\(\int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx\) [147]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx=\text {Int}\left (\frac {1}{x (a+a \cosh (x))^{3/2}},x\right ) \]

[Out]

Unintegrable(1/x/(a+a*cosh(x))^(3/2),x)

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx=\int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx \]

[In]

Int[1/(x*(a + a*Cosh[x])^(3/2)),x]

[Out]

Defer[Int][1/(x*(a + a*Cosh[x])^(3/2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 8.83 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx=\int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx \]

[In]

Integrate[1/(x*(a + a*Cosh[x])^(3/2)),x]

[Out]

Integrate[1/(x*(a + a*Cosh[x])^(3/2)), x]

Maple [N/A] (verified)

Not integrable

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86

\[\int \frac {1}{x \left (a +a \cosh \left (x \right )\right )^{\frac {3}{2}}}d x\]

[In]

int(1/x/(a+a*cosh(x))^(3/2),x)

[Out]

int(1/x/(a+a*cosh(x))^(3/2),x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.57 \[ \int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

integrate(1/x/(a+a*cosh(x))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*cosh(x) + a)/(a^2*x*cosh(x)^2 + 2*a^2*x*cosh(x) + a^2*x), x)

Sympy [N/A]

Not integrable

Time = 9.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx=\int \frac {1}{x \left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/x/(a+a*cosh(x))**(3/2),x)

[Out]

Integral(1/(x*(a*(cosh(x) + 1))**(3/2)), x)

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

integrate(1/x/(a+a*cosh(x))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((a*cosh(x) + a)^(3/2)*x), x)

Giac [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

integrate(1/x/(a+a*cosh(x))^(3/2),x, algorithm="giac")

[Out]

integrate(1/((a*cosh(x) + a)^(3/2)*x), x)

Mupad [N/A]

Not integrable

Time = 1.84 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+a \cosh (x))^{3/2}} \, dx=\int \frac {1}{x\,{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2}} \,d x \]

[In]

int(1/(x*(a + a*cosh(x))^(3/2)),x)

[Out]

int(1/(x*(a + a*cosh(x))^(3/2)), x)