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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 14.21, size = 0, normalized size = 0.00 \[ \int \frac {\sinh (c+d x)}{x (a+b \cosh (c+d x))} \, dx \]

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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fricas [A]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sinh \left (d x + c\right )}{b x \cosh \left (d x + c\right ) + a x}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (d x + c\right )}{{\left (b \cosh \left (d x + c\right ) + a\right )} x}\,{d x} \]

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)

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maple [A]  time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (d x +c \right )}{x \left (a +b \cosh \left (d x +c \right )\right )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {\log \relax (x)}{b} - \frac {1}{2} \, \int \frac {4 \, {\left (a e^{\left (d x + c\right )} + b\right )}}{b^{2} x e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b x e^{\left (d x + c\right )} + b^{2} x}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {sinh}\left (c+d\,x\right )}{x\,\left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh {\left (c + d x \right )}}{x \left (a + b \cosh {\left (c + d x \right )}\right )}\, dx \]

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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