3.21 \(\int \frac{\cosh (c+d x)}{(e+f x) (a+b \text{csch}(c+d x))} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.107985, 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 = {} \[ \int \frac{\cosh (c+d x)}{(e+f x) (a+b \text{csch}(c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 93.6895, size = 0, normalized size = 0. \[ \int \frac{\cosh (c+d x)}{(e+f x) (a+b \text{csch}(c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cosh[c + d*x]/((e + f*x)*(a + b*Csch[c + d*x])),x]

[Out]

Integrate[Cosh[c + d*x]/((e + f*x)*(a + b*Csch[c + d*x])), x]

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Maple [A]  time = 0.303, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cosh \left ( dx+c \right ) }{ \left ( fx+e \right ) \left ( a+b{\rm csch} \left (dx+c\right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)/(f*x+e)/(a+b*csch(d*x+c)),x)

[Out]

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

[Out]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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