3.6.22 \(\int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx\) [522]

Optimal. Leaf size=240 \[ -\frac {6 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b^3}+\frac {4 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {3 x^2 \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}-\frac {3 \text {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b^4}+\frac {3 x^2 \text {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{b^2}+\frac {3 \text {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 x^2 \text {PolyLog}\left (2,e^{2 a+2 b x}\right )}{b^2}-\frac {3 x \text {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{b^3}+\frac {3 x \text {PolyLog}\left (3,e^{2 a+2 b x}\right )}{b^3}+\frac {3 \text {PolyLog}\left (4,-e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 \text {PolyLog}\left (4,e^{2 a+2 b x}\right )}{2 b^4} \]

[Out]

-6*x*arctanh(exp(2*b*x+2*a))/b^3+4*x^3*arctanh(exp(2*b*x+2*a))/b-3*x^2*csch(2*b*x+2*a)/b^2-2*x^3*coth(2*b*x+2*
a)*csch(2*b*x+2*a)/b-3/2*polylog(2,-exp(2*b*x+2*a))/b^4+3*x^2*polylog(2,-exp(2*b*x+2*a))/b^2+3/2*polylog(2,exp
(2*b*x+2*a))/b^4-3*x^2*polylog(2,exp(2*b*x+2*a))/b^2-3*x*polylog(3,-exp(2*b*x+2*a))/b^3+3*x*polylog(3,exp(2*b*
x+2*a))/b^3+3/2*polylog(4,-exp(2*b*x+2*a))/b^4-3/2*polylog(4,exp(2*b*x+2*a))/b^4

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Rubi [A]
time = 0.22, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5569, 4271, 4267, 2317, 2438, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3 \text {Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^4}+\frac {3 \text {Li}_2\left (e^{2 a+2 b x}\right )}{2 b^4}+\frac {3 \text {Li}_4\left (-e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 \text {Li}_4\left (e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 x \text {Li}_3\left (-e^{2 a+2 b x}\right )}{b^3}+\frac {3 x \text {Li}_3\left (e^{2 a+2 b x}\right )}{b^3}-\frac {6 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b^3}+\frac {3 x^2 \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}-\frac {3 x^2 \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}-\frac {3 x^2 \text {csch}(2 a+2 b x)}{b^2}+\frac {4 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {2 x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Csch[a + b*x]^3*Sech[a + b*x]^3,x]

[Out]

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

Rule 2317

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 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 2438

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 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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[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 4271

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

Rule 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

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]

Rule 6744

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] - Dist[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]

Rubi steps

\begin {align*} \int x^3 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx &=8 \int x^3 \text {csch}^3(2 a+2 b x) \, dx\\ &=-\frac {3 x^2 \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}-4 \int x^3 \text {csch}(2 a+2 b x) \, dx+\frac {6 \int x \text {csch}(2 a+2 b x) \, dx}{b^2}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b^3}+\frac {4 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {3 x^2 \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}-\frac {3 \int \log \left (1-e^{2 a+2 b x}\right ) \, dx}{b^3}+\frac {3 \int \log \left (1+e^{2 a+2 b x}\right ) \, dx}{b^3}+\frac {6 \int x^2 \log \left (1-e^{2 a+2 b x}\right ) \, dx}{b}-\frac {6 \int x^2 \log \left (1+e^{2 a+2 b x}\right ) \, dx}{b}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b^3}+\frac {4 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {3 x^2 \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}+\frac {3 x^2 \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}-\frac {3 x^2 \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}-\frac {3 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{2 b^4}+\frac {3 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{2 b^4}-\frac {6 \int x \text {Li}_2\left (-e^{2 a+2 b x}\right ) \, dx}{b^2}+\frac {6 \int x \text {Li}_2\left (e^{2 a+2 b x}\right ) \, dx}{b^2}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b^3}+\frac {4 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {3 x^2 \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}-\frac {3 \text {Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^4}+\frac {3 x^2 \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}+\frac {3 \text {Li}_2\left (e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 x^2 \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}-\frac {3 x \text {Li}_3\left (-e^{2 a+2 b x}\right )}{b^3}+\frac {3 x \text {Li}_3\left (e^{2 a+2 b x}\right )}{b^3}+\frac {3 \int \text {Li}_3\left (-e^{2 a+2 b x}\right ) \, dx}{b^3}-\frac {3 \int \text {Li}_3\left (e^{2 a+2 b x}\right ) \, dx}{b^3}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b^3}+\frac {4 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {3 x^2 \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}-\frac {3 \text {Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^4}+\frac {3 x^2 \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}+\frac {3 \text {Li}_2\left (e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 x^2 \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}-\frac {3 x \text {Li}_3\left (-e^{2 a+2 b x}\right )}{b^3}+\frac {3 x \text {Li}_3\left (e^{2 a+2 b x}\right )}{b^3}+\frac {3 \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{2 b^4}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b^3}+\frac {4 x^3 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {3 x^2 \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^3 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}-\frac {3 \text {Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^4}+\frac {3 x^2 \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}+\frac {3 \text {Li}_2\left (e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 x^2 \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}-\frac {3 x \text {Li}_3\left (-e^{2 a+2 b x}\right )}{b^3}+\frac {3 x \text {Li}_3\left (e^{2 a+2 b x}\right )}{b^3}+\frac {3 \text {Li}_4\left (-e^{2 a+2 b x}\right )}{2 b^4}-\frac {3 \text {Li}_4\left (e^{2 a+2 b x}\right )}{2 b^4}\\ \end {align*}

