3.2.22 \(\int \frac {a+b \text {arccosh}(c x)}{x^3 (d-c^2 d x^2)^{3/2}} \, dx\) [122]

3.2.22.1 Optimal result

Integrand size = 27, antiderivative size = 329 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x \sqrt {d-c^2 d x^2}}+\frac {3 c^2 (a+b \text {arccosh}(c x))}{2 d \sqrt {d-c^2 d x^2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}+\frac {3 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}}+\frac {3 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d \sqrt {d-c^2 d x^2}} \]

output

3/2*c^2*(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(1/2)+1/2*(-a-b*arccosh(c*x))/ 
d/x^2/(-c^2*d*x^2+d)^(1/2)+1/2*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/x/(-c^2*d 
*x^2+d)^(1/2)+3*c^2*(a+b*arccosh(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)/d/(-c^2*d*x^2+d)^(1/2)+b*c^2*arctanh(c*x) 
*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-3/2*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)/d/(-c^2* 
d*x^2+d)^(1/2)+3/2*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)/d/(-c^2*d*x^2+d)^(1/2)
 

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

Time = 3.68 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.33 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {1}{2} \left (-\frac {a \left (-1+3 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{d^2 x^2 \left (-1+c^2 x^2\right )}+\frac {3 a c^2 \log (x)}{d^{3/2}}-\frac {3 a c^2 \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{d^{3/2}}-\frac {b c^2 \left (-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{c x}+\left (-1+\frac {1}{c^2 x^2}\right ) \text {arccosh}(c x)-2 \text {arccosh}(c x) \cosh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )+3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-3 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )+2 \text {arccosh}(c x) \sinh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{d \sqrt {d-c^2 d x^2}}\right ) \]

input

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

output

(-((a*(-1 + 3*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(d^2*x^2*(-1 + c^2*x^2))) + (3 
*a*c^2*Log[x])/d^(3/2) - (3*a*c^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/d^ 
(3/2) - (b*c^2*(-((Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x)) + (-1 + 1/ 
(c^2*x^2))*ArcCosh[c*x] - 2*ArcCosh[c*x]*Cosh[ArcCosh[c*x]/2]^2 + (3*I)*Sq 
rt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] 
- (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcC 
osh[c*x]] - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Cosh[ArcCosh[c*x]/2 
]] + 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sinh[ArcCosh[c*x]/2]] + (3 
*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - 
 (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]] + 
 2*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^2))/(d*Sqrt[d - c^2*d*x^2]))/2
 

3.2.22.3 Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.77, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {6347, 25, 82, 264, 219, 6351, 25, 39, 219, 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.

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

\(\Big \downarrow \) 6347

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 82

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6351

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 39

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6361

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle \frac {3}{2} c^2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\right )-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (c \text {arctanh}(c x)-\frac {1}{x}\right )}{2 d \sqrt {d-c^2 d x^2}}\)

input

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

output

-1/2*(a + b*ArcCosh[c*x])/(d*x^2*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[-1 + c*x 
]*Sqrt[1 + c*x]*(-x^(-1) + c*ArcTanh[c*x]))/(2*d*Sqrt[d - c^2*d*x^2]) + (3 
*c^2*((a + b*ArcCosh[c*x])/(d*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[-1 + c*x]*Sqr 
t[1 + c*x]*ArcTanh[c*x])/(d*Sqrt[d - c^2*d*x^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]]))/(d*Sqrt[d - c^2*d*x 
^2])))/2
 

3.2.22.3.1 Defintions of rubi rules used

rule 25

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

rule 39

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( 
a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( 
IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

rule 82

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

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 6351

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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 
))   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[ 
b*c*(n/(2*f*(p + 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, m}, x] && EqQ[c^2*d + e, 0] & 
& GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || 
EqQ[n, 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]
 

3.2.22.4 Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.79

method	result	size
default	\(a \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )\)	\(588\)
parts	\(a \left (-\frac {1}{2 d \,x^{2} \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 c^{2} \left (\frac {1}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}\right )}{2}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )+\sqrt {c x -1}\, \sqrt {c x +1}\, c x -\operatorname {arccosh}\left (c x \right )\right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) x^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{2}}{\left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}-\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 \left (c^{2} x^{2}-1\right ) d^{2}}\right )\)	\(588\)

input

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

output

a*(-1/2/d/x^2/(-c^2*d*x^2+d)^(1/2)+3/2*c^2*(1/d/(-c^2*d*x^2+d)^(1/2)-1/d^( 
3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)))+b*(-1/2*(-d*(c^2*x^2-1)) 
^(1/2)*(3*c^2*x^2*arccosh(c*x)+(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-arccosh(c*x 
))/d^2/(c^2*x^2-1)/x^2-(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/ 
(c^2*x^2-1)/d^2*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^2+(-d*(c^2*x^2-1)) 
^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*ln((c*x-1)^(1/2)*(c*x+1 
)^(1/2)+c*x-1)*c^2+3/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2 
)/(c^2*x^2-1)/d^2*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2+3/2*I*( 
-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccosh( 
c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2-3/2*I*(-d*(c^2*x^2-1))^ 
(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*dilog(1-I*(c*x+(c*x-1)^( 
1/2)*(c*x+1)^(1/2)))*c^2-3/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1 
)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/ 
2)))*c^2)
 

3.2.22.5 Fricas [F]

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

input

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

output

integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^4*d^2*x^7 - 2*c^2*d^ 
2*x^5 + d^2*x^3), x)
 

3.2.22.6 Sympy [F]

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

input

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

output

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

3.2.22.7 Maxima [F]

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

input

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

output

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

3.2.22.8 Giac [F]

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

input

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

output

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

3.2.22.9 Mupad [F(-1)]

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

input

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

output

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