\(\int \frac {\sqrt {a+a \cosh (x)}}{x^3} \, dx\) [132]

Optimal result

Integrand size = 14, antiderivative size = 67 \[ \int \frac {\sqrt {a+a \cosh (x)}}{x^3} \, dx=-\frac {\sqrt {a+a \cosh (x)}}{2 x^2}+\frac {1}{8} \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )-\frac {\sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )}{4 x} \]

[Out]

-1/2*(a+a*cosh(x))^(1/2)/x^2+1/8*Chi(1/2*x)*sech(1/2*x)*(a+a*cosh(x))^(1/2)-1/4*(a+a*cosh(x))^(1/2)*tanh(1/2*x
)/x

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3400, 3378, 3382} \[ \int \frac {\sqrt {a+a \cosh (x)}}{x^3} \, dx=\frac {1}{8} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {\sqrt {a \cosh (x)+a}}{2 x^2}-\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}}{4 x} \]

[In]

Int[Sqrt[a + a*Cosh[x]]/x^3,x]

[Out]

-1/2*Sqrt[a + a*Cosh[x]]/x^2 + (Sqrt[a + a*Cosh[x]]*CoshIntegral[x/2]*Sech[x/2])/8 - (Sqrt[a + a*Cosh[x]]*Tanh
[x/2])/(4*x)

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3400

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(2*a)^IntPart[n]
*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n])), Int[(c + d*x)^m*Sin[e/2
+ a*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh \left (\frac {x}{2}\right )}{x^3} \, dx \\ & = -\frac {\sqrt {a+a \cosh (x)}}{2 x^2}+\frac {1}{4} \left (\sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\sinh \left (\frac {x}{2}\right )}{x^2} \, dx \\ & = -\frac {\sqrt {a+a \cosh (x)}}{2 x^2}-\frac {\sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )}{4 x}+\frac {1}{8} \left (\sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh \left (\frac {x}{2}\right )}{x} \, dx \\ & = -\frac {\sqrt {a+a \cosh (x)}}{2 x^2}+\frac {1}{8} \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )-\frac {\sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )}{4 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {a+a \cosh (x)}}{x^3} \, dx=\frac {\sqrt {a (1+\cosh (x))} \left (-4+x^2 \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )-2 x \tanh \left (\frac {x}{2}\right )\right )}{8 x^2} \]

[In]

Integrate[Sqrt[a + a*Cosh[x]]/x^3,x]

[Out]

(Sqrt[a*(1 + Cosh[x])]*(-4 + x^2*CoshIntegral[x/2]*Sech[x/2] - 2*x*Tanh[x/2]))/(8*x^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+a \cosh (x)}}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int \frac {\sqrt {a+a \cosh (x)}}{x^3} \, dx=\int \frac {\sqrt {a \left (\cosh {\left (x \right )} + 1\right )}}{x^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {a+a \cosh (x)}}{x^3} \, dx=\int { \frac {\sqrt {a \cosh \left (x\right ) + a}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(a*cosh(x) + a)/x^3, x)

Giac [F]

\[ \int \frac {\sqrt {a+a \cosh (x)}}{x^3} \, dx=\int { \frac {\sqrt {a \cosh \left (x\right ) + a}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(a*cosh(x) + a)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+a \cosh (x)}}{x^3} \, dx=\int \frac {\sqrt {a+a\,\mathrm {cosh}\left (x\right )}}{x^3} \,d x \]

[In]

int((a + a*cosh(x))^(1/2)/x^3,x)

[Out]

int((a + a*cosh(x))^(1/2)/x^3, x)