3.26 \(\int x (a+b \text{csch}^{-1}(c x))^3 \, dx\)
Optimal. Leaf size=117 \[ -\frac{3 b^3 \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )}{2 c^2}-\frac{3 b^2 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c^2}+\frac{3 b x \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c}+\frac{3 b \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c^2}+\frac{1}{2} x^2 \left (a+b \text{csch}^{-1}(c x)\right )^3 \]
[Out]
(3*b*(a + b*ArcCsch[c*x])^2)/(2*c^2) + (3*b*Sqrt[1 + 1/(c^2*x^2)]*x*(a + b*ArcCsch[c*x])^2)/(2*c) + (x^2*(a +
b*ArcCsch[c*x])^3)/2 - (3*b^2*(a + b*ArcCsch[c*x])*Log[1 - E^(2*ArcCsch[c*x])])/c^2 - (3*b^3*PolyLog[2, E^(2*A
rcCsch[c*x])])/(2*c^2)
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Rubi [A] time = 0.159845, antiderivative size = 117, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used =
{6286, 5452, 4184, 3716, 2190, 2279, 2391} \[ -\frac{3 b^3 \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )}{2 c^2}-\frac{3 b^2 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{c^2}+\frac{3 b x \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c}+\frac{3 b \left (a+b \text{csch}^{-1}(c x)\right )^2}{2 c^2}+\frac{1}{2} x^2 \left (a+b \text{csch}^{-1}(c x)\right )^3 \]
Antiderivative was successfully verified.
[In]
Int[x*(a + b*ArcCsch[c*x])^3,x]
[Out]
(3*b*(a + b*ArcCsch[c*x])^2)/(2*c^2) + (3*b*Sqrt[1 + 1/(c^2*x^2)]*x*(a + b*ArcCsch[c*x])^2)/(2*c) + (x^2*(a +
b*ArcCsch[c*x])^3)/2 - (3*b^2*(a + b*ArcCsch[c*x])*Log[1 - E^(2*ArcCsch[c*x])])/c^2 - (3*b^3*PolyLog[2, E^(2*A
rcCsch[c*x])])/(2*c^2)
Rule 6286
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, ArcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])
Rule 5452
Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Csch[a + b*x]^n)/(b*n), x] + Dist[(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 4184
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(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 3716
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] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Rule 2190
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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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 2279
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2391
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]
Rubi steps
\begin{align*} \int x \left (a+b \text{csch}^{-1}(c x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \coth (x) \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^2}\\ &=\frac{1}{2} x^2 \left (a+b \text{csch}^{-1}(c x)\right )^3-\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{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{\left (3 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^2}\\ &=\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{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}(c x)\right )}{c^2}\\ &=\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{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^2}\\ &=\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{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \text{csch}^{-1}(c x)}\right )}{2 c^2}\\ &=\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 \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.45747, size = 171, normalized size = 1.46 \[ \frac{3 b^3 \text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )+a \left (a c x \left (a c x+3 b \sqrt{\frac{1}{c^2 x^2}+1}\right )-6 b^2 \log \left (\frac{1}{c x}\right )\right )+3 b^2 \text{csch}^{-1}(c x)^2 \left (a c^2 x^2+b \left (c x \sqrt{\frac{1}{c^2 x^2}+1}-1\right )\right )+3 b \text{csch}^{-1}(c x) \left (a c x \left (a c x+2 b \sqrt{\frac{1}{c^2 x^2}+1}\right )-2 b^2 \log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right )\right )+b^3 c^2 x^2 \text{csch}^{-1}(c x)^3}{2 c^2} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[x*(a + b*ArcCsch[c*x])^3,x]
[Out]
(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*ArcC
sch[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*ArcCsch[c*x])])/(2*c^2)
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Maple [F] time = 0.191, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(a+b*arccsch(c*x))^3,x)
[Out]
int(x*(a+b*arccsch(c*x))^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arccsch(c*x))^3,x, algorithm="maxima")
[Out]
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*sqrt(1/(c^2*x^2) + 1)/c)*a^2*b + 3*(x*s
qrt(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*integrate(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*integrate(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
*integrate(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)*log(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*integrate(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*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)*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) + 8*c^2*integrate(1/2*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*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*x^2 + sqrt(c^2*x
^2 + 1) + 1), x) + 24*c^2*integrate(1/2*x^3*log(x)*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) + 12*c^2*integrate(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) + 24*integrate(1/2*sqrt(c^2*x^2 + 1)*x*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*integrate(1/2*x*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*integrate(1/2*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)^2 + 24*integrate(1/2*sqrt(c^2
*x^2 + 1)*x*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*integrate(1
/2*sqrt(c^2*x^2 + 1)*x*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*integrate(1/2*x*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*integrate(1/2*x*log(x)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqr
t(c^2*x^2 + 1) + 1), x)*log(c) + 24*integrate(1/2*x*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) + 2*(c^2*x^2 - 2*sqrt(c^2*x^2 + 1) + 1)*log(c)^3/c^2 + 2*(2*sqrt(c
^2*x^2 + 1) - log(c^2*x^2 + 1))*log(c)^3/c^2 + 2*(log(c^2*x^2 + 1) - 2*log(sqrt(c^2*x^2 + 1) + 1))*log(c)^3/c^
2 + 6*(c^2*x^2 - 2*sqrt(c^2*x^2 + 1) + 1)*log(c)^2*log(x)/c^2 - 6*(c^2*x^2 - 2*sqrt(c^2*x^2 + 1) + 1)*log(c)^2
*log(sqrt(c^2*x^2 + 1) + 1)/c^2 + 4*log(c)^3*log(sqrt(c^2*x^2 + 1) + 1)/c^2 - 6*log(c)^2*log(sqrt(c^2*x^2 + 1)
+ 1)^2/c^2 + 4*log(c)*log(sqrt(c^2*x^2 + 1) + 1)^3/c^2 - 3*(c^2*x^2 - 4*sqrt(c^2*x^2 + 1) + 3*log(sqrt(c^2*x^
2 + 1) + 1) - log(sqrt(c^2*x^2 + 1) - 1) + 1)*log(c)^2/c^2 + 3*(c^2*x^2 - 6*sqrt(c^2*x^2 + 1) + 6*log(sqrt(c^2
*x^2 + 1) + 1) + 1)*log(c)^2/c^2 + 8*integrate(1/2*sqrt(c^2*x^2 + 1)*x*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*integrate(1/2*sqrt(c^2*x^2 + 1)*x*log(x)^2*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) + 24*integrate(1/2*sqrt(c^2*x^2 + 1)*x*log(
x)*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) + 8*integrat
e(1/2*x*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*integrate(1/2*x*log(x)
^2*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) + 24*integrate
(1/2*x*log(x)*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))*
b^3
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x \operatorname{arcsch}\left (c x\right )^{3} + 3 \, a b^{2} x \operatorname{arcsch}\left (c x\right )^{2} + 3 \, a^{2} b x \operatorname{arcsch}\left (c x\right ) + a^{3} x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arccsch(c*x))^3,x, algorithm="fricas")
[Out]
integral(b^3*x*arccsch(c*x)^3 + 3*a*b^2*x*arccsch(c*x)^2 + 3*a^2*b*x*arccsch(c*x) + a^3*x, x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \operatorname{acsch}{\left (c x \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*acsch(c*x))**3,x)
[Out]
Integral(x*(a + b*acsch(c*x))**3, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}^{3} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(a+b*arccsch(c*x))^3,x, algorithm="giac")
[Out]
integrate((b*arccsch(c*x) + a)^3*x, x)