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Mathematica [A]
time = 4.45, size = 274, normalized size = 1.14 \begin {gather*} \frac {-b^3 x^3 \text {csch}^2(a+b x)+6 b x \log \left (1-e^{2 (a+b x)}\right )-4 b^3 x^3 \log \left (1-e^{2 (a+b x)}\right )-6 b x \log \left (1+e^{2 (a+b x)}\right )+4 b^3 x^3 \log \left (1+e^{2 (a+b x)}\right )+\left (-3+6 b^2 x^2\right ) \text {PolyLog}\left (2,-e^{2 (a+b x)}\right )+\left (3-6 b^2 x^2\right ) \text {PolyLog}\left (2,e^{2 (a+b x)}\right )-6 b x \text {PolyLog}\left (3,-e^{2 (a+b x)}\right )+6 b x \text {PolyLog}\left (3,e^{2 (a+b x)}\right )+3 \text {PolyLog}\left (4,-e^{2 (a+b x)}\right )-3 \text {PolyLog}\left (4,e^{2 (a+b x)}\right )-3 b^2 x^2 \text {csch}(a) \text {sech}(a)-b^3 x^3 \text {sech}^2(a+b x)+3 b^2 x^2 \text {csch}(a) \text {csch}(a+b x) \sinh (b x)+3 b^2 x^2 \text {sech}(a) \text {sech}(a+b x) \sinh (b x)}{2 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Csch[a + b*x]^3*Sech[a + b*x]^3,x]

[Out]

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

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Maple [A]
time = 2.06, size = 445, normalized size = 1.85

method	result	size
risch	\(-\frac {2 x^{2} {\mathrm e}^{2 b x +2 a} \left (2 b x \,{\mathrm e}^{4 b x +4 a}+3 \,{\mathrm e}^{4 b x +4 a}+2 b x -3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {3 \polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {3 \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {12 \polylog \left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {12 \polylog \left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \polylog \left (4, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{4}}-\frac {3 x \polylog \left (3, -{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{3}}+\frac {2 x^{3} \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}+\frac {3 x^{2} \polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}+\frac {3 a \ln \left (1-{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 x \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b^{3}}+\frac {12 x \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{b}-\frac {6 x^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {12 x \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{b}-\frac {6 x^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {3 a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}-\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}\)	\(445\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*csch(b*x+a)^3*sech(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-2*x^2*exp(2*b*x+2*a)*(2*b*x*exp(4*b*x+4*a)+3*exp(4*b*x+4*a)+2*b*x-3)/b^2/(exp(2*b*x+2*a)+1)^2/(exp(2*b*x+2*a)
-1)^2-3/2*polylog(2,-exp(2*b*x+2*a))/b^4+2/b^4*a^3*ln(exp(b*x+a)-1)+3*polylog(2,exp(b*x+a))/b^4-12*polylog(4,e
xp(b*x+a))/b^4+3*polylog(2,-exp(b*x+a))/b^4-12*polylog(4,-exp(b*x+a))/b^4+3/2*polylog(4,-exp(2*b*x+2*a))/b^4-3
*x*polylog(3,-exp(2*b*x+2*a))/b^3+3/b^3*ln(1-exp(b*x+a))*x+3/b^3*ln(exp(b*x+a)+1)*x+2*x^3*ln(exp(2*b*x+2*a)+1)
/b+3*x^2*polylog(2,-exp(2*b*x+2*a))/b^2+3/b^4*a*ln(1-exp(b*x+a))-3*x*ln(exp(2*b*x+2*a)+1)/b^3+12*x*polylog(3,e
xp(b*x+a))/b^3-2/b*ln(exp(b*x+a)+1)*x^3-6*x^2*polylog(2,-exp(b*x+a))/b^2+12*x*polylog(3,-exp(b*x+a))/b^3-2/b*l
n(1-exp(b*x+a))*x^3-6*x^2*polylog(2,exp(b*x+a))/b^2-3/b^4*a*ln(exp(b*x+a)-1)-2/b^4*ln(1-exp(b*x+a))*a^3

