3.125 \(\int \frac{\sqrt{a+a \cosh (c+d x)}}{x^2} \, dx\)

Optimal. Leaf size=110 \[ \frac{1}{2} d \sinh \left (\frac{c}{2}\right ) \text{Chi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}+\frac{1}{2} d \cosh \left (\frac{c}{2}\right ) \text{Shi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}-\frac{\sqrt{a \cosh (c+d x)+a}}{x} \]

[Out]

-(Sqrt[a + a*Cosh[c + d*x]]/x) + (d*Sqrt[a + a*Cosh[c + d*x]]*CoshIntegral[(d*x)/2]*Sech[c/2 + (d*x)/2]*Sinh[c
/2])/2 + (d*Cosh[c/2]*Sqrt[a + a*Cosh[c + d*x]]*Sech[c/2 + (d*x)/2]*SinhIntegral[(d*x)/2])/2

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Rubi [A]  time = 0.140128, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3319, 3297, 3303, 3298, 3301} \[ \frac{1}{2} d \sinh \left (\frac{c}{2}\right ) \text{Chi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}+\frac{1}{2} d \cosh \left (\frac{c}{2}\right ) \text{Shi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}-\frac{\sqrt{a \cosh (c+d x)+a}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Cosh[c + d*x]]/x^2,x]

[Out]

-(Sqrt[a + a*Cosh[c + d*x]]/x) + (d*Sqrt[a + a*Cosh[c + d*x]]*CoshIntegral[(d*x)/2]*Sech[c/2 + (d*x)/2]*Sinh[c
/2])/2 + (d*Cosh[c/2]*Sqrt[a + a*Cosh[c + d*x]]*Sech[c/2 + (d*x)/2]*SinhIntegral[(d*x)/2])/2

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 3297

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 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

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]

Rule 3301

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]

Rubi steps

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

Mathematica [A]  time = 0.136294, size = 75, normalized size = 0.68 \[ \frac{\sqrt{a (\cosh (c+d x)+1)} \left (d x \sinh \left (\frac{c}{2}\right ) \text{Chi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{1}{2} (c+d x)\right )+d x \cosh \left (\frac{c}{2}\right ) \text{Shi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{1}{2} (c+d x)\right )-2\right )}{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Cosh[c + d*x]]/x^2,x]

[Out]

(Sqrt[a*(1 + Cosh[c + d*x])]*(-2 + d*x*CoshIntegral[(d*x)/2]*Sech[(c + d*x)/2]*Sinh[c/2] + d*x*Cosh[c/2]*Sech[
(c + d*x)/2]*SinhIntegral[(d*x)/2]))/(2*x)

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt{a+a\cosh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(cosh(c + d*x) + 1))/x**2, x)

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Giac [A]  time = 1.21604, size = 92, normalized size = 0.84 \begin{align*} \frac{\sqrt{2}{\left (\sqrt{a} d x{\rm Ei}\left (\frac{1}{2} \, d x\right ) e^{\left (\frac{1}{2} \, c\right )} - \sqrt{a} d x{\rm Ei}\left (-\frac{1}{2} \, d x\right ) e^{\left (-\frac{1}{2} \, c\right )} - 2 \, \sqrt{a} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - 2 \, \sqrt{a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}\right )}}{4 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*(sqrt(a)*d*x*Ei(1/2*d*x)*e^(1/2*c) - sqrt(a)*d*x*Ei(-1/2*d*x)*e^(-1/2*c) - 2*sqrt(a)*e^(1/2*d*x +
1/2*c) - 2*sqrt(a)*e^(-1/2*d*x - 1/2*c))/x