\(\int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx\) [138]

Optimal result

Integrand size = 14, antiderivative size = 109 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx=-\frac {a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}}{x^2}+\frac {3}{16} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+\frac {9}{16} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )-\frac {3 a \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )}{2 x} \] Output:

-a*cosh(1/2*x)^2*(a+a*cosh(x))^(1/2)/x^2+3/16*a*(a+a*cosh(x))^(1/2)*Chi(1/ 
2*x)*sech(1/2*x)+9/16*a*(a+a*cosh(x))^(1/2)*Chi(3/2*x)*sech(1/2*x)-3/2*a*c 
osh(1/2*x)*(a+a*cosh(x))^(1/2)*sinh(1/2*x)/x
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx=\frac {(a (1+\cosh (x)))^{3/2} \left (3 x^2 \text {Chi}\left (\frac {x}{2}\right ) \text {sech}^3\left (\frac {x}{2}\right )+9 x^2 \text {Chi}\left (\frac {3 x}{2}\right ) \text {sech}^3\left (\frac {x}{2}\right )-8 \left (2+3 x \tanh \left (\frac {x}{2}\right )\right )\right )}{32 x^2} \] Input:

Integrate[(a + a*Cosh[x])^(3/2)/x^3,x]
 

Output:

((a*(1 + Cosh[x]))^(3/2)*(3*x^2*CoshIntegral[x/2]*Sech[x/2]^3 + 9*x^2*Cosh 
Integral[(3*x)/2]*Sech[x/2]^3 - 8*(2 + 3*x*Tanh[x/2])))/(32*x^2)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3800, 3042, 3795, 3042, 3782, 3793, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+a \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2}}{x^3}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \int \frac {\cosh ^3\left (\frac {x}{2}\right )}{x^3}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \int \frac {\sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3}{x^3}dx\)

\(\Big \downarrow \) 3795

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {9}{8} \int \frac {\cosh ^3\left (\frac {x}{2}\right )}{x}dx-\frac {3}{4} \int \frac {\cosh \left (\frac {x}{2}\right )}{x}dx-\frac {\cosh ^3\left (\frac {x}{2}\right )}{2 x^2}-\frac {3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )}{4 x}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (-\frac {3}{4} \int \frac {\sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )}{x}dx+\frac {9}{8} \int \frac {\sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3}{x}dx-\frac {\cosh ^3\left (\frac {x}{2}\right )}{2 x^2}-\frac {3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )}{4 x}\right )\)

\(\Big \downarrow \) 3782

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {9}{8} \int \frac {\sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )^3}{x}dx-\frac {3 \text {Chi}\left (\frac {x}{2}\right )}{4}-\frac {\cosh ^3\left (\frac {x}{2}\right )}{2 x^2}-\frac {3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )}{4 x}\right )\)

\(\Big \downarrow \) 3793

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {9}{8} \int \left (\frac {3 \cosh \left (\frac {x}{2}\right )}{4 x}+\frac {\cosh \left (\frac {3 x}{2}\right )}{4 x}\right )dx-\frac {3 \text {Chi}\left (\frac {x}{2}\right )}{4}-\frac {\cosh ^3\left (\frac {x}{2}\right )}{2 x^2}-\frac {3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )}{4 x}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (-\frac {3 \text {Chi}\left (\frac {x}{2}\right )}{4}+\frac {9}{8} \left (\frac {3 \text {Chi}\left (\frac {x}{2}\right )}{4}+\frac {\text {Chi}\left (\frac {3 x}{2}\right )}{4}\right )-\frac {\cosh ^3\left (\frac {x}{2}\right )}{2 x^2}-\frac {3 \sinh \left (\frac {x}{2}\right ) \cosh ^2\left (\frac {x}{2}\right )}{4 x}\right )\)

Input:

Int[(a + a*Cosh[x])^(3/2)/x^3,x]
 

Output:

2*a*Sqrt[a + a*Cosh[x]]*Sech[x/2]*(-1/2*Cosh[x/2]^3/x^2 - (3*CoshIntegral[ 
x/2])/4 + (9*((3*CoshIntegral[x/2])/4 + CoshIntegral[(3*x)/2]/4))/8 - (3*C 
osh[x/2]^2*Sinh[x/2])/(4*x))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[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 3795

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ 
b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) 
*(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2)))   Int[(c + 
d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* 
(m + 2)))   Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, 
c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
 

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (81) = 162\).

Time = 0.12 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.56 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx=\frac {1}{32} \, \sqrt {2} {\left (\frac {9 \, a^{\frac {3}{2}} x^{2} {\rm Ei}\left (\frac {3}{2} \, x\right ) + 3 \, a^{\frac {3}{2}} x^{2} {\rm Ei}\left (\frac {1}{2} \, x\right ) + a^{\frac {3}{2}} x^{2} {\rm Ei}\left (-\frac {1}{2} \, x\right ) - 6 \, a^{\frac {3}{2}} x e^{\left (\frac {3}{2} \, x\right )} - 6 \, a^{\frac {3}{2}} x e^{\left (\frac {1}{2} \, x\right )} + 2 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} - 4 \, a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )} - 12 \, a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} - 4 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )}}{x^{2}} + \frac {2 \, a^{\frac {3}{2}} x^{2} {\rm Ei}\left (-\frac {1}{2} \, x\right ) + 9 \, a^{\frac {3}{2}} x^{2} {\rm Ei}\left (-\frac {3}{2} \, x\right ) + 4 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} + 6 \, a^{\frac {3}{2}} x e^{\left (-\frac {3}{2} \, x\right )} - 8 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} - 4 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )}}{x^{2}}\right )} \] Input:

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

Output:

1/32*sqrt(2)*((9*a^(3/2)*x^2*Ei(3/2*x) + 3*a^(3/2)*x^2*Ei(1/2*x) + a^(3/2) 
*x^2*Ei(-1/2*x) - 6*a^(3/2)*x*e^(3/2*x) - 6*a^(3/2)*x*e^(1/2*x) + 2*a^(3/2 
)*x*e^(-1/2*x) - 4*a^(3/2)*e^(3/2*x) - 12*a^(3/2)*e^(1/2*x) - 4*a^(3/2)*e^ 
(-1/2*x))/x^2 + (2*a^(3/2)*x^2*Ei(-1/2*x) + 9*a^(3/2)*x^2*Ei(-3/2*x) + 4*a 
^(3/2)*x*e^(-1/2*x) + 6*a^(3/2)*x*e^(-3/2*x) - 8*a^(3/2)*e^(-1/2*x) - 4*a^ 
(3/2)*e^(-3/2*x))/x^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*a*(int(sqrt(cosh(x) + 1)/x**3,x) + int((sqrt(cosh(x) + 1)*cosh(x)) 
/x**3,x))