### 3.870 $$\int \frac{1}{x (a+b \cosh (x) \sinh (x))} \, dx$$

Optimal. Leaf size=19 $\text{Unintegrable}\left (\frac{1}{x \left (a+\frac{1}{2} b \sinh (2 x)\right )},x\right )$

[Out]

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

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Rubi [A]  time = 0.0939793, 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 = {} $\int \frac{1}{x (a+b \cosh (x) \sinh (x))} \, dx$

Veriﬁcation is Not applicable to the result.

[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*} \int \frac{1}{x (a+b \cosh (x) \sinh (x))} \, dx &=\int \frac{1}{x \left (a+\frac{1}{2} b \sinh (2 x)\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.964546, size = 0, normalized size = 0. $\int \frac{1}{x (a+b \cosh (x) \sinh (x))} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( a+b\cosh \left ( x \right ) \sinh \left ( x \right ) \right ) }}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cosh \left (x\right ) \sinh \left (x\right ) + a\right )} x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b x \cosh \left (x\right ) \sinh \left (x\right ) + a x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (a + b \sinh{\left (x \right )} \cosh{\left (x \right )}\right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \cosh \left (x\right ) \sinh \left (x\right ) + a\right )} x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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