3.4.1 \(\int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx\) [301]

3.4.1.1 Optimal result

Integrand size = 26, antiderivative size = 92 \[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {\sqrt {-1+c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c^2 \sqrt {1-c x}}-\frac {\sqrt {-1+c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{b c^2 \sqrt {1-c x}} \]

output

Chi((a+b*arccosh(c*x))/b)*cosh(a/b)*(c*x-1)^(1/2)/b/c^2/(-c*x+1)^(1/2)-Shi 
((a+b*arccosh(c*x))/b)*sinh(a/b)*(c*x-1)^(1/2)/b/c^2/(-c*x+1)^(1/2)
 

3.4.1.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )}{c^2 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]

input

Integrate[x/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])),x]
 

output

(Sqrt[1 - c^2*x^2]*(-(Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]]) + Sinh[a 
/b]*SinhIntegral[a/b + ArcCosh[c*x]]))/(c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(b 
+ b*c*x))
 

3.4.1.3 Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6367, 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.

\(\displaystyle \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx\)

\(\Big \downarrow \) 6367

\(\displaystyle \frac {\sqrt {c x-1} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c^2 \sqrt {1-c x}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {c x-1} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b c^2 \sqrt {1-c x}}\)

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

\(\displaystyle \frac {\sqrt {c x-1} \left (\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{b c^2 \sqrt {1-c x}}\)

input

Int[x/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])),x]
 

output

(Sqrt[-1 + c*x]*(Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b] - Sinh[a/b 
]*SinhIntegral[(a + b*ArcCosh[c*x])/b]))/(b*c^2*Sqrt[1 - c*x])
 

3.4.1.3.1 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 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 6367

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x 
)^p*(-1 + c*x)^p)]   Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p 
 + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && Eq 
Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
 

3.4.1.4 Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}+\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}\right )}{2 b \left (c^{2} x^{2}-1\right ) c^{2}}\)	\(116\)

input

int(x/(a+b*arccosh(c*x))/(-c^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

output

1/2*(-c^2*x^2+1)^(1/2)*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(Ei(1,a 
rccosh(c*x)+a/b)*exp((-b*arccosh(c*x)+a)/b)+Ei(1,-arccosh(c*x)-a/b)*exp(-( 
a+b*arccosh(c*x))/b))/b/(c^2*x^2-1)/c^2
 

3.4.1.5 Fricas [F]

\[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]

input

integrate(x/(a+b*arccosh(c*x))/(-c^2*x^2+1)^(1/2),x, algorithm="fricas")
 

output

integral(-sqrt(-c^2*x^2 + 1)*x/(a*c^2*x^2 + (b*c^2*x^2 - b)*arccosh(c*x) - 
 a), x)
 

3.4.1.6 Sympy [F]

\[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {x}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

input

integrate(x/(a+b*acosh(c*x))/(-c**2*x**2+1)**(1/2),x)
 

output

Integral(x/(sqrt(-(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))), x)
 

3.4.1.7 Maxima [F]

\[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]

input

integrate(x/(a+b*arccosh(c*x))/(-c^2*x^2+1)^(1/2),x, algorithm="maxima")
 

output

integrate(x/(sqrt(-c^2*x^2 + 1)*(b*arccosh(c*x) + a)), x)
 

3.4.1.8 Giac [F]

\[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]

input

integrate(x/(a+b*arccosh(c*x))/(-c^2*x^2+1)^(1/2),x, algorithm="giac")
 

output

integrate(x/(sqrt(-c^2*x^2 + 1)*(b*arccosh(c*x) + a)), x)
 

3.4.1.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {x}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \]

input

int(x/((a + b*acosh(c*x))*(1 - c^2*x^2)^(1/2)),x)
 

output

int(x/((a + b*acosh(c*x))*(1 - c^2*x^2)^(1/2)), x)