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Maxima [A]
time = 0.28, size = 381, normalized size = 1.59 \begin {gather*} -\frac {2 \, {\left ({\left (2 \, b x^{3} e^{\left (6 \, a\right )} + 3 \, x^{2} e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )} + {\left (2 \, b x^{3} e^{\left (2 \, a\right )} - 3 \, x^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}\right )}}{b^{2} e^{\left (8 \, b x + 8 \, a\right )} - 2 \, b^{2} e^{\left (4 \, b x + 4 \, a\right )} + b^{2}} + \frac {2 \, {\left (4 \, b^{3} x^{3} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )}) + 3 \, {\rm Li}_{4}(-e^{\left (2 \, b x + 2 \, a\right )})\right )}}{3 \, b^{4}} - \frac {2 \, {\left (b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})\right )}}{b^{4}} - \frac {2 \, {\left (b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})\right )}}{b^{4}} - \frac {3 \, {\left (2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right )\right )}}{2 \, b^{4}} + \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{4}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csch(b*x+a)^3*sech(b*x+a)^3,x, algorithm="maxima")

[Out]

-2*((2*b*x^3*e^(6*a) + 3*x^2*e^(6*a))*e^(6*b*x) + (2*b*x^3*e^(2*a) - 3*x^2*e^(2*a))*e^(2*b*x))/(b^2*e^(8*b*x +
8*a) - 2*b^2*e^(4*b*x + 4*a) + b^2) + 2/3*(4*b^3*x^3*log(e^(2*b*x + 2*a) + 1) + 6*b^2*x^2*dilog(-e^(2*b*x + 2
*a)) - 6*b*x*polylog(3, -e^(2*b*x + 2*a)) + 3*polylog(4, -e^(2*b*x + 2*a)))/b^4 - 2*(b^3*x^3*log(e^(b*x + a) +
1) + 3*b^2*x^2*dilog(-e^(b*x + a)) - 6*b*x*polylog(3, -e^(b*x + a)) + 6*polylog(4, -e^(b*x + a)))/b^4 - 2*(b^
3*x^3*log(-e^(b*x + a) + 1) + 3*b^2*x^2*dilog(e^(b*x + a)) - 6*b*x*polylog(3, e^(b*x + a)) + 6*polylog(4, e^(b
*x + a)))/b^4 - 3/2*(2*b*x*log(e^(2*b*x + 2*a) + 1) + dilog(-e^(2*b*x + 2*a)))/b^4 + 3*(b*x*log(e^(b*x + a) +
1) + dilog(-e^(b*x + a)))/b^4 + 3*(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^4

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Fricas [C] Result contains complex when optimal does not.
time = 0.57, size = 6764, normalized size = 28.18 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csch(b*x+a)^3*sech(b*x+a)^3,x, algorithm="fricas")

[Out]

