3.5.39 \(\int x^3 \cosh (a+b x) \coth ^2(a+b x) \, dx\) [439]

Optimal. Leaf size=143 \[ -\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {6 \cosh (a+b x)}{b^4}-\frac {3 x^2 \cosh (a+b x)}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {6 x \text {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {6 \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac {6 \text {PolyLog}\left (3,e^{a+b x}\right )}{b^4}+\frac {6 x \sinh (a+b x)}{b^3}+\frac {x^3 \sinh (a+b x)}{b} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5558, 3377, 2718, 5527, 4267, 2611, 2320, 6724} \begin {gather*} \frac {6 \text {Li}_3\left (-e^{a+b x}\right )}{b^4}-\frac {6 \text {Li}_3\left (e^{a+b x}\right )}{b^4}-\frac {6 \cosh (a+b x)}{b^4}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {6 x \sinh (a+b x)}{b^3}-\frac {3 x^2 \cosh (a+b x)}{b^2}-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {x^3 \sinh (a+b x)}{b}-\frac {x^3 \text {csch}(a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Cosh[a + b*x]*Coth[a + b*x]^2,x]

[Out]

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

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 2718

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, 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 5527

Int[Coth[(a_.) + (b_.)*(x_)^(n_.)]^(q_.)*Csch[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[(-
x^(m - n + 1))*(Csch[a + b*x^n]^p/(b*n*p)), x] + Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Csch[a + b*x^n]^p, x]
, x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]

Rule 5558

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 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^3 \cosh (a+b x) \coth ^2(a+b x) \, dx &=\int x^3 \cosh (a+b x) \, dx+\int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx\\ &=-\frac {x^3 \text {csch}(a+b x)}{b}+\frac {x^3 \sinh (a+b x)}{b}+\frac {3 \int x^2 \text {csch}(a+b x) \, dx}{b}-\frac {3 \int x^2 \sinh (a+b x) \, dx}{b}\\ &=-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \cosh (a+b x)}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}+\frac {x^3 \sinh (a+b x)}{b}+\frac {6 \int x \cosh (a+b x) \, dx}{b^2}-\frac {6 \int x \log \left (1-e^{a+b x}\right ) \, dx}{b^2}+\frac {6 \int x \log \left (1+e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \cosh (a+b x)}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {6 x \sinh (a+b x)}{b^3}+\frac {x^3 \sinh (a+b x)}{b}+\frac {6 \int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^3}-\frac {6 \int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^3}-\frac {6 \int \sinh (a+b x) \, dx}{b^3}\\ &=-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {6 \cosh (a+b x)}{b^4}-\frac {3 x^2 \cosh (a+b x)}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {6 x \sinh (a+b x)}{b^3}+\frac {x^3 \sinh (a+b x)}{b}+\frac {6 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac {6 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {6 \cosh (a+b x)}{b^4}-\frac {3 x^2 \cosh (a+b x)}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {6 \text {Li}_3\left (-e^{a+b x}\right )}{b^4}-\frac {6 \text {Li}_3\left (e^{a+b x}\right )}{b^4}+\frac {6 x \sinh (a+b x)}{b^3}+\frac {x^3 \sinh (a+b x)}{b}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 225, normalized size = 1.57 \begin {gather*} \frac {\text {csch}\left (\frac {1}{2} (a+b x)\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \left (-6 b x-3 b^3 x^3+6 b x \cosh (2 (a+b x))+b^3 x^3 \cosh (2 (a+b x))-12 b^2 x^2 \tanh ^{-1}(\cosh (a+b x)+\sinh (a+b x)) \sinh (a+b x)-12 b x \text {PolyLog}(2,-\cosh (a+b x)-\sinh (a+b x)) \sinh (a+b x)+12 b x \text {PolyLog}(2,\cosh (a+b x)+\sinh (a+b x)) \sinh (a+b x)+12 \text {PolyLog}(3,-\cosh (a+b x)-\sinh (a+b x)) \sinh (a+b x)-12 \text {PolyLog}(3,\cosh (a+b x)+\sinh (a+b x)) \sinh (a+b x)-6 \sinh (2 (a+b x))-3 b^2 x^2 \sinh (2 (a+b x))\right )}{4 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Cosh[a + b*x]*Coth[a + b*x]^2,x]

