3.1.26 \(\int x (a+b \text {csch}^{-1}(c x))^3 \, dx\) [26]

3.1.26.1 Optimal result

Integrand size = 12, antiderivative size = 117 \[ \int x \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx=\frac {3 b \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b \sqrt {1+\frac {1}{c^2 x^2}} x \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c}+\frac {1}{2} x^2 \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )}{c^2}-\frac {3 b^3 \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )}{2 c^2} \]

output

3/2*b*(a+b*arccsch(c*x))^2/c^2+1/2*x^2*(a+b*arccsch(c*x))^3-3*b^2*(a+b*arc 
csch(c*x))*ln(1-(1/c/x+(1+1/c^2/x^2)^(1/2))^2)/c^2-3/2*b^3*polylog(2,(1/c/ 
x+(1+1/c^2/x^2)^(1/2))^2)/c^2+3/2*b*x*(a+b*arccsch(c*x))^2*(1+1/c^2/x^2)^( 
1/2)/c
 

3.1.26.2 Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.46 \[ \int x \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx=\frac {3 b^2 \left (a c^2 x^2+b \left (-1+c \sqrt {1+\frac {1}{c^2 x^2}} x\right )\right ) \text {csch}^{-1}(c x)^2+b^3 c^2 x^2 \text {csch}^{-1}(c x)^3+3 b \text {csch}^{-1}(c x) \left (a c x \left (2 b \sqrt {1+\frac {1}{c^2 x^2}}+a c x\right )-2 b^2 \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )+a \left (a c x \left (3 b \sqrt {1+\frac {1}{c^2 x^2}}+a c x\right )-6 b^2 \log \left (\frac {1}{c x}\right )\right )+3 b^3 \operatorname {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(c x)}\right )}{2 c^2} \]

input

Integrate[x*(a + b*ArcCsch[c*x])^3,x]
 

output

(3*b^2*(a*c^2*x^2 + b*(-1 + c*Sqrt[1 + 1/(c^2*x^2)]*x))*ArcCsch[c*x]^2 + b 
^3*c^2*x^2*ArcCsch[c*x]^3 + 3*b*ArcCsch[c*x]*(a*c*x*(2*b*Sqrt[1 + 1/(c^2*x 
^2)] + a*c*x) - 2*b^2*Log[1 - E^(-2*ArcCsch[c*x])]) + a*(a*c*x*(3*b*Sqrt[1 
 + 1/(c^2*x^2)] + a*c*x) - 6*b^2*Log[1/(c*x)]) + 3*b^3*PolyLog[2, E^(-2*Ar 
cCsch[c*x])])/(2*c^2)
 

3.1.26.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.60 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6840, 5975, 3042, 25, 4672, 26, 3042, 26, 4199, 25, 2620, 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.

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

\(\Big \downarrow \) 6840

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

\(\Big \downarrow \) 5975

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

Int[x*(a + b*ArcCsch[c*x])^3,x]
 

output

-((-1/2*(c^2*x^2*(a + b*ArcCsch[c*x])^3) - (3*b*(c*Sqrt[1 + 1/(c^2*x^2)]*x 
*(a + b*ArcCsch[c*x])^2 + (2*I)*b*(((-1/2*I)*(a + b*ArcCsch[c*x])^2)/b - ( 
2*I)*(-1/2*((a + b*ArcCsch[c*x])*Log[1 - E^(2*ArcCsch[c*x])]) - (b*PolyLog 
[2, E^(2*ArcCsch[c*x])])/4))))/2)/c^2)
 

3.1.26.3.1 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 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 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 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

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])
 

3.1.26.4 Maple [F]

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

input

int(x*(a+b*arccsch(c*x))^3,x)
 

output

int(x*(a+b*arccsch(c*x))^3,x)
 

3.1.26.5 Fricas [F]

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

input

integrate(x*(a+b*arccsch(c*x))^3,x, algorithm="fricas")
 

output

integral(b^3*x*arccsch(c*x)^3 + 3*a*b^2*x*arccsch(c*x)^2 + 3*a^2*b*x*arccs 
ch(c*x) + a^3*x, x)
 

3.1.26.6 Sympy [F]

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

input

integrate(x*(a+b*acsch(c*x))**3,x)
 

output

Integral(x*(a + b*acsch(c*x))**3, x)
 

3.1.26.7 Maxima [F]

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

input

integrate(x*(a+b*arccsch(c*x))^3,x, algorithm="maxima")
 

output

3/2*a*b^2*x^2*arccsch(c*x)^2 + 1/2*a^3*x^2 + 3/2*(x^2*arccsch(c*x) + x*sqr 
t(1/(c^2*x^2) + 1)/c)*a^2*b + 3*(x*sqrt(1/(c^2*x^2) + 1)*arccsch(c*x)/c + 
log(x)/c^2)*a*b^2 - 1/4*(24*c^2*integrate(1/2*x^3*log(x)/(sqrt(c^2*x^2 + 1 
)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x)*log(c)^2 - 24*c^2*integra 
te(1/2*x^3*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 
 + sqrt(c^2*x^2 + 1) + 1), x)*log(c)^2 - 2*x^2*log(sqrt(c^2*x^2 + 1) + 1)^ 
3 + 24*c^2*integrate(1/2*sqrt(c^2*x^2 + 1)*x^3*log(x)^2/(sqrt(c^2*x^2 + 1) 
*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x)*log(c) - 48*c^2*integrate( 
1/2*sqrt(c^2*x^2 + 1)*x^3*log(x)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(c^2*x^2 
+ 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x)*log(c) + 24*c^2*integr 
ate(1/2*sqrt(c^2*x^2 + 1)*x^3*log(sqrt(c^2*x^2 + 1) + 1)^2/(sqrt(c^2*x^2 + 
 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x)*log(c) + 24*c^2*integra 
te(1/2*x^3*log(x)^2/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 
1) + 1), x)*log(c) - 48*c^2*integrate(1/2*x^3*log(x)*log(sqrt(c^2*x^2 + 1) 
 + 1)/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x)*lo 
g(c) + 24*c^2*integrate(1/2*x^3*log(sqrt(c^2*x^2 + 1) + 1)^2/(sqrt(c^2*x^2 
 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x)*log(c) + 8*c^2*integr 
ate(1/2*sqrt(c^2*x^2 + 1)*x^3*log(x)^3/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^ 
2 + sqrt(c^2*x^2 + 1) + 1), x) - 24*c^2*integrate(1/2*sqrt(c^2*x^2 + 1)*x^ 
3*log(x)^2*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*...
 

3.1.26.8 Giac [F]

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

input

integrate(x*(a+b*arccsch(c*x))^3,x, algorithm="giac")
 

output

integrate((b*arccsch(c*x) + a)^3*x, x)
 

3.1.26.9 Mupad [F(-1)]

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

input

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

output

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