3.4.77 \(\int x^3 \sinh ^2(a+b x) \tanh (a+b x) \, dx\) [377]

Optimal. Leaf size=185 \[ \frac {3 x}{8 b^3}+\frac {x^3}{4 b}+\frac {x^4}{4}-\frac {x^3 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {3 x^2 \text {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{2 b^2}+\frac {3 x \text {PolyLog}\left (3,-e^{2 (a+b x)}\right )}{2 b^3}-\frac {3 \text {PolyLog}\left (4,-e^{2 (a+b x)}\right )}{4 b^4}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {3 x \sinh ^2(a+b x)}{4 b^3}+\frac {x^3 \sinh ^2(a+b x)}{2 b} \]

[Out]

3/8*x/b^3+1/4*x^3/b+1/4*x^4-x^3*ln(1+exp(2*b*x+2*a))/b-3/2*x^2*polylog(2,-exp(2*b*x+2*a))/b^2+3/2*x*polylog(3,
-exp(2*b*x+2*a))/b^3-3/4*polylog(4,-exp(2*b*x+2*a))/b^4-3/8*cosh(b*x+a)*sinh(b*x+a)/b^4-3/4*x^2*cosh(b*x+a)*si
nh(b*x+a)/b^2+3/4*x*sinh(b*x+a)^2/b^3+1/2*x^3*sinh(b*x+a)^2/b

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5557, 5480, 3392, 30, 2715, 8, 3799, 2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3 \text {Li}_4\left (-e^{2 (a+b x)}\right )}{4 b^4}-\frac {3 \sinh (a+b x) \cosh (a+b x)}{8 b^4}+\frac {3 x \text {Li}_3\left (-e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 x \sinh ^2(a+b x)}{4 b^3}-\frac {3 x^2 \text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x^2 \sinh (a+b x) \cosh (a+b x)}{4 b^2}-\frac {x^3 \log \left (e^{2 (a+b x)}+1\right )}{b}+\frac {x^3 \sinh ^2(a+b x)}{2 b}+\frac {3 x}{8 b^3}+\frac {x^3}{4 b}+\frac {x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Sinh[a + b*x]^2*Tanh[a + b*x],x]

[Out]

(3*x)/(8*b^3) + x^3/(4*b) + x^4/4 - (x^3*Log[1 + E^(2*(a + b*x))])/b - (3*x^2*PolyLog[2, -E^(2*(a + b*x))])/(2
*b^2) + (3*x*PolyLog[3, -E^(2*(a + b*x))])/(2*b^3) - (3*PolyLog[4, -E^(2*(a + b*x))])/(4*b^4) - (3*Cosh[a + b*
x]*Sinh[a + b*x])/(8*b^4) - (3*x^2*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (3*x*Sinh[a + b*x]^2)/(4*b^3) + (x^3
*Sinh[a + b*x]^2)/(2*b)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3392

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

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5480

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

Rule 5557

Int[((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Int
[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[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]

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

________________________________________________________________________________________

Mathematica [A]
time = 2.09, size = 260, normalized size = 1.41 \begin {gather*} \frac {1}{16} \left (-\frac {4 \left (-2 b^4 e^{2 a} x^4+4 b^3 x^3 \log \left (1+e^{2 (a+b x)}\right )+4 b^3 e^{2 a} x^3 \log \left (1+e^{2 (a+b x)}\right )+6 b^2 \left (1+e^{2 a}\right ) x^2 \text {PolyLog}\left (2,-e^{2 (a+b x)}\right )-6 b \left (1+e^{2 a}\right ) x \text {PolyLog}\left (3,-e^{2 (a+b x)}\right )+3 \text {PolyLog}\left (4,-e^{2 (a+b x)}\right )+3 e^{2 a} \text {PolyLog}\left (4,-e^{2 (a+b x)}\right )\right )}{b^4 \left (1+e^{2 a}\right )}+\frac {\cosh (2 b x) \left (2 b x \left (3+2 b^2 x^2\right ) \cosh (2 a)-3 \left (1+2 b^2 x^2\right ) \sinh (2 a)\right )}{b^4}+\frac {\left (-3 \left (1+2 b^2 x^2\right ) \cosh (2 a)+2 b x \left (3+2 b^2 x^2\right ) \sinh (2 a)\right ) \sinh (2 b x)}{b^4}-4 x^4 \tanh (a)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sinh[a + b*x]^2*Tanh[a + b*x],x]

