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

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0734448, 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 = {} \[ \int \frac{\coth ^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 180., size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [A]  time = 0.999, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm coth} \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)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(coth(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*} 2 \,{\left (a^{2} e^{c} + b^{2} e^{c}\right )} \int -\frac{e^{\left (d x\right )}}{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} + \frac{2}{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 )}} - \int -\frac{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} -{\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 )}}\,{d x} - \int \frac{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} +{\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 )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

[Out]

Timed out