[Out]

(Csch[(a + b*x)/2]*Sech[(a + b*x)/2]*(-6*b*x - 3*b^3*x^3 + 6*b*x*Cosh[2*(a + b*x)] + b^3*x^3*Cosh[2*(a + b*x)]
- 12*b^2*x^2*ArcTanh[Cosh[a + b*x] + Sinh[a + b*x]]*Sinh[a + b*x] - 12*b*x*PolyLog[2, -Cosh[a + b*x] - Sinh[a
+ b*x]]*Sinh[a + b*x] + 12*b*x*PolyLog[2, Cosh[a + b*x] + Sinh[a + b*x]]*Sinh[a + b*x] + 12*PolyLog[3, -Cosh[
a + b*x] - Sinh[a + b*x]]*Sinh[a + b*x] - 12*PolyLog[3, Cosh[a + b*x] + Sinh[a + b*x]]*Sinh[a + b*x] - 6*Sinh[
2*(a + b*x)] - 3*b^2*x^2*Sinh[2*(a + b*x)]))/(4*b^4)

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Maple [A]
time = 2.84, size = 241, normalized size = 1.69

method	result	size
risch	\(\frac {\left (b^{3} x^{3}-3 b^{2} x^{2}+6 b x -6\right ) {\mathrm e}^{b x +a}}{2 b^{4}}-\frac {\left (b^{3} x^{3}+3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x -a}}{2 b^{4}}-\frac {2 x^{3} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}-\frac {6 a^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{4}}+\frac {6 x \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {6 \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b^{2}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{b^{4}}-\frac {6 x \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{4}}\)	\(241\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^3*x^3-3*b^2*x^2+6*b*x-6)/b^4*exp(b*x+a)-1/2*(b^3*x^3+3*b^2*x^2+6*b*x+6)/b^4*exp(-b*x-a)-2/b*x^3*exp(b*x
+a)/(exp(2*b*x+2*a)-1)-6/b^4*a^2*arctanh(exp(b*x+a))+3/b^2*ln(1-exp(b*x+a))*x^2-3/b^4*ln(1-exp(b*x+a))*a^2+6*x
*polylog(2,exp(b*x+a))/b^3-6*polylog(3,exp(b*x+a))/b^4-3/b^2*ln(exp(b*x+a)+1)*x^2+3/b^4*ln(exp(b*x+a)+1)*a^2-6
*x*polylog(2,-exp(b*x+a))/b^3+6*polylog(3,-exp(b*x+a))/b^4

