3.5.54 \(\int x^2 \coth ^2(a+b x) \text {csch}(a+b x) \, dx\) [454]

Optimal. Leaf size=123 \[ -\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {x \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {x \text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {\text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {\text {PolyLog}\left (3,e^{a+b x}\right )}{b^3} \]

[Out]

-x^2*arctanh(exp(b*x+a))/b-arctanh(cosh(b*x+a))/b^3-x*csch(b*x+a)/b^2-1/2*x^2*coth(b*x+a)*csch(b*x+a)/b-x*poly
log(2,-exp(b*x+a))/b^2+x*polylog(2,exp(b*x+a))/b^2+polylog(3,-exp(b*x+a))/b^3-polylog(3,exp(b*x+a))/b^3

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Rubi [A]
time = 0.18, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5565, 4267, 2611, 2320, 6724, 4271, 3855} \begin {gather*} \frac {\text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {\text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3855

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

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 5565

Int[Coth[(a_.) + (b_.)*(x_)]^(p_)*Csch[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d
*x)^m*Csch[a + b*x]*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csch[a + b*x]^3*Coth[a + b*x]^(p - 2), x] /; F
reeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 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 ^2(a+b x) \text {csch}(a+b x) \, dx &=\int x^2 \text {csch}(a+b x) \, dx+\int x^2 \text {csch}^3(a+b x) \, dx\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {1}{2} \int x^2 \text {csch}(a+b x) \, dx+\frac {\int \text {csch}(a+b x) \, dx}{b^2}-\frac {2 \int x \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {2 \int x \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {2 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {2 \int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}-\frac {2 \int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}+\frac {\int x \log \left (1-e^{a+b x}\right ) \, dx}{b}-\frac {\int x \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {\int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}+\frac {\int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {2 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {2 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {\text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac {x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^3}-\frac {x \text {csch}(a+b x)}{b^2}-\frac {x^2 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {x \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {\text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {\text {Li}_3\left (e^{a+b x}\right )}{b^3}\\ \end {align*}

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Mathematica [A]
time = 2.07, size = 224, normalized size = 1.82 \begin {gather*} -\frac {16 \tanh ^{-1}(\cosh (a+b x)+\sinh (a+b x))+8 b^2 x^2 \tanh ^{-1}(\cosh (a+b x)+\sinh (a+b x))+8 b x \text {csch}(a)+b^2 x^2 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )+8 b x \text {PolyLog}(2,-\cosh (a+b x)-\sinh (a+b x))-8 b x \text {PolyLog}(2,\cosh (a+b x)+\sinh (a+b x))-8 \text {PolyLog}(3,-\cosh (a+b x)-\sinh (a+b x))+8 \text {PolyLog}(3,\cosh (a+b x)+\sinh (a+b x))+b^2 x^2 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )-4 b x \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )-4 b x \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )}{8 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(16*ArcTanh[Cosh[a + b*x] + Sinh[a + b*x]] + 8*b^2*x^2*ArcTanh[Cosh[a + b*x] + Sinh[a + b*x]] + 8*b*x*Csc
h[a] + b^2*x^2*Csch[(a + b*x)/2]^2 + 8*b*x*PolyLog[2, -Cosh[a + b*x] - Sinh[a + b*x]] - 8*b*x*PolyLog[2, Cosh[
a + b*x] + Sinh[a + b*x]] - 8*PolyLog[3, -Cosh[a + b*x] - Sinh[a + b*x]] + 8*PolyLog[3, Cosh[a + b*x] + Sinh[a
+ b*x]] + b^2*x^2*Sech[(a + b*x)/2]^2 - 4*b*x*Csch[a/2]*Csch[(a + b*x)/2]*Sinh[(b*x)/2] - 4*b*x*Sech[a/2]*Sec
h[(a + b*x)/2]*Sinh[(b*x)/2])/b^3

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Maple [A]
time = 1.88, size = 210, normalized size = 1.71

method	result	size
risch	\(-\frac {x \,{\mathrm e}^{b x +a} \left (b x \,{\mathrm e}^{2 b x +2 a}+b x +2 \,{\mathrm e}^{2 b x +2 a}-2\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {a^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{2 b}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{2 b^{3}}+\frac {x \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{2 b}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{2 b^{3}}-\frac {x \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(210\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 197, normalized size = 1.60 \begin {gather*} -\frac {{\left (b x^{2} e^{\left (3 \, a\right )} + 2 \, x e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b x^{2} e^{a} - 2 \, x e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + 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 )})}{2 \, 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 )})}{2 \, 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*cosh(b*x+a)^2*csch(b*x+a)^3,x, algorithm="maxima")

[Out]

