3.226 \(\int \frac{\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\sinh (c+d x)}{x (a+b \cosh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0364959, 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 = {} \[ \int \frac{\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 13.0255, size = 0, normalized size = 0. \[ \int \frac{\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sinh[c + d*x]/(x*(a + b*Cosh[c + d*x])),x]

[Out]

Integrate[Sinh[c + d*x]/(x*(a + b*Cosh[c + d*x])), x]

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Maple [A]  time = 0.127, size = 0, normalized size = 0. \begin{align*} \int{\frac{\sinh \left ( dx+c \right ) }{x \left ( a+b\cosh \left ( dx+c \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)/x/(a+b*cosh(d*x+c)),x)

[Out]

int(sinh(d*x+c)/x/(a+b*cosh(d*x+c)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\log \left (x\right )}{b} - \frac{1}{2} \, \int \frac{4 \,{\left (a e^{\left (d x + c\right )} + b\right )}}{b^{2} x e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b x e^{\left (d x + c\right )} + b^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/x/(a+b*cosh(d*x+c)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/x/(a+b*cosh(d*x+c)),x, algorithm="fricas")

[Out]

integral(sinh(d*x + c)/(b*x*cosh(d*x + c) + a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/x/(a+b*cosh(d*x+c)),x)

[Out]

Integral(sinh(c + d*x)/(x*(a + b*cosh(c + d*x))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(d*x + c)/((b*cosh(d*x + c) + a)*x), x)