[Out]

((-4*(-2*b^4*E^(2*a)*x^4 + 4*b^3*x^3*Log[1 + E^(2*(a + b*x))] + 4*b^3*E^(2*a)*x^3*Log[1 + E^(2*(a + b*x))] + 6
*b^2*(1 + E^(2*a))*x^2*PolyLog[2, -E^(2*(a + b*x))] - 6*b*(1 + E^(2*a))*x*PolyLog[3, -E^(2*(a + b*x))] + 3*Pol
yLog[4, -E^(2*(a + b*x))] + 3*E^(2*a)*PolyLog[4, -E^(2*(a + b*x))]))/(b^4*(1 + E^(2*a))) + (Cosh[2*b*x]*(2*b*x
*(3 + 2*b^2*x^2)*Cosh[2*a] - 3*(1 + 2*b^2*x^2)*Sinh[2*a]))/b^4 + ((-3*(1 + 2*b^2*x^2)*Cosh[2*a] + 2*b*x*(3 + 2
*b^2*x^2)*Sinh[2*a])*Sinh[2*b*x])/b^4 - 4*x^4*Tanh[a])/16

________________________________________________________________________________________

Maple [A]
time = 2.43, size = 189, normalized size = 1.02

method	result	size
risch	\(\frac {x^{4}}{4}+\frac {\left (4 b^{3} x^{3}-6 b^{2} x^{2}+6 b x -3\right ) {\mathrm e}^{2 b x +2 a}}{32 b^{4}}+\frac {\left (4 b^{3} x^{3}+6 b^{2} x^{2}+6 b x +3\right ) {\mathrm e}^{-2 b x -2 a}}{32 b^{4}}+\frac {2 a^{3} x}{b^{3}}+\frac {3 a^{4}}{2 b^{4}}-\frac {x^{3} \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}-\frac {3 x^{2} \polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{2}}+\frac {3 x \polylog \left (3, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{3}}-\frac {3 \polylog \left (4, -{\mathrm e}^{2 b x +2 a}\right )}{4 b^{4}}-\frac {2 a^{3} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4+1/32*(4*b^3*x^3-6*b^2*x^2+6*b*x-3)/b^4*exp(2*b*x+2*a)+1/32*(4*b^3*x^3+6*b^2*x^2+6*b*x+3)/b^4*exp(-2*b*
x-2*a)+2/b^3*a^3*x+3/2/b^4*a^4-x^3*ln(exp(2*b*x+2*a)+1)/b-3/2*x^2*polylog(2,-exp(2*b*x+2*a))/b^2+3/2*x*polylog
(3,-exp(2*b*x+2*a))/b^3-3/4*polylog(4,-exp(2*b*x+2*a))/b^4-2/b^4*a^3*ln(exp(b*x+a))

________________________________________________________________________________________

Maxima [A]
time = 0.32, size = 181, normalized size = 0.98 \begin {gather*} \frac {1}{2} \, x^{4} - \frac {{\left (8 \, b^{4} x^{4} e^{\left (2 \, a\right )} - {\left (4 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 6 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (2 \, b x\right )} - {\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-2 \, a\right )}}{32 \, b^{4}} - \frac {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 )})}{3 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^4 - 1/32*(8*b^4*x^4*e^(2*a) - (4*b^3*x^3*e^(4*a) - 6*b^2*x^2*e^(4*a) + 6*b*x*e^(4*a) - 3*e^(4*a))*e^(2*b
*x) - (4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(-2*b*x))*e^(-2*a)/b^4 - 1/3*(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

