3.1.16 \(\int x^2 (a+b \text {csch}^{-1}(c x))^2 \, dx\) [16]

3.1.16.1 Optimal result

Integrand size = 14, antiderivative size = 122 \[ \int x^2 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx=\frac {b^2 x}{3 c^2}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {2 b \left (a+b \text {csch}^{-1}(c x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}-\frac {b^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c x)}\right )}{3 c^3} \]

output

1/3*b^2*x/c^2+1/3*x^3*(a+b*arccsch(c*x))^2-2/3*b*(a+b*arccsch(c*x))*arctan 
h(1/c/x+(1+1/c^2/x^2)^(1/2))/c^3-1/3*b^2*polylog(2,-1/c/x-(1+1/c^2/x^2)^(1 
/2))/c^3+1/3*b^2*polylog(2,1/c/x+(1+1/c^2/x^2)^(1/2))/c^3+1/3*b*x^2*(a+b*a 
rccsch(c*x))*(1+1/c^2/x^2)^(1/2)/c
 

3.1.16.2 Mathematica [A] (verified)

Time = 1.43 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.84 \[ \int x^2 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx=\frac {b^2 c x+a b c^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2+a^2 c^3 x^3+b^2 c^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2 \text {csch}^{-1}(c x)+2 a b c^3 x^3 \text {csch}^{-1}(c x)+b^2 c^3 x^3 \text {csch}^{-1}(c x)^2+b^2 \text {csch}^{-1}(c x) \log \left (1-e^{-\text {csch}^{-1}(c x)}\right )-b^2 \text {csch}^{-1}(c x) \log \left (1+e^{-\text {csch}^{-1}(c x)}\right )+\frac {a b c \sqrt {1+\frac {1}{c^2 x^2}} x \log \left (-c x+\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}+b^2 \operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c x)}\right )-b^2 \operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c x)}\right )}{3 c^3} \]

input

Integrate[x^2*(a + b*ArcCsch[c*x])^2,x]
 

output

(b^2*c*x + a*b*c^2*Sqrt[1 + 1/(c^2*x^2)]*x^2 + a^2*c^3*x^3 + b^2*c^2*Sqrt[ 
1 + 1/(c^2*x^2)]*x^2*ArcCsch[c*x] + 2*a*b*c^3*x^3*ArcCsch[c*x] + b^2*c^3*x 
^3*ArcCsch[c*x]^2 + b^2*ArcCsch[c*x]*Log[1 - E^(-ArcCsch[c*x])] - b^2*ArcC 
sch[c*x]*Log[1 + E^(-ArcCsch[c*x])] + (a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Log[- 
(c*x) + Sqrt[1 + c^2*x^2]])/Sqrt[1 + c^2*x^2] + b^2*PolyLog[2, -E^(-ArcCsc 
h[c*x])] - b^2*PolyLog[2, E^(-ArcCsch[c*x])])/(3*c^3)
 

3.1.16.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6840, 5975, 3042, 26, 4673, 26, 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.

\(\displaystyle \int x^2 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx\)

\(\Big \downarrow \) 6840

\(\displaystyle -\frac {\int c^4 \sqrt {1+\frac {1}{c^2 x^2}} x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)}{c^3}\)

\(\Big \downarrow \) 5975

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4673

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

Int[x^2*(a + b*ArcCsch[c*x])^2,x]
 

output

-((-1/3*(c^3*x^3*(a + b*ArcCsch[c*x])^2) - ((2*I)/3)*b*((-1/2*I)*b*c*x - ( 
I/2)*c^2*Sqrt[1 + 1/(c^2*x^2)]*x^2*(a + b*ArcCsch[c*x]) + ((2*I)*(a + b*Ar 
cCsch[c*x])*ArcTanh[E^ArcCsch[c*x]] + I*b*PolyLog[2, -E^ArcCsch[c*x]] - I* 
b*PolyLog[2, E^ArcCsch[c*x]])/2))/c^3)
 

3.1.16.3.1 Defintions of rubi rules used

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 4673

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> 
 Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 
 x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

rule 5975

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

rule 6840

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
-(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, A 
rcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G 
tQ[n, 0] || LtQ[m, -1])
 

3.1.16.4 Maple [F]

\[\int x^{2} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{2}d x\]

input

int(x^2*(a+b*arccsch(c*x))^2,x)
 

output

int(x^2*(a+b*arccsch(c*x))^2,x)
 

3.1.16.5 Fricas [F]

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

input

integrate(x^2*(a+b*arccsch(c*x))^2,x, algorithm="fricas")
 

output

integral(b^2*x^2*arccsch(c*x)^2 + 2*a*b*x^2*arccsch(c*x) + a^2*x^2, x)
 

3.1.16.6 Sympy [F]

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

input

integrate(x**2*(a+b*acsch(c*x))**2,x)
 

output

Integral(x**2*(a + b*acsch(c*x))**2, x)
 

3.1.16.7 Maxima [F]

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

input

integrate(x^2*(a+b*arccsch(c*x))^2,x, algorithm="maxima")
 

output

1/3*a^2*x^3 + 1/6*(4*x^3*arccsch(c*x) + (2*sqrt(1/(c^2*x^2) + 1)/(c^2*(1/( 
c^2*x^2) + 1) - c^2) - log(sqrt(1/(c^2*x^2) + 1) + 1)/c^2 + log(sqrt(1/(c^ 
2*x^2) + 1) - 1)/c^2)/c)*a*b + 1/3*(x^3*log(sqrt(c^2*x^2 + 1) + 1)^2 - 3*i 
ntegrate(-1/3*(3*c^2*x^4*log(c)^2 + 3*x^2*log(c)^2 + 3*(c^2*x^4 + x^2)*log 
(x)^2 + 6*(c^2*x^4*log(c) + x^2*log(c))*log(x) - 2*(3*c^2*x^4*log(c) + 3*x 
^2*log(c) + 3*(c^2*x^4 + x^2)*log(x) + (c^2*x^4*(3*log(c) + 1) + 3*x^2*log 
(c) + 3*(c^2*x^4 + x^2)*log(x))*sqrt(c^2*x^2 + 1))*log(sqrt(c^2*x^2 + 1) + 
 1) + 3*(c^2*x^4*log(c)^2 + x^2*log(c)^2 + (c^2*x^4 + x^2)*log(x)^2 + 2*(c 
^2*x^4*log(c) + x^2*log(c))*log(x))*sqrt(c^2*x^2 + 1))/(c^2*x^2 + (c^2*x^2 
 + 1)^(3/2) + 1), x))*b^2
 

3.1.16.8 Giac [F]

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

input

integrate(x^2*(a+b*arccsch(c*x))^2,x, algorithm="giac")
 

output

integrate((b*arccsch(c*x) + a)^2*x^2, x)
 

3.1.16.9 Mupad [F(-1)]

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

input

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

output

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