3.1.28 \(\int \frac {(a+b \text {csch}^{-1}(c x))^3}{x} \, dx\) [28]

3.1.28.1 Optimal result
3.1.28.2 Mathematica [A] (verified)
3.1.28.3 Rubi [C] (verified)
3.1.28.4 Maple [F]
3.1.28.5 Fricas [F]
3.1.28.6 Sympy [F]
3.1.28.7 Maxima [F]
3.1.28.8 Giac [F]
3.1.28.9 Mupad [F(-1)]

3.1.28.1 Optimal result

Integrand size = 14, antiderivative size = 110 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x} \, dx=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}-\left (a+b \text {csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-\frac {3}{2} b \left (a+b \text {csch}^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right )-\frac {3}{4} b^3 \operatorname {PolyLog}\left (4,e^{2 \text {csch}^{-1}(c x)}\right ) \]

output
1/4*(a+b*arccsch(c*x))^4/b-(a+b*arccsch(c*x))^3*ln(1-(1/c/x+(1+1/c^2/x^2)^ 
(1/2))^2)-3/2*b*(a+b*arccsch(c*x))^2*polylog(2,(1/c/x+(1+1/c^2/x^2)^(1/2)) 
^2)+3/2*b^2*(a+b*arccsch(c*x))*polylog(3,(1/c/x+(1+1/c^2/x^2)^(1/2))^2)-3/ 
4*b^3*polylog(4,(1/c/x+(1+1/c^2/x^2)^(1/2))^2)
 
3.1.28.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x} \, dx=\frac {1}{4} \left (6 a^2 b \text {csch}^{-1}(c x)^2+4 a b^2 \text {csch}^{-1}(c x)^3+b^3 \text {csch}^{-1}(c x)^4-12 a^2 b \text {csch}^{-1}(c x) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-12 a b^2 \text {csch}^{-1}(c x)^2 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-4 b^3 \text {csch}^{-1}(c x)^3 \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+4 a^3 \log (c x)-6 b \left (a+b \text {csch}^{-1}(c x)\right )^2 \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )+6 b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right )-3 b^3 \operatorname {PolyLog}\left (4,e^{2 \text {csch}^{-1}(c x)}\right )\right ) \]

input
Integrate[(a + b*ArcCsch[c*x])^3/x,x]
 
output
(6*a^2*b*ArcCsch[c*x]^2 + 4*a*b^2*ArcCsch[c*x]^3 + b^3*ArcCsch[c*x]^4 - 12 
*a^2*b*ArcCsch[c*x]*Log[1 - E^(2*ArcCsch[c*x])] - 12*a*b^2*ArcCsch[c*x]^2* 
Log[1 - E^(2*ArcCsch[c*x])] - 4*b^3*ArcCsch[c*x]^3*Log[1 - E^(2*ArcCsch[c* 
x])] + 4*a^3*Log[c*x] - 6*b*(a + b*ArcCsch[c*x])^2*PolyLog[2, E^(2*ArcCsch 
[c*x])] + 6*b^2*(a + b*ArcCsch[c*x])*PolyLog[3, E^(2*ArcCsch[c*x])] - 3*b^ 
3*PolyLog[4, E^(2*ArcCsch[c*x])])/4
 
3.1.28.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6840, 3042, 26, 4199, 25, 2620, 3011, 7163, 2720, 7143}

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.

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

\(\Big \downarrow \) 6840

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

\(\displaystyle i \left (-2 i \left (\frac {3}{2} b \left (b \int \left (a+b \text {csch}^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )d\text {csch}^{-1}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^3\right )-\frac {i \left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}\right )\)

\(\Big \downarrow \) 7163

\(\displaystyle i \left (-2 i \left (\frac {3}{2} b \left (b \left (\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right )d\text {csch}^{-1}(c x)\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^3\right )-\frac {i \left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle i \left (-2 i \left (\frac {3}{2} b \left (b \left (\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{4} b \int e^{-2 \text {csch}^{-1}(c x)} \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right )de^{2 \text {csch}^{-1}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^3\right )-\frac {i \left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle i \left (-2 i \left (\frac {3}{2} b \left (b \left (\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{4} b \operatorname {PolyLog}\left (4,e^{2 \text {csch}^{-1}(c x)}\right )\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^2\right )-\frac {1}{2} \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )^3\right )-\frac {i \left (a+b \text {csch}^{-1}(c x)\right )^4}{4 b}\right )\)

input
Int[(a + b*ArcCsch[c*x])^3/x,x]
 
output
I*(((-1/4*I)*(a + b*ArcCsch[c*x])^4)/b - (2*I)*(-1/2*((a + b*ArcCsch[c*x]) 
^3*Log[1 - E^(2*ArcCsch[c*x])]) + (3*b*(-1/2*((a + b*ArcCsch[c*x])^2*PolyL 
og[2, E^(2*ArcCsch[c*x])]) + b*(((a + b*ArcCsch[c*x])*PolyLog[3, E^(2*ArcC 
sch[c*x])])/2 - (b*PolyLog[4, E^(2*ArcCsch[c*x])])/4)))/2))
 

