\(\int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx\) [870]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx=\text {Int}\left (\frac {1}{x \left (a+\frac {1}{2} b \sinh (2 x)\right )},x\right ) \]

[Out]

Unintegrable(1/x/(a+1/2*b*sinh(2*x)),x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx=\int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx \]

[In]

Int[1/(x*(a + b*Cosh[x]*Sinh[x])),x]

[Out]

Defer[Int][1/(x*(a + (b*Sinh[2*x])/2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (a+\frac {1}{2} b \sinh (2 x)\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx=\int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx \]

[In]

Integrate[1/(x*(a + b*Cosh[x]*Sinh[x])),x]

[Out]

Integrate[1/(x*(a + b*Cosh[x]*Sinh[x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

\[\int \frac {1}{x \left (a +b \cosh \left (x \right ) \sinh \left (x \right )\right )}d x\]

[In]

int(1/x/(a+b*cosh(x)*sinh(x)),x)

[Out]

int(1/x/(a+b*cosh(x)*sinh(x)),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) \sinh \left (x\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(a+b*cosh(x)*sinh(x)),x, algorithm="fricas")

[Out]

integral(1/(b*x*cosh(x)*sinh(x) + a*x), x)

Sympy [N/A]

Not integrable

Time = 66.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx=\int \frac {1}{x \left (a + b \sinh {\left (x \right )} \cosh {\left (x \right )}\right )}\, dx \]

[In]

integrate(1/x/(a+b*cosh(x)*sinh(x)),x)

[Out]

Integral(1/(x*(a + b*sinh(x)*cosh(x))), x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) \sinh \left (x\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(a+b*cosh(x)*sinh(x)),x, algorithm="maxima")

[Out]

integrate(1/((b*cosh(x)*sinh(x) + a)*x), x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right ) \sinh \left (x\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(a+b*cosh(x)*sinh(x)),x, algorithm="giac")

[Out]

integrate(1/((b*cosh(x)*sinh(x) + a)*x), x)

Mupad [N/A]

Not integrable

Time = 2.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x (a+b \cosh (x) \sinh (x))} \, dx=\int \frac {1}{x\,\left (a+b\,\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\right )} \,d x \]

[In]

int(1/(x*(a + b*cosh(x)*sinh(x))),x)

[Out]

int(1/(x*(a + b*cosh(x)*sinh(x))), x)