3.6.2 \(\int x^2 \text {csch}^2(a+b x) \text {sech}^3(a+b x) \, dx\) [502]
Optimal. Leaf size=206 \[ -\frac {3 x^2 \text {ArcTan}\left (e^{a+b x}\right )}{b}+\frac {\text {ArcTan}(\sinh (a+b x))}{b^3}-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}-\frac {2 \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {3 i x \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {3 i x \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {2 \text {PolyLog}\left (2,e^{a+b x}\right )}{b^3}-\frac {3 i \text {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac {3 i \text {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {x \text {sech}(a+b x)}{b^2}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b} \]
[Out]
-3*x^2*arctan(exp(b*x+a))/b+arctan(sinh(b*x+a))/b^3-4*x*arctanh(exp(b*x+a))/b^2-3/2*x^2*csch(b*x+a)/b-2*polylo
g(2,-exp(b*x+a))/b^3+3*I*x*polylog(2,-I*exp(b*x+a))/b^2-3*I*x*polylog(2,I*exp(b*x+a))/b^2+2*polylog(2,exp(b*x+
a))/b^3-3*I*polylog(3,-I*exp(b*x+a))/b^3+3*I*polylog(3,I*exp(b*x+a))/b^3-x*sech(b*x+a)/b^2+1/2*x^2*csch(b*x+a)
*sech(b*x+a)^2/b
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Rubi [A]
time = 0.32, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 29, number of rules used = 18, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {2701, 294,
327, 213, 5570, 14, 5313, 12, 4265, 2611, 2320, 6724, 4267, 2317, 2438, 2702, 6406, 3855}
\begin {gather*} \frac {\text {ArcTan}(\sinh (a+b x))}{b^3}-\frac {3 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 {3 i \text {Li}_3\left (-i e^{a+b x}\right )}{b^3}+\frac {3 i \text {Li}_3\left (i e^{a+b x}\right )}{b^3}+\frac {3 i x \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac {3 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 \text {sech}(a+b x)}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^2*Csch[a + b*x]^2*Sech[a + b*x]^3,x]
[Out]
(-3*x^2*ArcTan[E^(a + b*x)])/b + ArcTan[Sinh[a + b*x]]/b^3 - (4*x*ArcTanh[E^(a + b*x)])/b^2 - (3*x^2*Csch[a +
b*x])/(2*b) - (2*PolyLog[2, -E^(a + b*x)])/b^3 + ((3*I)*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - ((3*I)*x*PolyLog
[2, I*E^(a + b*x)])/b^2 + (2*PolyLog[2, E^(a + b*x)])/b^3 - ((3*I)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + ((3*I)*
PolyLog[3, I*E^(a + b*x)])/b^3 - (x*Sech[a + b*x])/b^2 + (x^2*Csch[a + b*x]*Sech[a + b*x]^2)/(2*b)
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 294
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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]
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 2702
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Rule 3855
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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 6406
Int[ArcTanh[u_], x_Symbol] :> Simp[x*ArcTanh[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/(1 - u^2)), x], x] /; I
nverseFunctionFreeQ[u, x]
