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

Optimal. Leaf size=83 \[ \cosh \left (\frac{c}{2}\right ) \text{Chi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}+\sinh \left (\frac{c}{2}\right ) \text{Shi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a} \]

[Out]

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

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Rubi [A]  time = 0.131785, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3319, 3303, 3298, 3301} \[ \cosh \left (\frac{c}{2}\right ) \text{Chi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a}+\sinh \left (\frac{c}{2}\right ) \text{Shi}\left (\frac{d x}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{a \cosh (c+d x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Cosh[c + d*x]]/x,x]

[Out]

Cosh[c/2]*Sqrt[a + a*Cosh[c + d*x]]*CoshIntegral[(d*x)/2]*Sech[c/2 + (d*x)/2] + Sqrt[a + a*Cosh[c + d*x]]*Sech
[c/2 + (d*x)/2]*Sinh[c/2]*SinhIntegral[(d*x)/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 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} \, 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} \, dx\\ &=\left (\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{\cosh \left (\frac{d x}{2}\right )}{x} \, 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 ) \sinh \left (\frac{c}{2}\right )\right ) \int \frac{\sinh \left (\frac{d x}{2}\right )}{x} \, dx\\ &=\cosh \left (\frac{c}{2}\right ) \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 )+\sqrt{a+a \cosh (c+d x)} \text{sech}\left (\frac{c}{2}+\frac{d x}{2}\right ) \sinh \left (\frac{c}{2}\right ) \text{Shi}\left (\frac{d x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0750043, size = 54, normalized size = 0.65 \[ \text{sech}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cosh (c+d x)+1)} \left (\cosh \left (\frac{c}{2}\right ) \text{Chi}\left (\frac{d x}{2}\right )+\sinh \left (\frac{c}{2}\right ) \text{Shi}\left (\frac{d x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\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,x)

[Out]

int((a+a*cosh(d*x+c))^(1/2)/x,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}\,{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,x, algorithm="maxima")

[Out]

integrate(sqrt(a*cosh(d*x + c) + a)/x, 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,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}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.29986, size = 43, normalized size = 0.52 \begin{align*} \frac{1}{2} \, \sqrt{2}{\left (\sqrt{a}{\rm Ei}\left (\frac{1}{2} \, d x\right ) e^{\left (\frac{1}{2} \, c\right )} + \sqrt{a}{\rm Ei}\left (-\frac{1}{2} \, d x\right ) e^{\left (-\frac{1}{2} \, c\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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