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

Optimal. Leaf size=67 \[ -\frac {\sqrt {a+a \cosh (x)}}{2 x^2}+\frac {1}{8} \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )-\frac {\sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )}{4 x} \]

[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

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Rubi [A]
time = 0.08, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3400, 3378, 3382} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\sqrt {a+a \cosh (x)}}{x^3} \, dx &=\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*}

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Mathematica [A]
time = 0.06, size = 44, normalized size = 0.66 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [F]
time = 0.38, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a +a \cosh \left (x \right )}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cosh(x) + a)/x^3, x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a \left (\cosh {\left (x \right )} + 1\right )}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cosh(x) + a)/x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+a\,\mathrm {cosh}\left (x\right )}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*cosh(x))^(1/2)/x^3,x)

[Out]

int((a + a*cosh(x))^(1/2)/x^3, x)

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