________________________________________________________________________________________

Fricas [C] Result contains complex when optimal does not.
time = 0.40, size = 966, normalized size = 5.22 \begin {gather*} \frac {4 \, b^{3} x^{3} + {\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} \sinh \left (b x + a\right )^{4} + 6 \, b^{2} x^{2} + 8 \, {\left (b^{4} x^{4} - 2 \, a^{4}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (4 \, b^{4} x^{4} - 8 \, a^{4} + 3 \, {\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} + 6 \, b x - 96 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )^{2}\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 96 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )^{2}\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 32 \, {\left (a^{3} \cosh \left (b x + a\right )^{2} + 2 \, a^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a^{3} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + 32 \, {\left (a^{3} \cosh \left (b x + a\right )^{2} + 2 \, a^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a^{3} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - 32 \, {\left ({\left (b^{3} x^{3} + a^{3}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{3} x^{3} + a^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{3} x^{3} + a^{3}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - 32 \, {\left ({\left (b^{3} x^{3} + a^{3}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{3} x^{3} + a^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{3} x^{3} + a^{3}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) - 192 \, {\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}\right )} {\rm polylog}\left (4, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 192 \, {\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}\right )} {\rm polylog}\left (4, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 192 \, {\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}\right )} {\rm polylog}\left (3, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 192 \, {\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}\right )} {\rm polylog}\left (3, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 4 \, {\left ({\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} \cosh \left (b x + a\right )^{3} + 4 \, {\left (b^{4} x^{4} - 2 \, a^{4}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3}{32 \, {\left (b^{4} \cosh \left (b x + a\right )^{2} + 2 \, b^{4} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{4} \sinh \left (b x + a\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(4*b^3*x^3 + (4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*cosh(b*x + a)^4 + 4*(4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*
cosh(b*x + a)*sinh(b*x + a)^3 + (4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*sinh(b*x + a)^4 + 6*b^2*x^2 + 8*(b^4*x^4 -
2*a^4)*cosh(b*x + a)^2 + 2*(4*b^4*x^4 - 8*a^4 + 3*(4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*cosh(b*x + a)^2)*sinh(b
*x + a)^2 + 6*b*x - 96*(b^2*x^2*cosh(b*x + a)^2 + 2*b^2*x^2*cosh(b*x + a)*sinh(b*x + a) + b^2*x^2*sinh(b*x + a
)^2)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 96*(b^2*x^2*cosh(b*x + a)^2 + 2*b^2*x^2*cosh(b*x + a)*sinh(b*x
+ a) + b^2*x^2*sinh(b*x + a)^2)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + 32*(a^3*cosh(b*x + a)^2 + 2*a^3*c
osh(b*x + a)*sinh(b*x + a) + a^3*sinh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) + I) + 32*(a^3*cosh(b*x +
a)^2 + 2*a^3*cosh(b*x + a)*sinh(b*x + a) + a^3*sinh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) - I) - 32*((
b^3*x^3 + a^3)*cosh(b*x + a)^2 + 2*(b^3*x^3 + a^3)*cosh(b*x + a)*sinh(b*x + a) + (b^3*x^3 + a^3)*sinh(b*x + a)
^2)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) - 32*((b^3*x^3 + a^3)*cosh(b*x + a)^2 + 2*(b^3*x^3 + a^3)*cosh(
b*x + a)*sinh(b*x + a) + (b^3*x^3 + a^3)*sinh(b*x + a)^2)*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) - 192*(c
osh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2)*polylog(4, I*cosh(b*x + a) + I*sinh(b*x + a)
) - 192*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2)*polylog(4, -I*cosh(b*x + a) - I*si
nh(b*x + a)) + 192*(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)*polylog(3,
I*cosh(b*x + a) + I*sinh(b*x + a)) + 192*(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)*polylog(3, -I*cosh(b*x + a) - I*sinh(b*x + a)) + 4*((4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*cosh(b*x +
a)^3 + 4*(b^4*x^4 - 2*a^4)*cosh(b*x + a))*sinh(b*x + a) + 3)/(b^4*cosh(b*x + a)^2 + 2*b^4*cosh(b*x + a)*sinh(b
*x + a) + b^4*sinh(b*x + a)^2)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \sinh ^{3}{\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**3*sech(b*x+a)*sinh(b*x+a)**3,x)

[Out]

Integral(x**3*sinh(a + b*x)**3*sech(a + b*x), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________