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]

-(a+a*cosh(d*x+c))^(1/2)/x+1/2*d*cosh(1/2*c)*sech(1/2*d*x+1/2*c)*Shi(1/2*d*x)*(a+a*cosh(d*x+c))^(1/2)+1/2*d*Ch
i(1/2*d*x)*sech(1/2*d*x+1/2*c)*sinh(1/2*c)*(a+a*cosh(d*x+c))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 110, normalized size of antiderivative = 1.00, 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 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 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]

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 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])

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 = 0.14, 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)

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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

________________________________________________________________________________________

giac [A]  time = 0.14, size = 68, normalized size = 0.62 \[ \frac {\sqrt {2} {\left (\sqrt {a} d x {\rm Ei}\left (\frac {1}{2} \, d x\right ) e^{\left (\frac {1}{2} \, c\right )} - \sqrt {a} d x {\rm Ei}\left (-\frac {1}{2} \, d x\right ) e^{\left (-\frac {1}{2} \, c\right )} - 2 \, \sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 2 \, \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}\right )}}{4 \, x} \]

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

________________________________________________________________________________________

maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +a \cosh \left (d x +c \right )}}{x^{2}}\, dx \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \cosh \left (d x + c\right ) + a}}{x^{2}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}{x^{2}}\, dx \]

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)

________________________________________________________________________________________