3.4.45 \(\int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx\) [345]

3.4.45.1 Optimal result

Integrand size = 28, antiderivative size = 136 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^2 \sqrt {-1+c x}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}-\frac {\sqrt {-1+c x} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c^3 \sqrt {1-c x}}+\frac {\sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{b^2 c^3 \sqrt {1-c x}} \]

output

-x^2*(c*x-1)^(1/2)/b/c/(a+b*arccosh(c*x))/(-c*x+1)^(1/2)+cosh(2*a/b)*Shi(2 
*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)/b^2/c^3/(-c*x+1)^(1/2)-Chi(2*(a+b*arc 
cosh(c*x))/b)*sinh(2*a/b)*(c*x-1)^(1/2)/b^2/c^3/(-c*x+1)^(1/2)
 

3.4.45.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2} \left (b c^2 x^2+(a+b \text {arccosh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-(a+b \text {arccosh}(c x)) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))} \]

input

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

output

(Sqrt[1 - c^2*x^2]*(b*c^2*x^2 + (a + b*ArcCosh[c*x])*CoshIntegral[2*(a/b + 
 ArcCosh[c*x])]*Sinh[(2*a)/b] - (a + b*ArcCosh[c*x])*Cosh[(2*a)/b]*SinhInt 
egral[2*(a/b + ArcCosh[c*x])]))/(b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + 
 b*ArcCosh[c*x]))
 

3.4.45.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.81 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.89, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {6365, 6302, 25, 5971, 27, 3042, 26, 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.

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

\(\Big \downarrow \) 6365

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

\(\Big \downarrow \) 6302

\(\displaystyle \frac {2 \sqrt {c x-1} \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {1-c x}}-\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \sqrt {c x-1} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {1-c x}}-\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {2 \sqrt {c x-1} \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{2 (a+b \text {arccosh}(c x))}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {1-c x}}-\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {c x-1} \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {1-c x}}-\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}-\frac {\sqrt {c x-1} \int -\frac {i \sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {1-c x}}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}+\frac {i \sqrt {c x-1} \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^3 \sqrt {1-c x}}\)

\(\Big \downarrow \) 3784

\(\displaystyle -\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}+\frac {i \sqrt {c x-1} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))+\cosh \left (\frac {2 a}{b}\right ) \int -\frac {i \sinh \left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 c^3 \sqrt {1-c x}}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}+\frac {i \sqrt {c x-1} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\cosh \left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {2 a}{b}\right ) \int \frac {\sinh \left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 c^3 \sqrt {1-c x}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}+\frac {i \sqrt {c x-1} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {2 a}{b}\right ) \int -\frac {i \sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 c^3 \sqrt {1-c x}}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}+\frac {i \sqrt {c x-1} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-\cosh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))\right )}{b^2 c^3 \sqrt {1-c x}}\)

\(\Big \downarrow \) 3779

\(\displaystyle -\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}+\frac {i \sqrt {c x-1} \left (i \sinh \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 i (a+b \text {arccosh}(c x))}{b}+\frac {\pi }{2}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))-i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {1-c x}}\)

\(\Big \downarrow \) 3782

\(\displaystyle -\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} (a+b \text {arccosh}(c x))}+\frac {i \sqrt {c x-1} \left (i \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )-i \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^3 \sqrt {1-c x}}\)

input

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

output

-((x^2*Sqrt[-1 + c*x])/(b*c*Sqrt[1 - c*x]*(a + b*ArcCosh[c*x]))) + (I*Sqrt 
[-1 + c*x]*(I*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b] - I*C 
osh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b]))/(b^2*c^3*Sqrt[1 - 
c*x])
 

3.4.45.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6302

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

rule 6365

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) 
 + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/( 
b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], x] - Si 
mp[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])] 
   Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c 
, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]
 

3.4.45.4 Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.79

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-2 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{2} x^{2}+2 b \,c^{3} x^{3}+\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+a \,\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}} a \right )}{2 c^{3} \left (c^{2} x^{2}-1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\)	\(243\)

input

int(x^2/(a+b*arccosh(c*x))^2/(-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)*(-2*(c* 
x-1)^(1/2)*(c*x+1)^(1/2)*b*c^2*x^2+2*b*c^3*x^3+arccosh(c*x)*b*Ei(1,-2*arcc 
osh(c*x)-2*a/b)*exp(-(b*arccosh(c*x)+2*a)/b)-Ei(1,2*arccosh(c*x)+2*a/b)*ex 
p((-b*arccosh(c*x)+2*a)/b)*b*arccosh(c*x)+a*Ei(1,-2*arccosh(c*x)-2*a/b)*ex 
p(-(b*arccosh(c*x)+2*a)/b)-Ei(1,2*arccosh(c*x)+2*a/b)*exp((-b*arccosh(c*x) 
+2*a)/b)*a)/c^3/(c^2*x^2-1)/b^2/(a+b*arccosh(c*x))
 

3.4.45.5 Fricas [F]

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

input

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

output

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

3.4.45.6 Sympy [F]

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

input

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

output

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

3.4.45.7 Maxima [F]

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

input

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

output

-(c^3*x^5 - c*x^3 + (c^2*x^4 - x^2)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 
1)*sqrt(c*x - 1)*b^2*c^2*x + (b^2*c^3*x^2 - b^2*c)*sqrt(c*x + 1))*sqrt(-c* 
x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)*sqrt(c*x - 1)*a 
*b*c^2*x + (a*b*c^3*x^2 - a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)) + integrat 
e((2*c^5*x^6 - 5*c^3*x^4 + (2*c^3*x^4 - c*x^2)*(c*x + 1)*(c*x - 1) + 3*c*x 
^2 + 2*(2*c^4*x^5 - 3*c^2*x^3 + x)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 1 
)^(3/2)*(c*x - 1)*b^2*c^3*x^2 + 2*(b^2*c^4*x^3 - b^2*c^2*x)*(c*x + 1)*sqrt 
(c*x - 1) + (b^2*c^5*x^4 - 2*b^2*c^3*x^2 + b^2*c)*sqrt(c*x + 1))*sqrt(-c*x 
 + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)^(3/2)*(c*x - 1)* 
a*b*c^3*x^2 + 2*(a*b*c^4*x^3 - a*b*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c 
^5*x^4 - 2*a*b*c^3*x^2 + a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)), x)
 

3.4.45.8 Giac [F]

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

input

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

output

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

3.4.45.9 Mupad [F(-1)]

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

input

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

output

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