3.1.28.3.1 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 2620
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2720
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042
Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4199
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ 
.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp 
[2*I   Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[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])
 

rule 7143
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 
3.1.28.4 Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{3}}{x}d x\]

input
int((a+b*arccsch(c*x))^3/x,x)
 
output
int((a+b*arccsch(c*x))^3/x,x)
 
3.1.28.5 Fricas [F]

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

input
integrate((a+b*arccsch(c*x))^3/x,x, algorithm="fricas")
 
output
integral((b^3*arccsch(c*x)^3 + 3*a*b^2*arccsch(c*x)^2 + 3*a^2*b*arccsch(c* 
x) + a^3)/x, x)
 
3.1.28.6 Sympy [F]

\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{3}}{x}\, dx \]

input
integrate((a+b*acsch(c*x))**3/x,x)
 
output
Integral((a + b*acsch(c*x))**3/x, x)
 
3.1.28.7 Maxima [F]

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

input
integrate((a+b*arccsch(c*x))^3/x,x, algorithm="maxima")
 
output
b^3*log(x)*log(sqrt(c^2*x^2 + 1) + 1)^3 + a^3*log(x) - integrate((b^3*log( 
c)^3 - 3*a*b^2*log(c)^2 + 3*a^2*b*log(c) + (b^3*c^2*x^2 + b^3)*log(x)^3 + 
(b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c)^2 + 3*a^2*b*c^2*log(c))*x^2 + 3*(b^ 
3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x)^2 + 3*(b^3*log 
(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2 + (b^3*c^2*x^2 + b^3)*log(x 
) + sqrt(c^2*x^2 + 1)*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x 
^2 + (2*b^3*c^2*x^2 + b^3)*log(x)))*log(sqrt(c^2*x^2 + 1) + 1)^2 + 3*(b^3* 
log(c)^2 - 2*a*b^2*log(c) + a^2*b + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c) 
 + a^2*b*c^2)*x^2)*log(x) - 3*(b^3*log(c)^2 - 2*a*b^2*log(c) + a^2*b + (b^ 
3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c) + a^2*b*c^2)*x^2 + (b^3*c^2*x^2 + b^3) 
*log(x)^2 + 2*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log( 
x) + (b^3*log(c)^2 - 2*a*b^2*log(c) + a^2*b + (b^3*c^2*log(c)^2 - 2*a*b^2* 
c^2*log(c) + a^2*b*c^2)*x^2 + (b^3*c^2*x^2 + b^3)*log(x)^2 + 2*(b^3*log(c) 
 - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x))*sqrt(c^2*x^2 + 1))*lo 
g(sqrt(c^2*x^2 + 1) + 1) + (b^3*log(c)^3 - 3*a*b^2*log(c)^2 + 3*a^2*b*log( 
c) + (b^3*c^2*x^2 + b^3)*log(x)^3 + (b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c) 
^2 + 3*a^2*b*c^2*log(c))*x^2 + 3*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a 
*b^2*c^2)*x^2)*log(x)^2 + 3*(b^3*log(c)^2 - 2*a*b^2*log(c) + a^2*b + (b^3* 
c^2*log(c)^2 - 2*a*b^2*c^2*log(c) + a^2*b*c^2)*x^2)*log(x))*sqrt(c^2*x^2 + 
 1))/(c^2*x^3 + (c^2*x^3 + x)*sqrt(c^2*x^2 + 1) + x), x)
 
3.1.28.8 Giac [F]

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

input
integrate((a+b*arccsch(c*x))^3/x,x, algorithm="giac")
 
output
integrate((b*arccsch(c*x) + a)^3/x, x)
 
3.1.28.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3}{x} \,d x \]

input
int((a + b*asinh(1/(c*x)))^3/x,x)
 
output
int((a + b*asinh(1/(c*x)))^3/x, x)