3.137 \(\int \frac{(a+a \cosh (x))^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=79 \[ \frac{3}{4} a \text{Shi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{3}{4} a \text{Shi}\left (\frac{3 x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{2 a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}}{x} \]

[Out]

(-2*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/x + (3*a*Sqrt[a + a*Cosh[x]]*Sech[x/2]*SinhIntegral[x/2])/4 + (3*a*Sqrt
[a + a*Cosh[x]]*Sech[x/2]*SinhIntegral[(3*x)/2])/4

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Rubi [A]  time = 0.13079, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3319, 3313, 3298} \[ \frac{3}{4} a \text{Shi}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{3}{4} a \text{Shi}\left (\frac{3 x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{2 a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cosh[x])^(3/2)/x^2,x]

[Out]

(-2*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/x + (3*a*Sqrt[a + a*Cosh[x]]*Sech[x/2]*SinhIntegral[x/2])/4 + (3*a*Sqrt
[a + a*Cosh[x]]*Sech[x/2]*SinhIntegral[(3*x)/2])/4

Rule 3319

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])

Rule 3313

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

Rule 3298

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

Rubi steps

\begin{align*} \int \frac{(a+a \cosh (x))^{3/2}}{x^2} \, dx &=\left (2 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \frac{\cosh ^3\left (\frac{x}{2}\right )}{x^2} \, dx\\ &=-\frac{2 a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}}{x}+\left (3 i a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \left (-\frac{i \sinh \left (\frac{x}{2}\right )}{4 x}-\frac{i \sinh \left (\frac{3 x}{2}\right )}{4 x}\right ) \, dx\\ &=-\frac{2 a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}}{x}+\frac{1}{4} \left (3 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \frac{\sinh \left (\frac{x}{2}\right )}{x} \, dx+\frac{1}{4} \left (3 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \frac{\sinh \left (\frac{3 x}{2}\right )}{x} \, dx\\ &=-\frac{2 a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}}{x}+\frac{3}{4} a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right ) \text{Shi}\left (\frac{x}{2}\right )+\frac{3}{4} a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right ) \text{Shi}\left (\frac{3 x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0838189, size = 53, normalized size = 0.67 \[ -\frac{a \text{sech}\left (\frac{x}{2}\right ) \sqrt{a (\cosh (x)+1)} \left (-3 x \text{Shi}\left (\frac{x}{2}\right )-3 x \text{Shi}\left (\frac{3 x}{2}\right )+8 \cosh ^3\left (\frac{x}{2}\right )\right )}{4 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Sqrt[a*(1 + Cosh[x])]*Sech[x/2]*(8*Cosh[x/2]^3 - 3*x*SinhIntegral[x/2] - 3*x*SinhIntegral[(3*x)/2]))/(4*x)

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+a\cosh \left ( x \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \cosh \left (x\right ) + a\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.26839, size = 115, normalized size = 1.46 \begin{align*} \frac{\sqrt{2}{\left (3 \, a^{\frac{3}{2}} x{\rm Ei}\left (\frac{3}{2} \, x\right ) + 3 \, a^{\frac{3}{2}} x{\rm Ei}\left (\frac{1}{2} \, x\right ) - 3 \, a^{\frac{3}{2}} x{\rm Ei}\left (-\frac{1}{2} \, x\right ) - 3 \, a^{\frac{3}{2}} x{\rm Ei}\left (-\frac{3}{2} \, x\right ) - 2 \, a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, x\right )} - 6 \, a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} - 6 \, a^{\frac{3}{2}} e^{\left (-\frac{1}{2} \, x\right )} - 2 \, a^{\frac{3}{2}} e^{\left (-\frac{3}{2} \, x\right )}\right )}}{8 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*(3*a^(3/2)*x*Ei(3/2*x) + 3*a^(3/2)*x*Ei(1/2*x) - 3*a^(3/2)*x*Ei(-1/2*x) - 3*a^(3/2)*x*Ei(-3/2*x) -
2*a^(3/2)*e^(3/2*x) - 6*a^(3/2)*e^(1/2*x) - 6*a^(3/2)*e^(-1/2*x) - 2*a^(3/2)*e^(-3/2*x))/x