3.1.12 \(\int x^2 \coth ^3(a+b x) \, dx\) [12]

Optimal. Leaf size=114 \[ \frac {x^2}{2 b}-\frac {x^3}{3}-\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \coth ^2(a+b x)}{2 b}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\log (\sinh (a+b x))}{b^3}+\frac {x \text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^2}-\frac {\text {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3} \]

[Out]

1/2*x^2/b-1/3*x^3-x*coth(b*x+a)/b^2-1/2*x^2*coth(b*x+a)^2/b+x^2*ln(1-exp(2*b*x+2*a))/b+ln(sinh(b*x+a))/b^3+x*p
olylog(2,exp(2*b*x+2*a))/b^2-1/2*polylog(3,exp(2*b*x+2*a))/b^3

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Rubi [A]
time = 0.15, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3801, 3556, 30, 3797, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {\text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3}+\frac {\log (\sinh (a+b x))}{b^3}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {x \coth (a+b x)}{b^2}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {x^2 \coth ^2(a+b x)}{2 b}+\frac {x^2}{2 b}-\frac {x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Coth[a + b*x]^3,x]

[Out]

x^2/(2*b) - x^3/3 - (x*Coth[a + b*x])/b^2 - (x^2*Coth[a + b*x]^2)/(2*b) + (x^2*Log[1 - E^(2*(a + b*x))])/b + L
og[Sinh[a + b*x]]/b^3 + (x*PolyLog[2, E^(2*(a + b*x))])/b^2 - PolyLog[3, E^(2*(a + b*x))]/(2*b^3)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2221

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]
- 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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 2611

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] + Dist[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 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3797

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))/(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] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3801

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

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps

\begin {align*} \int x^2 \coth ^3(a+b x) \, dx &=-\frac {x^2 \coth ^2(a+b x)}{2 b}+\frac {\int x \coth ^2(a+b x) \, dx}{b}+\int x^2 \coth (a+b x) \, dx\\ &=-\frac {x^3}{3}-\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \coth ^2(a+b x)}{2 b}-2 \int \frac {e^{2 (a+b x)} x^2}{1-e^{2 (a+b x)}} \, dx+\frac {\int \coth (a+b x) \, dx}{b^2}+\frac {\int x \, dx}{b}\\ &=\frac {x^2}{2 b}-\frac {x^3}{3}-\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \coth ^2(a+b x)}{2 b}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\log (\sinh (a+b x))}{b^3}-\frac {2 \int x \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b}\\ &=\frac {x^2}{2 b}-\frac {x^3}{3}-\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \coth ^2(a+b x)}{2 b}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\log (\sinh (a+b x))}{b^3}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {\int \text {Li}_2\left (e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=\frac {x^2}{2 b}-\frac {x^3}{3}-\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \coth ^2(a+b x)}{2 b}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\log (\sinh (a+b x))}{b^3}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {\text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^3}\\ &=\frac {x^2}{2 b}-\frac {x^3}{3}-\frac {x \coth (a+b x)}{b^2}-\frac {x^2 \coth ^2(a+b x)}{2 b}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\log (\sinh (a+b x))}{b^3}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {\text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3}\\ \end {align*}

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Mathematica [A]
time = 1.49, size = 133, normalized size = 1.17 \begin {gather*} \frac {1}{3} x^3 \coth (a)-\frac {x^2 \text {csch}^2(a+b x)}{2 b}+\frac {-\frac {4 b e^{2 a} x \left (3+b^2 x^2\right )}{-1+e^{2 a}}+6 \left (1+b^2 x^2\right ) \log \left (1-e^{2 (a+b x)}\right )+6 b x \text {PolyLog}\left (2,e^{2 (a+b x)}\right )-3 \text {PolyLog}\left (3,e^{2 (a+b x)}\right )}{6 b^3}+\frac {x \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Coth[a + b*x]^3,x]

[Out]

