3.2.12 \(\int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx\) [112]

3.2.12.1 Optimal result

Integrand size = 27, antiderivative size = 238 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 x \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 d x^2}+\frac {c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {d-c^2 d x^2}}+\frac {i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 \sqrt {d-c^2 d x^2}} \]

output

1/2*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/x/(-c^2*d*x^2+d)^(1/2)+c^2*(a+b*arccos 
h(c*x))*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2 
)/(-c^2*d*x^2+d)^(1/2)-1/2*I*b*c^2*polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1) 
^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/2*I*b*c^2*poly 
log(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-c 
^2*d*x^2+d)^(1/2)-1/2*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/d/x^2
 

3.2.12.2 Mathematica [A] (warning: unable to verify)

Time = 0.89 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.30 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\frac {1}{2} \left (-\frac {a \sqrt {d-c^2 d x^2}}{d x^2}+\frac {a c^2 \log (x)}{\sqrt {d}}-\frac {a c^2 \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{\sqrt {d}}+\frac {b (1+c x) \left (c x \sqrt {\frac {-1+c x}{1+c x}}-\text {arccosh}(c x)+c x \text {arccosh}(c x)-i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )+i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )-i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )+i c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{x^2 \sqrt {d-c^2 d x^2}}\right ) \]

input

Integrate[(a + b*ArcCosh[c*x])/(x^3*Sqrt[d - c^2*d*x^2]),x]
 

output

(-((a*Sqrt[d - c^2*d*x^2])/(d*x^2)) + (a*c^2*Log[x])/Sqrt[d] - (a*c^2*Log[ 
d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/Sqrt[d] + (b*(1 + c*x)*(c*x*Sqrt[(-1 + c 
*x)/(1 + c*x)] - ArcCosh[c*x] + c*x*ArcCosh[c*x] - I*c^2*x^2*Sqrt[(-1 + c* 
x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + I*c^2*x^2*Sqrt[(-1 
+ c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[ 
(-1 + c*x)/(1 + c*x)]*PolyLog[2, (-I)/E^ArcCosh[c*x]] + I*c^2*x^2*Sqrt[(-1 
 + c*x)/(1 + c*x)]*PolyLog[2, I/E^ArcCosh[c*x]]))/(x^2*Sqrt[d - c^2*d*x^2] 
))/2
 

3.2.12.3 Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.71, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6347, 15, 6361, 3042, 4668, 2715, 2838}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx\)

\(\Big \downarrow \) 6347

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6361

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle \frac {c^2 \sqrt {c x-1} \sqrt {c x+1} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )\right )}{2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{2 d x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 x \sqrt {d-c^2 d x^2}}\)

input

Int[(a + b*ArcCosh[c*x])/(x^3*Sqrt[d - c^2*d*x^2]),x]
 

output

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c 
^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*d*x^2) + (c^2*Sqrt[-1 + c*x]*Sqrt[1 + c 
*x]*(2*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]] - I*b*PolyLog[2, (-I)*E 
^ArcCosh[c*x]] + I*b*PolyLog[2, I*E^ArcCosh[c*x]]))/(2*Sqrt[d - c^2*d*x^2] 
)
 

3.2.12.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 6347

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + 
b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1 
)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp 
[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)]   Int[( 
f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^ 
(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[n, 0] && ILtQ[m, -1]
 

rule 6361

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

3.2.12.4 Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.81

method	result	size
default	\(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{2 d \,x^{2}}-\frac {a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}+b \left (-\frac {\left (c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 d \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}\right )\)	\(431\)
parts	\(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{2 d \,x^{2}}-\frac {a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 \sqrt {d}}+b \left (-\frac {\left (c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 d \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d \left (c^{2} x^{2}-1\right )}\right )\)	\(431\)

input

int((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
 

output

-1/2*a/d/x^2*(-c^2*d*x^2+d)^(1/2)-1/2*a*c^2/d^(1/2)*ln((2*d+2*d^(1/2)*(-c^ 
2*d*x^2+d)^(1/2))/x)+b*(-1/2*(c^2*x^2*arccosh(c*x)+(c*x-1)^(1/2)*(c*x+1)^( 
1/2)*c*x-arccosh(c*x))*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)/x^2+1/2*I*(-d* 
(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*arccosh(c*x)* 
ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2-1/2*I*(-d*(c^2*x^2-1))^(1/2) 
*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*arccosh(c*x)*ln(1-I*(c*x+(c*x-1 
)^(1/2)*(c*x+1)^(1/2)))*c^2+1/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c* 
x+1)^(1/2)/d/(c^2*x^2-1)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2- 
1/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*x^2-1)*dil 
og(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2)
 

3.2.12.5 Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} x^{3}} \,d x } \]

input

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

output

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^2*d*x^5 - d*x^3), x 
)
 

3.2.12.6 Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{3} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]

input

integrate((a+b*acosh(c*x))/x**3/(-c**2*d*x**2+d)**(1/2),x)
 

output

Integral((a + b*acosh(c*x))/(x**3*sqrt(-d*(c*x - 1)*(c*x + 1))), x)
 

3.2.12.7 Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} x^{3}} \,d x } \]

input

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

output

-1/2*(c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/sqrt(d) 
+ sqrt(-c^2*d*x^2 + d)/(d*x^2))*a + b*integrate(log(c*x + sqrt(c*x + 1)*sq 
rt(c*x - 1))/(sqrt(-c^2*d*x^2 + d)*x^3), x)
 

3.2.12.8 Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} x^{3}} \,d x } \]

input

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

output

integrate((b*arccosh(c*x) + a)/(sqrt(-c^2*d*x^2 + d)*x^3), x)
 

3.2.12.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,\sqrt {d-c^2\,d\,x^2}} \,d x \]

input

int((a + b*acosh(c*x))/(x^3*(d - c^2*d*x^2)^(1/2)),x)
 

output

int((a + b*acosh(c*x))/(x^3*(d - c^2*d*x^2)^(1/2)), x)