\(\int \frac {(a+a \cosh (x))^{3/2}}{x} \, dx\) [136]

Optimal result

Integrand size = 14, antiderivative size = 55 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x} \, dx=\frac {3}{2} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+\frac {1}{2} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \]

[Out]

3/2*a*Chi(1/2*x)*sech(1/2*x)*(a+a*cosh(x))^(1/2)+1/2*a*Chi(3/2*x)*sech(1/2*x)*(a+a*cosh(x))^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3400, 3393, 3382} \[ \int \frac {(a+a \cosh (x))^{3/2}}{x} \, dx=\frac {3}{2} a \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {1}{2} a \text {Chi}\left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \]

[In]

Int[(a + a*Cosh[x])^(3/2)/x,x]

[Out]

(3*a*Sqrt[a + a*Cosh[x]]*CoshIntegral[x/2]*Sech[x/2])/2 + (a*Sqrt[a + a*Cosh[x]]*CoshIntegral[(3*x)/2]*Sech[x/
2])/2

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 3393

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

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 (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} \, dx \\ & = \left (2 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \left (\frac {3 \cosh \left (\frac {x}{2}\right )}{4 x}+\frac {\cosh \left (\frac {3 x}{2}\right )}{4 x}\right ) \, dx \\ & = \frac {1}{2} \left (a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh \left (\frac {3 x}{2}\right )}{x} \, dx+\frac {1}{2} \left (3 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh \left (\frac {x}{2}\right )}{x} \, dx \\ & = \frac {3}{2} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+\frac {1}{2} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x} \, dx=\frac {1}{2} a \sqrt {a (1+\cosh (x))} \left (3 \text {Chi}\left (\frac {x}{2}\right )+\text {Chi}\left (\frac {3 x}{2}\right )\right ) \text {sech}\left (\frac {x}{2}\right ) \]

[In]

Integrate[(a + a*Cosh[x])^(3/2)/x,x]

[Out]

(a*Sqrt[a*(1 + Cosh[x])]*(3*CoshIntegral[x/2] + CoshIntegral[(3*x)/2])*Sech[x/2])/2

Maple [F]

\[\int \frac {\left (a +a \cosh \left (x \right )\right )^{\frac {3}{2}}}{x}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {(a+a \cosh (x))^{3/2}}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {(a+a \cosh (x))^{3/2}}{x} \, dx=\int \frac {\left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}{x}\, dx \]

[In]

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

[Out]

Integral((a*(cosh(x) + 1))**(3/2)/x, x)

Maxima [F]

\[ \int \frac {(a+a \cosh (x))^{3/2}}{x} \, dx=\int { \frac {{\left (a \cosh \left (x\right ) + a\right )}^{\frac {3}{2}}}{x} \,d x } \]

[In]

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

[Out]

integrate((a*cosh(x) + a)^(3/2)/x, x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x} \, dx=\frac {1}{4} \, \sqrt {2} {\left (a^{\frac {3}{2}} {\rm Ei}\left (\frac {3}{2} \, x\right ) + 3 \, a^{\frac {3}{2}} {\rm Ei}\left (\frac {1}{2} \, x\right ) + 3 \, a^{\frac {3}{2}} {\rm Ei}\left (-\frac {1}{2} \, x\right ) + a^{\frac {3}{2}} {\rm Ei}\left (-\frac {3}{2} \, x\right )\right )} \]

[In]

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

[Out]

1/4*sqrt(2)*(a^(3/2)*Ei(3/2*x) + 3*a^(3/2)*Ei(1/2*x) + 3*a^(3/2)*Ei(-1/2*x) + a^(3/2)*Ei(-3/2*x))

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \cosh (x))^{3/2}}{x} \, dx=\int \frac {{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2}}{x} \,d x \]

[In]

int((a + a*cosh(x))^(3/2)/x,x)

[Out]

int((a + a*cosh(x))^(3/2)/x, x)