(x^3*Coth[a])/3 - (x^2*Csch[a + b*x]^2)/(2*b) + ((-4*b*E^(2*a)*x*(3 + b^2*x^2))/(-1 + E^(2*a)) + 6*(1 + b^2*x^
2)*Log[1 - E^(2*(a + b*x))] + 6*b*x*PolyLog[2, E^(2*(a + b*x))] - 3*PolyLog[3, E^(2*(a + b*x))])/(6*b^3) + (x*
Csch[a]*Csch[a + b*x]*Sinh[b*x])/b^2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(245\) vs. \(2(106)=212\).
time = 1.69, size = 246, normalized size = 2.16

method	result	size
risch	\(-\frac {x^{3}}{3}-\frac {2 x \left (b x \,{\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 b x +2 a}-1\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {2 a^{2} x}{b^{2}}+\frac {4 a^{3}}{3 b^{3}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}+\frac {2 \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}+\frac {2 \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b}+\frac {a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}-\frac {2 a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}\)	\(246\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*coth(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/3*x^3-2*x*(b*x*exp(2*b*x+2*a)+exp(2*b*x+2*a)-1)/b^2/(exp(2*b*x+2*a)-1)^2+2/b^2*a^2*x+4/3/b^3*a^3-2/b^3*ln(e
xp(b*x+a))-2/b^3*polylog(3,exp(b*x+a))-2/b^3*polylog(3,-exp(b*x+a))+1/b^3*ln(exp(b*x+a)+1)+1/b^3*ln(exp(b*x+a)
-1)+2/b^2*polylog(2,-exp(b*x+a))*x+1/b*ln(1-exp(b*x+a))*x^2+2/b^2*polylog(2,exp(b*x+a))*x+1/b*ln(exp(b*x+a)+1)
*x^2+1/b^3*a^2*ln(exp(b*x+a)-1)-2/b^3*a^2*ln(exp(b*x+a))-1/b^3*ln(1-exp(b*x+a))*a^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (105) = 210\).
time = 0.32, size = 226, normalized size = 1.98 \begin {gather*} -\frac {2}{3} \, x^{3} + \frac {b^{2} x^{3} e^{\left (4 \, b x + 4 \, a\right )} + b^{2} x^{3} - 2 \, {\left (b^{2} x^{3} e^{\left (2 \, a\right )} + 3 \, b x^{2} e^{\left (2 \, a\right )} + 3 \, x e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 6 \, x}{3 \, {\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} - \frac {2 \, x}{b^{2}} + \frac {b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})}{b^{3}} + \frac {b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})}{b^{3}} + \frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{b^{3}} + \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*coth(b*x+a)^3,x, algorithm="maxima")

[Out]

