\(\int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx\) [124]

Optimal result

Integrand size = 18, antiderivative size = 83 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\cosh \left (\frac {c}{2}\right ) \sqrt {a+a \cosh (c+d x)} \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right )+\sqrt {a+a \cosh (c+d x)} \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3400, 3384, 3379, 3382} \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\cosh \left (\frac {c}{2}\right ) \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a}+\sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \]

[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 3379

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 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 3384

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 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*} \text {integral}& = \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] (verified)

Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\sqrt {a (1+\cosh (c+d x))} \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (\cosh \left (\frac {c}{2}\right ) \text {Chi}\left (\frac {d x}{2}\right )+\sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )\right ) \]

[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])

Maple [F]

\[\int \frac {\sqrt {a +a \cosh \left (d x +c \right )}}{x}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\int \frac {\sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\int { \frac {\sqrt {a \cosh \left (d x + c\right ) + a}}{x} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\frac {1}{2} \, \sqrt {2} {\left (\sqrt {a} {\rm Ei}\left (\frac {1}{2} \, d x\right ) e^{\left (\frac {1}{2} \, c\right )} + \sqrt {a} {\rm Ei}\left (-\frac {1}{2} \, d x\right ) e^{\left (-\frac {1}{2} \, c\right )}\right )} \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\int \frac {\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}}{x} \,d x \]

[In]

int((a + a*cosh(c + d*x))^(1/2)/x,x)

[Out]

int((a + a*cosh(c + d*x))^(1/2)/x, x)