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Maxima [A]
time = 0.33, size = 216, normalized size = 1.51 \begin {gather*} \frac {{\left (b^{3} x^{3} e^{\left (4 \, a\right )} - 3 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 6 \, b x e^{\left (4 \, a\right )} - 6 \, e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} - 6 \, {\left (b^{3} x^{3} e^{\left (2 \, a\right )} + 2 \, b x e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x\right )}}{2 \, {\left (b^{4} e^{\left (2 \, b x + 3 \, a\right )} - b^{4} e^{a}\right )}} - \frac {3 \, {\left (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 )})\right )}}{b^{4}} + \frac {3 \, {\left (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 )})\right )}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1055 vs. \(2 (136) = 272\).
time = 0.38, size = 1055, normalized size = 7.38 \begin {gather*} \frac {b^{3} x^{3} + {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \sinh \left (b x + a\right )^{4} + 3 \, b^{2} x^{2} - 6 \, {\left (b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{2} - 6 \, {\left (b^{3} x^{3} - {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right )^{2} + 2 \, b x\right )} \sinh \left (b x + a\right )^{2} + 6 \, b x + 12 \, {\left (b x \cosh \left (b x + a\right )^{3} + 3 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b x \sinh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right ) + {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\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 ) - 12 \, {\left (b x \cosh \left (b x + a\right )^{3} + 3 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b x \sinh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right ) + {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\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 ) - 6 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right )^{3} + 3 \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{2} x^{2} \sinh \left (b x + a\right )^{3} - b^{2} x^{2} \cosh \left (b x + a\right ) + {\left (3 \, b^{2} x^{2} \cosh \left (b x + a\right )^{2} - b^{2} x^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 6 \, {\left (a^{2} \cosh \left (b x + a\right )^{3} + 3 \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + a^{2} \sinh \left (b x + a\right )^{3} - a^{2} \cosh \left (b x + a\right ) + {\left (3 \, a^{2} \cosh \left (b x + a\right )^{2} - a^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 6 \, {\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{3} - {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) - {\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 )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 12 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 12 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 4 \, {\left ({\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right )^{3} - 3 \, {\left (b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 6}{2 \, {\left (b^{4} \cosh \left (b x + a\right )^{3} + 3 \, b^{4} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{4} \sinh \left (b x + a\right )^{3} - b^{4} \cosh \left (b x + a\right ) + {\left (3 \, b^{4} \cosh \left (b x + a\right )^{2} - b^{4}\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^3*x^3 + (b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*x + a)^4 + 4*(b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*
x + a)*sinh(b*x + a)^3 + (b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*sinh(b*x + a)^4 + 3*b^2*x^2 - 6*(b^3*x^3 + 2*b*x)*c
osh(b*x + a)^2 - 6*(b^3*x^3 - (b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*x + a)^2 + 2*b*x)*sinh(b*x + a)^2 + 6*b
*x + 12*(b*x*cosh(b*x + a)^3 + 3*b*x*cosh(b*x + a)*sinh(b*x + a)^2 + b*x*sinh(b*x + a)^3 - b*x*cosh(b*x + a) +
(3*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a))*dilog(cosh(b*x + a) + sinh(b*x + a)) - 12*(b*x*cosh(b*x + a)^3 +
3*b*x*cosh(b*x + a)*sinh(b*x + a)^2 + b*x*sinh(b*x + a)^3 - b*x*cosh(b*x + a) + (3*b*x*cosh(b*x + a)^2 - b*x)
*sinh(b*x + a))*dilog(-cosh(b*x + a) - sinh(b*x + a)) - 6*(b^2*x^2*cosh(b*x + a)^3 + 3*b^2*x^2*cosh(b*x + a)*s
inh(b*x + a)^2 + b^2*x^2*sinh(b*x + a)^3 - b^2*x^2*cosh(b*x + a) + (3*b^2*x^2*cosh(b*x + a)^2 - b^2*x^2)*sinh(
b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 6*(a^2*cosh(b*x + a)^3 + 3*a^2*cosh(b*x + a)*sinh(b*x + a)^
2 + a^2*sinh(b*x + a)^3 - a^2*cosh(b*x + a) + (3*a^2*cosh(b*x + a)^2 - a^2)*sinh(b*x + a))*log(cosh(b*x + a) +
sinh(b*x + a) - 1) + 6*((b^2*x^2 - a^2)*cosh(b*x + a)^3 + 3*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a)^2 + (
b^2*x^2 - a^2)*sinh(b*x + a)^3 - (b^2*x^2 - a^2)*cosh(b*x + a) - (b^2*x^2 - 3*(b^2*x^2 - a^2)*cosh(b*x + a)^2
- a^2)*sinh(b*x + a))*log(-cosh(b*x + a) - sinh(b*x + a) + 1) - 12*(cosh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x
+ a)^2 + sinh(b*x + a)^3 + (3*cosh(b*x + a)^2 - 1)*sinh(b*x + a) - cosh(b*x + a))*polylog(3, cosh(b*x + a) +
sinh(b*x + a)) + 12*(cosh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b*x + a)^3 + (3*cosh(b*x + a)^2
- 1)*sinh(b*x + a) - cosh(b*x + a))*polylog(3, -cosh(b*x + a) - sinh(b*x + a)) + 4*((b^3*x^3 - 3*b^2*x^2 + 6*b
*x - 6)*cosh(b*x + a)^3 - 3*(b^3*x^3 + 2*b*x)*cosh(b*x + a))*sinh(b*x + a) + 6)/(b^4*cosh(b*x + a)^3 + 3*b^4*c
osh(b*x + a)*sinh(b*x + a)^2 + b^4*sinh(b*x + a)^3 - b^4*cosh(b*x + a) + (3*b^4*cosh(b*x + a)^2 - b^4)*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**3*cosh(b*x+a)**3*csch(b*x+a)**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4369 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^3*cosh(b*x+a)^3*csch(b*x+a)^2,x, algorithm="giac")

[Out]

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

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

[Out]

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

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