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

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

3.2.38.2 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)
 

3.2.38.3 Rubi [A] (verified)

Time = 0.54 (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))
 

3.2.38.3.1 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])
 

3.2.38.4 Maple [F]

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

input

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

output

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

3.2.38.5 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)
 

3.2.38.6 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)
 

3.2.38.7 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)
 

3.2.38.8 Giac [B] (verification not implemented)

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

Time = 0.29 (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)
 

3.2.38.9 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)