-2/3*x^3 + 1/3*(b^2*x^3*e^(4*b*x + 4*a) + b^2*x^3 - 2*(b^2*x^3*e^(2*a) + 3*b*x^2*e^(2*a) + 3*x*e^(2*a))*e^(2*b
*x) + 6*x)/(b^2*e^(4*b*x + 4*a) - 2*b^2*e^(2*b*x + 2*a) + b^2) - 2*x/b^2 + (b^2*x^2*log(e^(b*x + a) + 1) + 2*b
*x*dilog(-e^(b*x + a)) - 2*polylog(3, -e^(b*x + a)))/b^3 + (b^2*x^2*log(-e^(b*x + a) + 1) + 2*b*x*dilog(e^(b*x
+ a)) - 2*polylog(3, e^(b*x + a)))/b^3 + log(e^(b*x + a) + 1)/b^3 + log(e^(b*x + a) - 1)/b^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1467 vs. \(2 (105) = 210\).
time = 0.37, size = 1467, normalized size = 12.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*coth(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/3*(b^3*x^3 + (b^3*x^3 + 2*a^3 + 6*b*x + 6*a)*cosh(b*x + a)^4 + 4*(b^3*x^3 + 2*a^3 + 6*b*x + 6*a)*cosh(b*x +
a)*sinh(b*x + a)^3 + (b^3*x^3 + 2*a^3 + 6*b*x + 6*a)*sinh(b*x + a)^4 + 2*a^3 - 2*(b^3*x^3 - 3*b^2*x^2 + 2*a^3
+ 3*b*x + 6*a)*cosh(b*x + a)^2 - 2*(b^3*x^3 - 3*b^2*x^2 + 2*a^3 - 3*(b^3*x^3 + 2*a^3 + 6*b*x + 6*a)*cosh(b*x
+ a)^2 + 3*b*x + 6*a)*sinh(b*x + a)^2 - 6*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sin
h(b*x + a)^4 - 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^2 + b*x + 4*(b*x*cosh(b*x
+ a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))*dilog(cosh(b*x + a) + sinh(b*x + a)) - 6*(b*x*cosh(b*x + a)^4 + 4*
b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 - b
*x)*sinh(b*x + a)^2 + b*x + 4*(b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))*dilog(-cosh(b*x + a) -
sinh(b*x + a)) - 3*((b^2*x^2 + 1)*cosh(b*x + a)^4 + 4*(b^2*x^2 + 1)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 +
1)*sinh(b*x + a)^4 + b^2*x^2 - 2*(b^2*x^2 + 1)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 + 1)*cosh(b*x + a)^2
+ 1)*sinh(b*x + a)^2 + 4*((b^2*x^2 + 1)*cosh(b*x + a)^3 - (b^2*x^2 + 1)*cosh(b*x + a))*sinh(b*x + a) + 1)*log
(cosh(b*x + a) + sinh(b*x + a) + 1) - 3*((a^2 + 1)*cosh(b*x + a)^4 + 4*(a^2 + 1)*cosh(b*x + a)*sinh(b*x + a)^3
+ (a^2 + 1)*sinh(b*x + a)^4 - 2*(a^2 + 1)*cosh(b*x + a)^2 + 2*(3*(a^2 + 1)*cosh(b*x + a)^2 - a^2 - 1)*sinh(b*
x + a)^2 + a^2 + 4*((a^2 + 1)*cosh(b*x + a)^3 - (a^2 + 1)*cosh(b*x + a))*sinh(b*x + a) + 1)*log(cosh(b*x + a)
+ sinh(b*x + a) - 1) - 3*((b^2*x^2 - a^2)*cosh(b*x + a)^4 + 4*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a)^3 +
(b^2*x^2 - a^2)*sinh(b*x + a)^4 + b^2*x^2 - 2*(b^2*x^2 - a^2)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 - a^2)
*cosh(b*x + a)^2 - a^2)*sinh(b*x + a)^2 - a^2 + 4*((b^2*x^2 - a^2)*cosh(b*x + a)^3 - (b^2*x^2 - a^2)*cosh(b*x
+ a))*sinh(b*x + a))*log(-cosh(b*x + a) - sinh(b*x + a) + 1) + 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x +
a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 -
cosh(b*x + a))*sinh(b*x + a) + 1)*polylog(3, cosh(b*x + a) + sinh(b*x + a)) + 6*(cosh(b*x + a)^4 + 4*cosh(b*x
+ a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(c
osh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*polylog(3, -cosh(b*x + a) - sinh(b*x + a)) + 4*((b^3*x^3 +
2*a^3 + 6*b*x + 6*a)*cosh(b*x + a)^3 - (b^3*x^3 - 3*b^2*x^2 + 2*a^3 + 3*b*x + 6*a)*cosh(b*x + a))*sinh(b*x + a
) + 6*a)/(b^3*cosh(b*x + a)^4 + 4*b^3*cosh(b*x + a)*sinh(b*x + a)^3 + b^3*sinh(b*x + a)^4 - 2*b^3*cosh(b*x + a
)^2 + b^3 + 2*(3*b^3*cosh(b*x + a)^2 - b^3)*sinh(b*x + a)^2 + 4*(b^3*cosh(b*x + a)^3 - b^3*cosh(b*x + a))*sinh
(b*x + a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \coth ^{3}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*coth(b*x+a)**3,x)

[Out]

Integral(x**2*coth(a + b*x)**3, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*coth(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(x^2*coth(b*x + a)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\mathrm {coth}\left (a+b\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*coth(a + b*x)^3,x)

[Out]

int(x^2*coth(a + b*x)^3, x)

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