-((b*x^2*e^(3*a) + 2*x*e^(3*a))*e^(3*b*x) + (b*x^2*e^a - 2*x*e^a)*e^(b*x))/(b^2*e^(4*b*x + 4*a) - 2*b^2*e^(2*b
*x + 2*a) + b^2) - 1/2*(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 + 1/2*(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 - 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. 1311 vs. \(2 (114) = 228\).
time = 0.49, size = 1311, normalized size = 10.66 \begin {gather*} -\frac {2 \, {\left (b^{2} x^{2} + 2 \, b x\right )} \cosh \left (b x + a\right )^{3} + 6 \, {\left (b^{2} x^{2} + 2 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} x^{2} + 2 \, b x\right )} \sinh \left (b x + a\right )^{3} + 2 \, {\left (b^{2} x^{2} - 2 \, b x\right )} \cosh \left (b x + a\right ) - 2 \, {\left (b x \cosh \left (b x + a\right )^{4} + 4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b x \sinh \left (b x + a\right )^{4} - 2 \, b x \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} + b x + 4 \, {\left (b x \cosh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\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 )^{4} + 4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b x \sinh \left (b x + a\right )^{4} - 2 \, b x \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} + b x + 4 \, {\left (b x \cosh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left ({\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} + 2\right )} \sinh \left (b x + a\right )^{4} + b^{2} x^{2} - 2 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - 3 \, {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left ({\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{3} - {\left (b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 2\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left ({\left (a^{2} + 2\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (a^{2} + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (a^{2} + 2\right )} \sinh \left (b x + a\right )^{4} - 2 \, {\left (a^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2\right )} \cosh \left (b x + a\right )^{2} - a^{2} - 2\right )} \sinh \left (b x + a\right )^{2} + a^{2} + 4 \, {\left ({\left (a^{2} + 2\right )} \cosh \left (b x + a\right )^{3} - {\left (a^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 2\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{4} + b^{2} x^{2} - 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - 3 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{2} - a^{2} + 4 \, {\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{3} - {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 2 \, {\left (b^{2} x^{2} + 3 \, {\left (b^{2} x^{2} + 2 \, b x\right )} \cosh \left (b x + a\right )^{2} - 2 \, b x\right )} \sinh \left (b x + a\right )}{2 \, {\left (b^{3} \cosh \left (b x + a\right )^{4} + 4 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{3} \sinh \left (b x + a\right )^{4} - 2 \, b^{3} \cosh \left (b x + a\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (b x + a\right )^{2} - b^{3}\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b^{3} \cosh \left (b x + a\right )^{3} - b^{3} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(b^2*x^2 + 2*b*x)*cosh(b*x + a)^3 + 6*(b^2*x^2 + 2*b*x)*cosh(b*x + a)*sinh(b*x + a)^2 + 2*(b^2*x^2 + 2
*b*x)*sinh(b*x + a)^3 + 2*(b^2*x^2 - 2*b*x)*cosh(b*x + a) - 2*(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)) + 2*(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))*dil
og(-cosh(b*x + a) - sinh(b*x + a)) + ((b^2*x^2 + 2)*cosh(b*x + a)^4 + 4*(b^2*x^2 + 2)*cosh(b*x + a)*sinh(b*x +
a)^3 + (b^2*x^2 + 2)*sinh(b*x + a)^4 + b^2*x^2 - 2*(b^2*x^2 + 2)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 +
2)*cosh(b*x + a)^2 + 2)*sinh(b*x + a)^2 + 4*((b^2*x^2 + 2)*cosh(b*x + a)^3 - (b^2*x^2 + 2)*cosh(b*x + a))*sinh
(b*x + a) + 2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - ((a^2 + 2)*cosh(b*x + a)^4 + 4*(a^2 + 2)*cosh(b*x + a)
*sinh(b*x + a)^3 + (a^2 + 2)*sinh(b*x + a)^4 - 2*(a^2 + 2)*cosh(b*x + a)^2 + 2*(3*(a^2 + 2)*cosh(b*x + a)^2 -
a^2 - 2)*sinh(b*x + a)^2 + a^2 + 4*((a^2 + 2)*cosh(b*x + a)^3 - (a^2 + 2)*cosh(b*x + a))*sinh(b*x + a) + 2)*lo
g(cosh(b*x + a) + sinh(b*x + a) - 1) - ((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) + 2*(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*(cos
h(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*polylog(3, cosh(b*x + a) + sinh(b*x + a)) - 2*(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)) +
2*(b^2*x^2 + 3*(b^2*x^2 + 2*b*x)*cosh(b*x + a)^2 - 2*b*x)*sinh(b*x + 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)*si
nh(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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cosh(b*x+a)**2*csch(b*x+a)**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

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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*cosh(b*x+a)^2*csch(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(x^2*cosh(b*x + a)^2*csch(b*x + a)^3, 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 )}^2}{{\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^2*cosh(a + b*x)^2)/sinh(a + b*x)^3,x)

[Out]

int((x^2*cosh(a + b*x)^2)/sinh(a + b*x)^3, x)

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