\(\int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx\) [126]

Optimal result

Integrand size = 18, antiderivative size = 151 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=-\frac {\sqrt {a+a \cosh (c+d x)}}{2 x^2}+\frac {1}{8} d^2 \cosh \left (\frac {c}{2}\right ) \sqrt {a+a \cosh (c+d x)} \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right )+\frac {1}{8} d^2 \sqrt {a+a \cosh (c+d x)} \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )-\frac {d \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{4 x} \] Output:

-1/2*(a+a*cosh(d*x+c))^(1/2)/x^2+1/8*d^2*cosh(1/2*c)*(a+a*cosh(d*x+c))^(1/ 
2)*Chi(1/2*d*x)*sech(1/2*d*x+1/2*c)+1/8*d^2*(a+a*cosh(d*x+c))^(1/2)*sech(1 
/2*d*x+1/2*c)*sinh(1/2*c)*Shi(1/2*d*x)-1/4*d*(a+a*cosh(d*x+c))^(1/2)*tanh( 
1/2*d*x+1/2*c)/x
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=\frac {\sqrt {a (1+\cosh (c+d x))} \left (-4+d^2 x^2 \cosh \left (\frac {c}{2}\right ) \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {1}{2} (c+d x)\right )+d^2 x^2 \text {sech}\left (\frac {1}{2} (c+d x)\right ) \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )-2 d x \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 x^2} \] Input:

Integrate[Sqrt[a + a*Cosh[c + d*x]]/x^3,x]
 

Output:

(Sqrt[a*(1 + Cosh[c + d*x])]*(-4 + d^2*x^2*Cosh[c/2]*CoshIntegral[(d*x)/2] 
*Sech[(c + d*x)/2] + d^2*x^2*Sech[(c + d*x)/2]*Sinh[c/2]*SinhIntegral[(d*x 
)/2] - 2*d*x*Tanh[(c + d*x)/2]))/(8*x^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.68 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3042, 3800, 3042, 3778, 26, 3042, 26, 3778, 3042, 3784, 26, 3042, 26, 3779, 3782}

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.

\(\displaystyle \int \frac {\sqrt {a \cosh (c+d x)+a}}{x^3} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}+\frac {1}{4} i d \int -\frac {i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x^2}dx\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {1}{4} d \int \frac {\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x^2}dx-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}+\frac {1}{4} d \int -\frac {i \sin \left (\frac {i c}{2}+\frac {i d x}{2}\right )}{x^2}dx\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}-\frac {1}{4} i d \int \frac {\sin \left (\frac {i c}{2}+\frac {i d x}{2}\right )}{x^2}dx\right )\)

\(\Big \downarrow \) 3778

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}-\frac {1}{4} i d \left (\frac {1}{2} i d \int \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}dx-\frac {i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}-\frac {1}{4} i d \left (\frac {1}{2} i d \int \frac {\sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{2}\right )}{x}dx-\frac {i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}\right )\right )\)

\(\Big \downarrow \) 3784

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}-\frac {1}{4} i d \left (\frac {1}{2} i d \left (\cosh \left (\frac {c}{2}\right ) \int \frac {\cosh \left (\frac {d x}{2}\right )}{x}dx-i \sinh \left (\frac {c}{2}\right ) \int \frac {i \sinh \left (\frac {d x}{2}\right )}{x}dx\right )-\frac {i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}-\frac {1}{4} i d \left (\frac {1}{2} i d \left (\sinh \left (\frac {c}{2}\right ) \int \frac {\sinh \left (\frac {d x}{2}\right )}{x}dx+\cosh \left (\frac {c}{2}\right ) \int \frac {\cosh \left (\frac {d x}{2}\right )}{x}dx\right )-\frac {i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}-\frac {1}{4} i d \left (\frac {1}{2} i d \left (\sinh \left (\frac {c}{2}\right ) \int -\frac {i \sin \left (\frac {i d x}{2}\right )}{x}dx+\cosh \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {i d x}{2}+\frac {\pi }{2}\right )}{x}dx\right )-\frac {i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}-\frac {1}{4} i d \left (\frac {1}{2} i d \left (\cosh \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {i d x}{2}+\frac {\pi }{2}\right )}{x}dx-i \sinh \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {i d x}{2}\right )}{x}dx\right )-\frac {i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}\right )\right )\)

