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

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

[Out]

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

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Rubi [A]  time = 0.0834457, 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 = {} \[ \int \frac{\cosh ^2(c+d x) \coth (c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 180.002, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [A]  time = 1.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}{\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(cosh(d*x+c)^2*coth(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(cosh(d*x+c)^2*coth(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{e^{\left (-c + \frac{d e}{f}\right )} E_{1}\left (\frac{{\left (f x + e\right )} d}{f}\right )}{2 \, b f} - \frac{e^{\left (c - \frac{d e}{f}\right )} E_{1}\left (-\frac{{\left (f x + e\right )} d}{f}\right )}{2 \, b f} - \frac{a \log \left (f x + e\right )}{b^{2} f} + \frac{1}{4} \, \int \frac{8 \,{\left (a^{2} b + b^{3} -{\left (a^{3} e^{c} + a b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{a b^{3} f x + a b^{3} e -{\left (a b^{3} f x e^{\left (2 \, c\right )} + a b^{3} e e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \,{\left (a^{2} b^{2} f x e^{c} + a^{2} b^{2} e e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \int \frac{1}{a f x + a e +{\left (a f x e^{c} + a e e^{c}\right )} e^{\left (d x\right )}}\,{d x} + \int -\frac{1}{a f x + a e -{\left (a f x e^{c} + a 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(cosh(d*x+c)^2*coth(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*e^(-c + d*e/f)*exp_integral_e(1, (f*x + e)*d/f)/(b*f) - 1/2*e^(c - d*e/f)*exp_integral_e(1, -(f*x + e)*d/
f)/(b*f) - a*log(f*x + e)/(b^2*f) + 1/4*integrate(8*(a^2*b + b^3 - (a^3*e^c + a*b^2*e^c)*e^(d*x))/(a*b^3*f*x +
a*b^3*e - (a*b^3*f*x*e^(2*c) + a*b^3*e*e^(2*c))*e^(2*d*x) - 2*(a^2*b^2*f*x*e^c + a^2*b^2*e*e^c)*e^(d*x)), x)
- integrate(1/(a*f*x + a*e + (a*f*x*e^c + a*e*e^c)*e^(d*x)), x) + integrate(-1/(a*f*x + a*e - (a*f*x*e^c + a*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{\cosh \left (d x + c\right )^{2} \coth \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(cosh(d*x+c)^2*coth(d*x+c)/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

[Out]

Integral(cosh(c + d*x)**2*coth(c + d*x)/((a + b*sinh(c + d*x))*(e + f*x)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x