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}^3(a+b x) \, dx &=-\frac {3 x^2 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}-2 \int x \left (-\frac {3 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac {3 \text {csch}(a+b x)}{2 b}+\frac {\text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}\right ) \, dx\\ &=-\frac {3 x^2 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}-2 \int \left (-\frac {3 x \left (\tan ^{-1}(\sinh (a+b x))+\text {csch}(a+b x)\right )}{2 b}+\frac {x \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}\right ) \, dx\\ &=-\frac {3 x^2 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}-\frac {\int x \text {csch}(a+b x) \text {sech}^2(a+b x) \, dx}{b}+\frac {3 \int x \left (\tan ^{-1}(\sinh (a+b x))+\text {csch}(a+b x)\right ) \, dx}{b}\\ &=-\frac {3 x^2 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac {x \tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}-\frac {x \text {sech}(a+b x)}{b^2}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}+\frac {\int \left (-\frac {\tanh ^{-1}(\cosh (a+b x))}{b}+\frac {\text {sech}(a+b x)}{b}\right ) \, dx}{b}+\frac {3 \int \left (x \tan ^{-1}(\sinh (a+b x))+x \text {csch}(a+b x)\right ) \, dx}{b}\\ &=-\frac {3 x^2 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac {x \tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}-\frac {x \text {sech}(a+b x)}{b^2}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}-\frac {\int \tanh ^{-1}(\cosh (a+b x)) \, dx}{b^2}+\frac {\int \text {sech}(a+b x) \, dx}{b^2}+\frac {3 \int x \tan ^{-1}(\sinh (a+b x)) \, dx}{b}+\frac {3 \int x \text {csch}(a+b x) \, dx}{b}\\ &=\frac {\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}-\frac {x \text {sech}(a+b x)}{b^2}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}-\frac {\int b x \text {csch}(a+b x) \, dx}{b^2}-\frac {3 \int \log \left (1-e^{a+b x}\right ) \, dx}{b^2}+\frac {3 \int \log \left (1+e^{a+b x}\right ) \, dx}{b^2}-\frac {3 \int b x^2 \text {sech}(a+b x) \, dx}{2 b}\\ &=\frac {\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}-\frac {x \text {sech}(a+b x)}{b^2}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}-\frac {3}{2} \int x^2 \text {sech}(a+b x) \, dx-\frac {3 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {3 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {\int x \text {csch}(a+b x) \, dx}{b}\\ &=-\frac {3 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac {\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}-\frac {3 \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {3 \text {Li}_2\left (e^{a+b x}\right )}{b^3}-\frac {x \text {sech}(a+b x)}{b^2}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}+\frac {\int \log \left (1-e^{a+b x}\right ) \, dx}{b^2}-\frac {\int \log \left (1+e^{a+b x}\right ) \, dx}{b^2}+\frac {(3 i) \int x \log \left (1-i e^{a+b x}\right ) \, dx}{b}-\frac {(3 i) \int x \log \left (1+i e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {3 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac {\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}-\frac {3 \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {3 i x \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac {3 i x \text {Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac {3 \text {Li}_2\left (e^{a+b x}\right )}{b^3}-\frac {x \text {sech}(a+b x)}{b^2}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}+\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {(3 i) \int \text {Li}_2\left (-i e^{a+b x}\right ) \, dx}{b^2}+\frac {(3 i) \int \text {Li}_2\left (i e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {3 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac {\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}-\frac {2 \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {3 i x \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac {3 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 {x \text {sech}(a+b x)}{b^2}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}-\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac {3 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b}+\frac {\tan ^{-1}(\sinh (a+b x))}{b^3}-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b}-\frac {2 \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {3 i x \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac {3 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 {3 i \text {Li}_3\left (-i e^{a+b x}\right )}{b^3}+\frac {3 i \text {Li}_3\left (i e^{a+b x}\right )}{b^3}-\frac {x \text {sech}(a+b x)}{b^2}+\frac {x^2 \text {csch}(a+b x) \text {sech}^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 7.12, size = 397, normalized size = 1.93 \begin {gather*} -\frac {x^2 \text {csch}(a)}{b}+\frac {2 \left (-a \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )-i \left ((i a+i b x) \left (\log \left (1-e^{i (i a+i b x)}\right )-\log \left (1+e^{i (i a+i b x)}\right )\right )+i \left (\text {PolyLog}\left (2,-e^{i (i a+i b x)}\right )-\text {PolyLog}\left (2,e^{i (i a+i b x)}\right )\right )\right )\right )}{b^3}-\frac {i \left (4 i \text {ArcTan}\left (e^{a+b x}\right )+3 b^2 x^2 \log \left (1-i e^{a+b x}\right )-3 b^2 x^2 \log \left (1+i e^{a+b x}\right )-6 b x \text {PolyLog}\left (2,-i e^{a+b x}\right )+6 b x \text {PolyLog}\left (2,i e^{a+b x}\right )+6 \text {PolyLog}\left (3,-i e^{a+b x}\right )-6 \text {PolyLog}\left (3,i e^{a+b x}\right )\right )}{2 b^3}-\frac {x \text {sech}(a) \text {sech}(a+b x) (2 \cosh (a)+b x \sinh (a))}{2 b^2}+\frac {x^2 \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {a}{2}+\frac {b x}{2}\right ) \sinh \left (\frac {b x}{2}\right )}{2 b}+\frac {x^2 \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {a}{2}+\frac {b x}{2}\right ) \sinh \left (\frac {b x}{2}\right )}{2 b}-\frac {x^2 \text {sech}(a) \text {sech}^2(a+b x) \sinh (b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^2*Csch[a + b*x]^2*Sech[a + b*x]^3,x]
[Out]
-((x^2*Csch[a])/b) + (2*(-(a*Log[Tanh[(a + b*x)/2]]) - I*((I*a + I*b*x)*(Log[1 - E^(I*(I*a + I*b*x))] - Log[1
+ E^(I*(I*a + I*b*x))]) + I*(PolyLog[2, -E^(I*(I*a + I*b*x))] - PolyLog[2, E^(I*(I*a + I*b*x))]))))/b^3 - ((I/
2)*((4*I)*ArcTan[E^(a + b*x)] + 3*b^2*x^2*Log[1 - I*E^(a + b*x)] - 3*b^2*x^2*Log[1 + I*E^(a + b*x)] - 6*b*x*Po
lyLog[2, (-I)*E^(a + b*x)] + 6*b*x*PolyLog[2, I*E^(a + b*x)] + 6*PolyLog[3, (-I)*E^(a + b*x)] - 6*PolyLog[3, I
*E^(a + b*x)]))/b^3 - (x*Sech[a]*Sech[a + b*x]*(2*Cosh[a] + b*x*Sinh[a]))/(2*b^2) + (x^2*Csch[a/2]*Csch[a/2 +
(b*x)/2]*Sinh[(b*x)/2])/(2*b) + (x^2*Sech[a/2]*Sech[a/2 + (b*x)/2]*Sinh[(b*x)/2])/(2*b) - (x^2*Sech[a]*Sech[a
+ b*x]^2*Sinh[b*x])/(2*b)
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \mathrm {csch}\left (b x +a \right )^{2} \mathrm {sech}\left (b x +a \right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*csch(b*x+a)^2*sech(b*x+a)^3,x)
[Out]
int(x^2*csch(b*x+a)^2*sech(b*x+a)^3,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)^3,x, algorithm="maxima")
