3.5.89 \(\int x^2 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx\) [489]

Optimal. Leaf size=157 \[ -\frac {2 x^2 \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {2 i x \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {2 i x \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {2 \text {PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac {2 i \text {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {PolyLog}\left (3,i e^{a+b x}\right )}{b^3} \]

[Out]

-2*x^2*arctan(exp(b*x+a))/b-4*x*arctanh(exp(b*x+a))/b^2-x^2*csch(b*x+a)/b-2*polylog(2,-exp(b*x+a))/b^3+2*I*x*p
olylog(2,-I*exp(b*x+a))/b^2-2*I*x*polylog(2,I*exp(b*x+a))/b^2+2*polylog(2,exp(b*x+a))/b^3-2*I*polylog(3,-I*exp
(b*x+a))/b^3+2*I*polylog(3,I*exp(b*x+a))/b^3

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Rubi [A]
time = 0.17, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {2701, 327, 213, 5570, 14, 5313, 12, 4265, 2611, 2320, 6724, 4267, 2317, 2438} \begin {gather*} -\frac {2 x^2 \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {2 \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {2 \text {Li}_2\left (e^{a+b x}\right )}{b^3}-\frac {2 i \text {Li}_3\left (-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {Li}_3\left (i e^{a+b x}\right )}{b^3}+\frac {2 i x \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac {2 i x \text {Li}_2\left (i e^{a+b x}\right )}{b^2}-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Csch[a + b*x]^2*Sech[a + b*x],x]

[Out]

(-2*x^2*ArcTan[E^(a + b*x)])/b - (4*x*ArcTanh[E^(a + b*x)])/b^2 - (x^2*Csch[a + b*x])/b - (2*PolyLog[2, -E^(a
+ b*x)])/b^3 + ((2*I)*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - ((2*I)*x*PolyLog[2, I*E^(a + b*x)])/b^2 + (2*PolyL
og[2, E^(a + b*x)])/b^3 - ((2*I)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + ((2*I)*PolyLog[3, I*E^(a + b*x)])/b^3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[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 5313

Int[((a_.) + ArcTan[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcTan[
u])/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(1 + u^2)), x], x]
, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m +
1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 5570

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

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

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Mathematica [A]
time = 1.27, size = 312, normalized size = 1.99 \begin {gather*} \frac {-2 b^2 x^2 \text {csch}(a)+4 a \log \left (1-e^{-a-b x}\right )+4 b x \log \left (1-e^{-a-b x}\right )-4 a \log \left (1+e^{-a-b x}\right )-4 b x \log \left (1+e^{-a-b x}\right )-2 i b^2 x^2 \log \left (1-i e^{a+b x}\right )+2 i b^2 x^2 \log \left (1+i e^{a+b x}\right )-4 a \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )+4 \text {PolyLog}\left (2,-e^{-a-b x}\right )-4 \text {PolyLog}\left (2,e^{-a-b x}\right )+4 i b x \text {PolyLog}\left (2,-i e^{a+b x}\right )-4 i b x \text {PolyLog}\left (2,i e^{a+b x}\right )-4 i \text {PolyLog}\left (3,-i e^{a+b x}\right )+4 i \text {PolyLog}\left (3,i e^{a+b x}\right )+b^2 x^2 \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )+b^2 x^2 \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )}{2 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Csch[a + b*x]^2*Sech[a + b*x],x]

[Out]

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

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Maple [F]
time = 1.07, size = 0, normalized size = 0.00 \[\int x^{2} \mathrm {csch}\left (b x +a \right )^{2} \mathrm {sech}\left (b x +a \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*csch(b*x+a)^2*sech(b*x+a),x)

[Out]

int(x^2*csch(b*x+a)^2*sech(b*x+a),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*csch(b*x+a)^2*sech(b*x+a),x, algorithm="maxima")

[Out]

-2*x^2*e^(b*x + a)/(b*e^(2*b*x + 2*a) - b) - 2*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^3 + 2*(b*x*l
og(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^3 - 8*integrate(1/4*x^2*e^(b*x + a)/(e^(2*b*x + 2*a) + 1), x)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 966 vs. \(2 (129) = 258\).
time = 0.43, size = 966, normalized size = 6.15 \begin {gather*} -\frac {2 \, b^{2} x^{2} \cosh \left (b x + a\right ) + 2 \, b^{2} x^{2} \sinh \left (b x + a\right ) - 2 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\left (i \, b x \cosh \left (b x + a\right )^{2} + 2 i \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, b x \sinh \left (b x + a\right )^{2} - i \, b x\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 2 \, {\left (-i \, b x \cosh \left (b x + a\right )^{2} - 2 i \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, b x \sinh \left (b x + a\right )^{2} + i \, b x\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 2 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 2 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (-i \, a^{2} \cosh \left (b x + a\right )^{2} - 2 i \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, a^{2} \sinh \left (b x + a\right )^{2} + i \, a^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) - {\left (i \, a^{2} \cosh \left (b x + a\right )^{2} + 2 i \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, a^{2} \sinh \left (b x + a\right )^{2} - i \, a^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + 2 \, {\left (a \cosh \left (b x + a\right )^{2} + 2 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a \sinh \left (b x + a\right )^{2} - a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left (-i \, b^{2} x^{2} + {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \sinh \left (b x + a\right )^{2} + i \, a^{2}\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - {\left (i \, b^{2} x^{2} + {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \sinh \left (b x + a\right )^{2} - i \, a^{2}\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )^{2} - b x - a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left (-i \, \cosh \left (b x + a\right )^{2} - 2 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )^{2} + i\right )} {\rm polylog}\left (3, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 2 \, {\left (i \, \cosh \left (b x + a\right )^{2} + 2 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )^{2} - i\right )} {\rm polylog}\left (3, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right )}{b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} - b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*csch(b*x+a)^2*sech(b*x+a),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*csch(b*x+a)**2*sech(b*x+a),x)

[Out]

Integral(x**2*csch(a + b*x)**2*sech(a + b*x), 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*csch(b*x+a)^2*sech(b*x+a),x, algorithm="giac")

[Out]

sage0*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(cosh(a + b*x)*sinh(a + b*x)^2),x)

[Out]

int(x^2/(cosh(a + b*x)*sinh(a + b*x)^2), x)

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