-(2*(2*b^3*x^3 + 3*b^2*x^2)*cosh(b*x + a)^6 + 40*(2*b^3*x^3 + 3*b^2*x^2)*cosh(b*x + a)^3*sinh(b*x + a)^3 + 30*
(2*b^3*x^3 + 3*b^2*x^2)*cosh(b*x + a)^2*sinh(b*x + a)^4 + 12*(2*b^3*x^3 + 3*b^2*x^2)*cosh(b*x + a)*sinh(b*x +
a)^5 + 2*(2*b^3*x^3 + 3*b^2*x^2)*sinh(b*x + a)^6 + 2*(2*b^3*x^3 - 3*b^2*x^2)*cosh(b*x + a)^2 + 2*(2*b^3*x^3 +
15*(2*b^3*x^3 + 3*b^2*x^2)*cosh(b*x + a)^4 - 3*b^2*x^2)*sinh(b*x + a)^2 + 3*((2*b^2*x^2 - 1)*cosh(b*x + a)^8 +
56*(2*b^2*x^2 - 1)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(2*b^2*x^2 - 1)*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*(
2*b^2*x^2 - 1)*cosh(b*x + a)*sinh(b*x + a)^7 + (2*b^2*x^2 - 1)*sinh(b*x + a)^8 - 2*(2*b^2*x^2 - 1)*cosh(b*x +
a)^4 + 2*(35*(2*b^2*x^2 - 1)*cosh(b*x + a)^4 - 2*b^2*x^2 + 1)*sinh(b*x + a)^4 + 2*b^2*x^2 + 8*(7*(2*b^2*x^2 -
1)*cosh(b*x + a)^5 - (2*b^2*x^2 - 1)*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*(2*b^2*x^2 - 1)*cosh(b*x + a)^6 - 3
*(2*b^2*x^2 - 1)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 8*((2*b^2*x^2 - 1)*cosh(b*x + a)^7 - (2*b^2*x^2 - 1)*cosh(
b*x + a)^3)*sinh(b*x + a) - 1)*dilog(cosh(b*x + a) + sinh(b*x + a)) - 3*((2*b^2*x^2 - 1)*cosh(b*x + a)^8 + 56*
(2*b^2*x^2 - 1)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(2*b^2*x^2 - 1)*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*(2*b^
2*x^2 - 1)*cosh(b*x + a)*sinh(b*x + a)^7 + (2*b^2*x^2 - 1)*sinh(b*x + a)^8 - 2*(2*b^2*x^2 - 1)*cosh(b*x + a)^4
+ 2*(35*(2*b^2*x^2 - 1)*cosh(b*x + a)^4 - 2*b^2*x^2 + 1)*sinh(b*x + a)^4 + 2*b^2*x^2 + 8*(7*(2*b^2*x^2 - 1)*c
osh(b*x + a)^5 - (2*b^2*x^2 - 1)*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*(2*b^2*x^2 - 1)*cosh(b*x + a)^6 - 3*(2*
b^2*x^2 - 1)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 8*((2*b^2*x^2 - 1)*cosh(b*x + a)^7 - (2*b^2*x^2 - 1)*cosh(b*x
+ a)^3)*sinh(b*x + a) - 1)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 3*((2*b^2*x^2 - 1)*cosh(b*x + a)^8 + 56*
(2*b^2*x^2 - 1)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(2*b^2*x^2 - 1)*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*(2*b^
2*x^2 - 1)*cosh(b*x + a)*sinh(b*x + a)^7 + (2*b^2*x^2 - 1)*sinh(b*x + a)^8 - 2*(2*b^2*x^2 - 1)*cosh(b*x + a)^4
+ 2*(35*(2*b^2*x^2 - 1)*cosh(b*x + a)^4 - 2*b^2*x^2 + 1)*sinh(b*x + a)^4 + 2*b^2*x^2 + 8*(7*(2*b^2*x^2 - 1)*c
osh(b*x + a)^5 - (2*b^2*x^2 - 1)*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*(2*b^2*x^2 - 1)*cosh(b*x + a)^6 - 3*(2*
b^2*x^2 - 1)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 8*((2*b^2*x^2 - 1)*cosh(b*x + a)^7 - (2*b^2*x^2 - 1)*cosh(b*x
+ a)^3)*sinh(b*x + a) - 1)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + 3*((2*b^2*x^2 - 1)*cosh(b*x + a)^8 + 56
*(2*b^2*x^2 - 1)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(2*b^2*x^2 - 1)*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*(2*b
^2*x^2 - 1)*cosh(b*x + a)*sinh(b*x + a)^7 + (2*b^2*x^2 - 1)*sinh(b*x + a)^8 - 2*(2*b^2*x^2 - 1)*cosh(b*x + a)^
4 + 2*(35*(2*b^2*x^2 - 1)*cosh(b*x + a)^4 - 2*b^2*x^2 + 1)*sinh(b*x + a)^4 + 2*b^2*x^2 + 8*(7*(2*b^2*x^2 - 1)*
