\(\int \frac {x (a+b \text {arccosh}(c x))^2}{(d-c^2 d x^2)^{3/2}} \, dx\) [196]

Optimal result

Integrand size = 27, antiderivative size = 196 \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {(a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}} \] Output:

(a+b*arccosh(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+4*b*(c*x-1)^(1/2)*(c*x+1)^ 
(1/2)*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^2/d/(- 
c^2*d*x^2+d)^(1/2)+2*b^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*polylog(2,-c*x-(c*x-1 
)^(1/2)*(c*x+1)^(1/2))/c^2/d/(-c^2*d*x^2+d)^(1/2)-2*b^2*(c*x-1)^(1/2)*(c*x 
+1)^(1/2)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^2/d/(-c^2*d*x^2+d)^ 
(1/2)
 

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.12 \[ \int \frac {x (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {a^2+2 a b \left (\text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right )\right )-b^2 \left (-\text {arccosh}(c x) \left (\text {arccosh}(c x)-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\log \left (1-e^{-\text {arccosh}(c x)}\right )-\log \left (1+e^{-\text {arccosh}(c x)}\right )\right )\right )+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

(a^2 + 2*a*b*(ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(Log[Cos 
h[ArcCosh[c*x]/2]] - Log[Sinh[ArcCosh[c*x]/2]])) - b^2*(-(ArcCosh[c*x]*(Ar 
cCosh[c*x] - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(Log[1 - E^(-ArcCosh[c 
*x])] - Log[1 + E^(-ArcCosh[c*x])]))) + 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + 
c*x)*PolyLog[2, -E^(-ArcCosh[c*x])] - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c* 
x)*PolyLog[2, E^(-ArcCosh[c*x])]))/(c^2*d*Sqrt[d - c^2*d*x^2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.80 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.64, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6329, 25, 6304, 6318, 3042, 26, 4670, 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.

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

\(\Big \downarrow \) 6329

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6304

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

\(\Big \downarrow \) 6318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 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 4670

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

rule 6304

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( 
(d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A 
rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + 
 d1*e2, 0] && IntegerQ[p]
 

rule 6318

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

rule 6329

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + 
 c*x)^p)]   Int[(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, p}, x] && EqQ[c^2*d + e, 0] && 
GtQ[n, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.74

method	result	size
default	\(\frac {a^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right )^{2}\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )+\operatorname {arccosh}\left (c x \right )\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\)	\(341\)
parts	\(\frac {a^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-2 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {arccosh}\left (c x \right )^{2}\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )+\operatorname {arccosh}\left (c x \right )\right )}{d^{2} c^{2} \left (c^{2} x^{2}-1\right )}\)	\(341\)

Input:

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

Output:

a^2/c^2/d/(-c^2*d*x^2+d)^(1/2)-b^2*(-d*(c^2*x^2-1))^(1/2)*(2*(c*x-1)^(1/2) 
*(c*x+1)^(1/2)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-2*(c*x-1 
)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+2 
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-2 
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+ar 
ccosh(c*x)^2)/d^2/c^2/(c^2*x^2-1)-2*a*b*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^(1 
/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-(c*x-1)^(1/2)*(c*x 
+1)^(1/2)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)+arccosh(c*x))/d^2/c^2/(c^2 
*x^2-1)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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