\(\int \frac {a+b \text {arccosh}(c x)}{x (d-c^2 d x^2)^{5/2}} \, dx\) [125]

Optimal result

Integrand size = 27, antiderivative size = 305 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b c x}{6 d^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {a+b \text {arccosh}(c x)}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {7 b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d^3 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-1/6*b*c*x/d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/3*(a+b*a 
rccosh(c*x))/d/(-c^2*d*x^2+d)^(3/2)+(a+b*arccosh(c*x))/d^2/(-c^2*d*x^2+d)^ 
(1/2)-2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))*arctan(c*x+(c*x-1)^(1/2)*( 
c*x+1)^(1/2))/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+7/6*b*(c*x-1)^(1/2)*(c*x+1)^ 
(1/2)*arctanh(c*x)/d^2/(-c^2*d*x^2+d)^(1/2)+I*b*(-c^2*d*x^2+d)^(1/2)*polyl 
og(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2) 
-I*b*(-c^2*d*x^2+d)^(1/2)*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))/d 
^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 6.45 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.23 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {a \left (-4+3 c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 d^3 \left (-1+c^2 x^2\right )^2}+\frac {a \log (x)}{d^{5/2}}-\frac {a \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{d^{5/2}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (14 \text {arccosh}(c x) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )-\text {csch}^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\frac {1}{2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \text {csch}^4\left (\frac {1}{2} \text {arccosh}(c x)\right )-24 i \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )+24 i \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+28 \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-28 \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-24 i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )+24 i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )-\text {sech}^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\frac {8 \text {arccosh}(c x) \sinh ^4\left (\frac {1}{2} \text {arccosh}(c x)\right )}{\left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3}-14 \text {arccosh}(c x) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{24 d^2 \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^(5/2)),x]
 

Output:

-1/3*(a*(-4 + 3*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(d^3*(-1 + c^2*x^2)^2) + (a* 
Log[x])/d^(5/2) - (a*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/d^(5/2) + (b*Sq 
rt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(14*ArcCosh[c*x]*Coth[ArcCosh[c*x]/2] - 
 Csch[ArcCosh[c*x]/2]^2 - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c* 
x]*Csch[ArcCosh[c*x]/2]^4)/2 - (24*I)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x 
]] + (24*I)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + 28*Log[Cosh[ArcCosh[c 
*x]/2]] - 28*Log[Sinh[ArcCosh[c*x]/2]] - (24*I)*PolyLog[2, (-I)/E^ArcCosh[ 
c*x]] + (24*I)*PolyLog[2, I/E^ArcCosh[c*x]] - Sech[ArcCosh[c*x]/2]^2 - (8* 
ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^4)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c* 
x)^3) - 14*ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]))/(24*d^2*Sqrt[d - c^2*d*x^2] 
)
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.87, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {6351, 39, 215, 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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 6351

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

\(\Big \downarrow \) 39

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6351

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 39

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6361

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^(5/2)),x]
 

Output:

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

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 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 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 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]
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.84

method	result	size
default	\(\frac {a}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 \,\operatorname {arccosh}\left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3}}+\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(560\)
parts	\(\frac {a}{3 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {a}{d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-\sqrt {c x -1}\, \sqrt {c x +1}\, c x -8 \,\operatorname {arccosh}\left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3}}+\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {7 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(560\)

Input:

int((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*a/d/(-c^2*d*x^2+d)^(3/2)+a/d^2/(-c^2*d*x^2+d)^(1/2)-a/d^(5/2)*ln((2*d+ 
2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+b*(-1/6*(-d*(c^2*x^2-1))^(1/2)*(6*c^2*x 
^2*arccosh(c*x)-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-8*arccosh(c*x))/(c^2*x^2-1 
)^2/d^3+7/6*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^ 
2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)-7/6*(-d*(c^2*x^2-1))^(1/2)*(c*x 
-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/ 
2))-I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*a 
rccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-I*(-d*(c^2*x^2-1))^( 
1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*dilog(1-I*(c*x+(c*x-1)^(1 
/2)*(c*x+1)^(1/2)))+I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d 
^3/(c^2*x^2-1)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+I*(-d*(c^2*x^2 
-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*arccosh(c*x)*ln(1+I 
*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))))
 

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^6*d^3*x^7 - 3*c^4*d 
^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((a+b*acosh(c*x))/x/(-c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral((a + b*acosh(c*x))/(x*(-d*(c*x - 1)*(c*x + 1))**(5/2)), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")
 

Output:

-1/3*a*(3*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(5/2) 
- 3/(sqrt(-c^2*d*x^2 + d)*d^2) - 1/((-c^2*d*x^2 + d)^(3/2)*d)) + b*integra 
te(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/((-c^2*d*x^2 + d)^(5/2)*x), x)
 

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/x/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:

int((a + b*acosh(c*x))/(x*(d - c^2*d*x^2)^(5/2)),x)
 

Output:

int((a + b*acosh(c*x))/(x*(d - c^2*d*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{5}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) b \,c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{5}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{3}+\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) b +3 \sqrt {-c^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a \,c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a -4 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}+4 \sqrt {-c^{2} x^{2}+1}\, a +3 a \,c^{2} x^{2}-4 a}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int((a+b*acosh(c*x))/x/(-c^2*d*x^2+d)^(5/2),x)
 

Output:

(3*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**5 
 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**3 + sqrt( - c**2*x**2 + 1)*x),x)*b*c** 
2*x**2 - 3*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c 
**4*x**5 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**3 + sqrt( - c**2*x**2 + 1)*x), 
x)*b + 3*sqrt( - c**2*x**2 + 1)*log(tan(asin(c*x)/2))*a*c**2*x**2 - 3*sqrt 
( - c**2*x**2 + 1)*log(tan(asin(c*x)/2))*a - 4*sqrt( - c**2*x**2 + 1)*a*c* 
*2*x**2 + 4*sqrt( - c**2*x**2 + 1)*a + 3*a*c**2*x**2 - 4*a)/(3*sqrt(d)*sqr 
t( - c**2*x**2 + 1)*d**2*(c**2*x**2 - 1))