cosh(b*x + a)^5 - (2*b^2*x^2 - 1)*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*(2*b^2*x^2 - 1)*cosh(b*x + a)^6 - 3*(2
*b^2*x^2 - 1)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 8*((2*b^2*x^2 - 1)*cosh(b*x + a)^7 - (2*b^2*x^2 - 1)*cosh(b*x
+ a)^3)*sinh(b*x + a) - 1)*dilog(-cosh(b*x + a) - sinh(b*x + a)) + ((2*b^3*x^3 - 3*b*x)*cosh(b*x + a)^8 + 56*
(2*b^3*x^3 - 3*b*x)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(2*b^3*x^3 - 3*b*x)*cosh(b*x + a)^2*sinh(b*x + a)^6 +
8*(2*b^3*x^3 - 3*b*x)*cosh(b*x + a)*sinh(b*x + a)^7 + (2*b^3*x^3 - 3*b*x)*sinh(b*x + a)^8 + 2*b^3*x^3 - 2*(2*
b^3*x^3 - 3*b*x)*cosh(b*x + a)^4 - 2*(2*b^3*x^3 - 35*(2*b^3*x^3 - 3*b*x)*cosh(b*x + a)^4 - 3*b*x)*sinh(b*x + a
)^4 + 8*(7*(2*b^3*x^3 - 3*b*x)*cosh(b*x + a)^5 - (2*b^3*x^3 - 3*b*x)*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*(2*
b^3*x^3 - 3*b*x)*cosh(b*x + a)^6 - 3*(2*b^3*x^3 - 3*b*x)*cosh(b*x + a)^2)*sinh(b*x + a)^2 - 3*b*x + 8*((2*b^3*
x^3 - 3*b*x)*cosh(b*x + a)^7 - (2*b^3*x^3 - 3*b*x)*cosh(b*x + a)^3)*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*
x + a) + 1) + ((2*a^3 - 3*a)*cosh(b*x + a)^8 + 56*(2*a^3 - 3*a)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(2*a^3 -
3*a)*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*(2*a^3 - 3*a)*cosh(b*x + a)*sinh(b*x + a)^7 + (2*a^3 - 3*a)*sinh(b*x
+ a)^8 - 2*(2*a^3 - 3*a)*cosh(b*x + a)^4 + 2*(35*(2*a^3 - 3*a)*cosh(b*x + a)^4 - 2*a^3 + 3*a)*sinh(b*x + a)^4
+ 8*(7*(2*a^3 - 3*a)*cosh(b*x + a)^5 - (2*a^3 - 3*a)*cosh(b*x + a))*sinh(b*x + a)^3 + 2*a^3 + 4*(7*(2*a^3 - 3*
a)*cosh(b*x + a)^6 - 3*(2*a^3 - 3*a)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 8*((2*a^3 - 3*a)*cosh(b*x + a)^7 - (2*
a^3 - 3*a)*cosh(b*x + a)^3)*sinh(b*x + a) - 3*a)*log(cosh(b*x + a) + sinh(b*x + a) + I) + ((2*a^3 - 3*a)*cosh(
b*x + a)^8 + 56*(2*a^3 - 3*a)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(2*a^3 - 3*a)*cosh(b*x + a)^2*sinh(b*x + a)
^6 + 8*(2*a^3 - 3*a)*cosh(b*x + a)*sinh(b*x + a)^7 + (2*a^3 - 3*a)*sinh(b*x + a)^8 - 2*(2*a^3 - 3*a)*cosh(b*x
+ a)^4 + 2*(35*(2*a^3 - 3*a)*cosh(b*x + a)^4 - 2*a^3 + 3*a)*sinh(b*x + a)^4 + 8*(7*(2*a^3 - 3*a)*cosh(b*x + a)
^5 - (2*a^3 - 3*a)*cosh(b*x + a))*sinh(b*x + a)^3 + 2*a^3 + 4*(7*(2*a^3 - 3*a)*cosh(b*x + a)^6 - 3*(2*a^3 - 3*
a)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 8*((2*a^3 - 3*a)*cosh(b*x + a)^7 - (2*a^3 - 3*a)*cosh(b*x + a)^3)*sinh(b
*x + a) - 3*a)*log(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^{3} \operatorname {csch}^{3}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*csch(b*x+a)**3*sech(b*x+a)**3,x)

[Out]

Integral(x**3*csch(a + b*x)**3*sech(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^3*csch(b*x+a)^3*sech(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(x^3*csch(b*x + a)^3*sech(b*x + a)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(cosh(a + b*x)^3*sinh(a + b*x)^3),x)

[Out]

int(x^3/(cosh(a + b*x)^3*sinh(a + b*x)^3), x)

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