[Out]
-96*b^2*integrate(1/32*x^2*e^(b*x + a)/(b^2*e^(2*b*x + 2*a) + b^2), x) - (2*b*x^2*e^(3*b*x + 3*a) + (3*b*x^2*e
^(5*a) + 2*x*e^(5*a))*e^(5*b*x) + (3*b*x^2*e^a - 2*x*e^a)*e^(b*x))/(b^2*e^(6*b*x + 6*a) + b^2*e^(4*b*x + 4*a)
- b^2*e^(2*b*x + 2*a) - b^2) - 2*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^3 + 2*(b*x*log(-e^(b*x + a
) + 1) + dilog(e^(b*x + a)))/b^3 + 2*arctan(e^(b*x + a))/b^3
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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. 3825 vs. \(2 (174) = 348\).
time = 0.45, size = 3825, normalized size = 18.57 \begin {gather*} \text {Too large to display} \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)^3,x, algorithm="fricas")
[Out]
-1/2*(4*b^2*x^2*cosh(b*x + a)^3 + 2*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)^5 + 10*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)
*sinh(b*x + a)^4 + 2*(3*b^2*x^2 + 2*b*x)*sinh(b*x + a)^5 + 4*(b^2*x^2 + 5*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)^2)
*sinh(b*x + a)^3 + 4*(3*b^2*x^2*cosh(b*x + a) + 5*(3*b^2*x^2 + 2*b*x)*cosh(b*x + a)^3)*sinh(b*x + a)^2 + 2*(3*
b^2*x^2 - 2*b*x)*cosh(b*x + a) - 4*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + (15*
cosh(b*x + a)^2 + 1)*sinh(b*x + a)^4 + cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a)^3
+ (15*cosh(b*x + a)^4 + 6*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - cosh(b*x + a)^2 + 2*(3*cosh(b*x + a)^5 + 2*c
osh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) - 1)*dilog(cosh(b*x + a) + sinh(b*x + a)) + 6*(I*b*x*cosh(b*x +
a)^6 + 6*I*b*x*cosh(b*x + a)*sinh(b*x + a)^5 + I*b*x*sinh(b*x + a)^6 + I*b*x*cosh(b*x + a)^4 + (15*I*b*x*cosh(
b*x + a)^2 + I*b*x)*sinh(b*x + a)^4 - I*b*x*cosh(b*x + a)^2 + 4*(5*I*b*x*cosh(b*x + a)^3 + I*b*x*cosh(b*x + a)
)*sinh(b*x + a)^3 + (15*I*b*x*cosh(b*x + a)^4 + 6*I*b*x*cosh(b*x + a)^2 - I*b*x)*sinh(b*x + a)^2 - I*b*x + 2*(
3*I*b*x*cosh(b*x + a)^5 + 2*I*b*x*cosh(b*x + a)^3 - I*b*x*cosh(b*x + a))*sinh(b*x + a))*dilog(I*cosh(b*x + a)
+ I*sinh(b*x + a)) + 6*(-I*b*x*cosh(b*x + a)^6 - 6*I*b*x*cosh(b*x + a)*sinh(b*x + a)^5 - I*b*x*sinh(b*x + a)^6
- I*b*x*cosh(b*x + a)^4 + (-15*I*b*x*cosh(b*x + a)^2 - I*b*x)*sinh(b*x + a)^4 + I*b*x*cosh(b*x + a)^2 + 4*(-5
*I*b*x*cosh(b*x + a)^3 - I*b*x*cosh(b*x + a))*sinh(b*x + a)^3 + (-15*I*b*x*cosh(b*x + a)^4 - 6*I*b*x*cosh(b*x
+ a)^2 + I*b*x)*sinh(b*x + a)^2 + I*b*x + 2*(-3*I*b*x*cosh(b*x + a)^5 - 2*I*b*x*cosh(b*x + a)^3 + I*b*x*cosh(b
*x + a))*sinh(b*x + a))*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + 4*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(
b*x + a)^5 + sinh(b*x + a)^6 + (15*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^4 + cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)
^3 + cosh(b*x + a))*sinh(b*x + a)^3 + (15*cosh(b*x + a)^4 + 6*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - cosh(b*x
+ a)^2 + 2*(3*cosh(b*x + a)^5 + 2*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) - 1)*dilog(-cosh(b*x + a) - s
inh(b*x + a)) + 4*(b*x*cosh(b*x + a)^6 + 6*b*x*cosh(b*x + a)*sinh(b*x + a)^5 + b*x*sinh(b*x + a)^6 + b*x*cosh(
b*x + a)^4 + (15*b*x*cosh(b*x + a)^2 + b*x)*sinh(b*x + a)^4 - b*x*cosh(b*x + a)^2 + 4*(5*b*x*cosh(b*x + a)^3 +
b*x*cosh(b*x + a))*sinh(b*x + a)^3 + (15*b*x*cosh(b*x + a)^4 + 6*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^2 -
