e^(d*x)), x) + 2*integrate(1/2*(b*d*f*x + b*d*e - a*f)/(a^2*d*f^2*x^2 + 2*a^2*d*e*f*x + a^2*d*e^2 + (a^2*d*f
^2*x^2*e^c + 2*a^2*d*e*f*x*e^c + a^2*d*e^2*e^c)*e^(d*x)), x) - 2*integrate(-(a*b*e^(d*x + c) - b^2)/(a^2*b*f*x
+ a^2*b*e - (a^2*b*f*x*e^(2*c) + a^2*b*e*e^(2*c))*e^(2*d*x) - 2*(a^3*f*x*e^c + a^3*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 )}{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)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out