\(\Big \downarrow \) 3779

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}-\frac {1}{4} i d \left (\frac {1}{2} i d \left (\sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )+\cosh \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {i d x}{2}+\frac {\pi }{2}\right )}{x}dx\right )-\frac {i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}\right )\right )\)

\(\Big \downarrow \) 3782

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (-\frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 x^2}-\frac {1}{4} i d \left (\frac {1}{2} i d \left (\cosh \left (\frac {c}{2}\right ) \text {Chi}\left (\frac {d x}{2}\right )+\sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )\right )-\frac {i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}\right )\right )\)

Input:

Int[Sqrt[a + a*Cosh[c + d*x]]/x^3,x]
 

Output:

Sqrt[a + a*Cosh[c + d*x]]*Sech[c/2 + (d*x)/2]*(-1/2*Cosh[c/2 + (d*x)/2]/x^ 
2 - (I/4)*d*(((-I)*Sinh[c/2 + (d*x)/2])/x + (I/2)*d*(Cosh[c/2]*CoshIntegra 
l[(d*x)/2] + Sinh[c/2]*SinhIntegral[(d*x)/2])))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

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

rule 3778

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] - Simp[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 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> 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 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 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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 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]

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

Input:

int((a+a*cosh(d*x+c))^(1/2)/x^3,x)
 

Output:

int((a+a*cosh(d*x+c))^(1/2)/x^3,x)
 

Fricas [F(-2)]

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

integrate((a+a*cosh(d*x+c))^(1/2)/x^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((a+a*cosh(d*x+c))**(1/2)/x**3,x)
 

Output:

Integral(sqrt(a*(cosh(c + d*x) + 1))/x**3, x)
 

Maxima [F]

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

integrate((a+a*cosh(d*x+c))^(1/2)/x^3,x, algorithm="maxima")
 

Output:

integrate(sqrt(a*cosh(d*x + c) + a)/x^3, x)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x^3} \, dx=\frac {\sqrt {2} {\left (\sqrt {a} d^{2} x^{2} {\rm Ei}\left (\frac {1}{2} \, d x\right ) e^{\left (\frac {1}{2} \, c\right )} + \sqrt {a} d^{2} x^{2} {\rm Ei}\left (-\frac {1}{2} \, d x\right ) e^{\left (-\frac {1}{2} \, c\right )} - 2 \, \sqrt {a} d x e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + 2 \, \sqrt {a} d x e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} - 4 \, \sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 4 \, \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}\right )}}{16 \, x^{2}} \] Input:

integrate((a+a*cosh(d*x+c))^(1/2)/x^3,x, algorithm="giac")
 

Output:

1/16*sqrt(2)*(sqrt(a)*d^2*x^2*Ei(1/2*d*x)*e^(1/2*c) + sqrt(a)*d^2*x^2*Ei(- 
1/2*d*x)*e^(-1/2*c) - 2*sqrt(a)*d*x*e^(1/2*d*x + 1/2*c) + 2*sqrt(a)*d*x*e^ 
(-1/2*d*x - 1/2*c) - 4*sqrt(a)*e^(1/2*d*x + 1/2*c) - 4*sqrt(a)*e^(-1/2*d*x 
 - 1/2*c))/x^2
 

Mupad [F(-1)]

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

int((a + a*cosh(c + d*x))^(1/2)/x^3,x)
 

Output:

int((a + a*cosh(c + d*x))^(1/2)/x^3, x)
 

Reduce [F]

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

int((a+a*cosh(d*x+c))^(1/2)/x^3,x)
 

Output:

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