b*x + 2*(3*b*x*cosh(b*x + a)^5 + 2*b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))*log(cosh(b*x + a)
+ sinh(b*x + a) + 1) - ((-3*I*a^2 + 2*I)*cosh(b*x + a)^6 - 6*(3*I*a^2 - 2*I)*cosh(b*x + a)*sinh(b*x + a)^5 + (
-3*I*a^2 + 2*I)*sinh(b*x + a)^6 + (-3*I*a^2 + 2*I)*cosh(b*x + a)^4 - (15*(3*I*a^2 - 2*I)*cosh(b*x + a)^2 + 3*I
*a^2 - 2*I)*sinh(b*x + a)^4 - 4*(5*(3*I*a^2 - 2*I)*cosh(b*x + a)^3 + (3*I*a^2 - 2*I)*cosh(b*x + a))*sinh(b*x +
a)^3 + (3*I*a^2 - 2*I)*cosh(b*x + a)^2 - (15*(3*I*a^2 - 2*I)*cosh(b*x + a)^4 + 6*(3*I*a^2 - 2*I)*cosh(b*x + a
)^2 - 3*I*a^2 + 2*I)*sinh(b*x + a)^2 + 3*I*a^2 - 2*(3*(3*I*a^2 - 2*I)*cosh(b*x + a)^5 + 2*(3*I*a^2 - 2*I)*cosh
(b*x + a)^3 + (-3*I*a^2 + 2*I)*cosh(b*x + a))*sinh(b*x + a) - 2*I)*log(cosh(b*x + a) + sinh(b*x + a) + I) - ((
3*I*a^2 - 2*I)*cosh(b*x + a)^6 - 6*(-3*I*a^2 + 2*I)*cosh(b*x + a)*sinh(b*x + a)^5 + (3*I*a^2 - 2*I)*sinh(b*x +
a)^6 + (3*I*a^2 - 2*I)*cosh(b*x + a)^4 - (15*(-3*I*a^2 + 2*I)*cosh(b*x + a)^2 - 3*I*a^2 + 2*I)*sinh(b*x + a)^
4 - 4*(5*(-3*I*a^2 + 2*I)*cosh(b*x + a)^3 + (-3*I*a^2 + 2*I)*cosh(b*x + a))*sinh(b*x + a)^3 + (-3*I*a^2 + 2*I)
*cosh(b*x + a)^2 - (15*(-3*I*a^2 + 2*I)*cosh(b*x + a)^4 + 6*(-3*I*a^2 + 2*I)*cosh(b*x + a)^2 + 3*I*a^2 - 2*I)*
sinh(b*x + a)^2 - 3*I*a^2 - 2*(3*(-3*I*a^2 + 2*I)*cosh(b*x + a)^5 + 2*(-3*I*a^2 + 2*I)*cosh(b*x + a)^3 + (3*I*
a^2 - 2*I)*cosh(b*x + a))*sinh(b*x + a) + 2*I)*log(cosh(b*x + a) + sinh(b*x + a) - I) + 4*(a*cosh(b*x + a)^6 +
6*a*cosh(b*x + a)*sinh(b*x + a)^5 + a*sinh(b*x + a)^6 + a*cosh(b*x + a)^4 + (15*a*cosh(b*x + a)^2 + a)*sinh(b
*x + a)^4 + 4*(5*a*cosh(b*x + a)^3 + a*cosh(b*x + a))*sinh(b*x + a)^3 - a*cosh(b*x + a)^2 + (15*a*cosh(b*x + a
)^4 + 6*a*cosh(b*x + a)^2 - a)*sinh(b*x + a)^2 + 2*(3*a*cosh(b*x + a)^5 + 2*a*cosh(b*x + a)^3 - a*cosh(b*x + a
))*sinh(b*x + a) - a)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 3*((-I*b^2*x^2 + I*a^2)*cosh(b*x + a)^6 + 6*(-I
*b^2*x^2 + I*a^2)*cosh(b*x + a)*sinh(b*x + a)^5 + (-I*b^2*x^2 + I*a^2)*sinh(b*x + a)^6 + (-I*b^2*x^2 + I*a^2)*
cosh(b*x + a)^4 + (-I*b^2*x^2 + 15*(-I*b^2*x^2 + I*a^2)*cosh(b*x + a)^2 + I*a^2)*sinh(b*x + a)^4 + I*b^2*x^2 +
4*(5*(-I*b^2*x^2 + I*a^2)*cosh(b*x + a)^3 + (-I*b^2*x^2 + I*a^2)*cosh(b*x + a))*sinh(b*x + a)^3 + (I*b^2*x^2
- I*a^2)*cosh(b*x + a)^2 + (15*(-I*b^2*x^2 + I*a^2)*cosh(b*x + a)^4 + I*b^2*x^2 + 6*(-I*b^2*x^2 + I*a^2)*cosh(
b*x + a)^2 - I*a^2)*sinh(b*x + a)^2 - I*a^2 + 2*(3*(-I*b^2*x^2 + I*a^2)*cosh(b*x + a)^5 + 2*(-I*b^2*x^2 + I*a^
2)*cosh(b*x + a)^3 + (I*b^2*x^2 - I*a^2)*cosh(b*x + a))*sinh(b*x + a))*log(I*cosh(b*x + a) + I*sinh(b*x + a) +
1) + 3*((I*b^2*x^2 - I*a^2)*cosh(b*x + a)^6 + ...
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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}^{3}{\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)**3,x)
[Out]
Integral(x**2*csch(a + b*x)**2*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^2*csch(b*x+a)^2*sech(b*x+a)^3,x, algorithm="giac")
[Out]
integrate(x^2*csch(b*x + a)^2*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^2}{{\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^2/(cosh(a + b*x)^3*sinh(a + b*x)^2),x)
[Out]
int(x^2/(cosh(a + b*x)^3*sinh(a + b*x)^2), x)
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