3.6.23 \(\int x^2 \text {csch}^3(a+b x) \text {sech}^3(a+b x) \, dx\) [523]
Optimal. Leaf size=149 \[ \frac {4 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (2 a+2 b x))}{b^3}-\frac {2 x \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}+\frac {2 x \text {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{b^2}-\frac {2 x \text {PolyLog}\left (2,e^{2 a+2 b x}\right )}{b^2}-\frac {\text {PolyLog}\left (3,-e^{2 a+2 b x}\right )}{b^3}+\frac {\text {PolyLog}\left (3,e^{2 a+2 b x}\right )}{b^3} \]
[Out]
4*x^2*arctanh(exp(2*b*x+2*a))/b-arctanh(cosh(2*b*x+2*a))/b^3-2*x*csch(2*b*x+2*a)/b^2-2*x^2*coth(2*b*x+2*a)*csc
h(2*b*x+2*a)/b+2*x*polylog(2,-exp(2*b*x+2*a))/b^2-2*x*polylog(2,exp(2*b*x+2*a))/b^2-polylog(3,-exp(2*b*x+2*a))
/b^3+polylog(3,exp(2*b*x+2*a))/b^3
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Rubi [A]
time = 0.15, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5569, 4271,
3855, 4267, 2611, 2320, 6724} \begin {gather*} -\frac {\text {Li}_3\left (-e^{2 a+2 b x}\right )}{b^3}+\frac {\text {Li}_3\left (e^{2 a+2 b x}\right )}{b^3}-\frac {\tanh ^{-1}(\cosh (2 a+2 b x))}{b^3}+\frac {2 x \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}-\frac {2 x \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}-\frac {2 x \text {csch}(2 a+2 b x)}{b^2}+\frac {4 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {2 x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^2*Csch[a + b*x]^3*Sech[a + b*x]^3,x]
[Out]
(4*x^2*ArcTanh[E^(2*a + 2*b*x)])/b - ArcTanh[Cosh[2*a + 2*b*x]]/b^3 - (2*x*Csch[2*a + 2*b*x])/b^2 - (2*x^2*Cot
h[2*a + 2*b*x]*Csch[2*a + 2*b*x])/b + (2*x*PolyLog[2, -E^(2*a + 2*b*x)])/b^2 - (2*x*PolyLog[2, E^(2*a + 2*b*x)
])/b^2 - PolyLog[3, -E^(2*a + 2*b*x)]/b^3 + PolyLog[3, E^(2*a + 2*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 5569
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
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}^3(a+b x) \text {sech}^3(a+b x) \, dx &=8 \int x^2 \text {csch}^3(2 a+2 b x) \, dx\\ &=-\frac {2 x \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}-4 \int x^2 \text {csch}(2 a+2 b x) \, dx+\frac {2 \int \text {csch}(2 a+2 b x) \, dx}{b^2}\\ &=\frac {4 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (2 a+2 b x))}{b^3}-\frac {2 x \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}+\frac {4 \int x \log \left (1-e^{2 a+2 b x}\right ) \, dx}{b}-\frac {4 \int x \log \left (1+e^{2 a+2 b x}\right ) \, dx}{b}\\ &=\frac {4 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (2 a+2 b x))}{b^3}-\frac {2 x \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}+\frac {2 x \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}-\frac {2 x \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}-\frac {2 \int \text {Li}_2\left (-e^{2 a+2 b x}\right ) \, dx}{b^2}+\frac {2 \int \text {Li}_2\left (e^{2 a+2 b x}\right ) \, dx}{b^2}\\ &=\frac {4 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (2 a+2 b x))}{b^3}-\frac {2 x \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}+\frac {2 x \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}-\frac {2 x \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}-\frac {\text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{b^3}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{b^3}\\ &=\frac {4 x^2 \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\tanh ^{-1}(\cosh (2 a+2 b x))}{b^3}-\frac {2 x \text {csch}(2 a+2 b x)}{b^2}-\frac {2 x^2 \coth (2 a+2 b x) \text {csch}(2 a+2 b x)}{b}+\frac {2 x \text {Li}_2\left (-e^{2 a+2 b x}\right )}{b^2}-\frac {2 x \text {Li}_2\left (e^{2 a+2 b x}\right )}{b^2}-\frac {\text {Li}_3\left (-e^{2 a+2 b x}\right )}{b^3}+\frac {\text {Li}_3\left (e^{2 a+2 b x}\right )}{b^3}\\ \end {align*}
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Mathematica [A]
time = 3.88, size = 192, normalized size = 1.29 \begin {gather*} -\frac {4 \tanh ^{-1}\left (e^{2 (a+b x)}\right )+b^2 x^2 \text {csch}^2(a+b x)+4 b^2 x^2 \log \left (1-e^{2 (a+b x)}\right )-4 b^2 x^2 \log \left (1+e^{2 (a+b x)}\right )-4 b x \text {PolyLog}\left (2,-e^{2 (a+b x)}\right )+4 b x \text {PolyLog}\left (2,e^{2 (a+b x)}\right )+2 \text {PolyLog}\left (3,-e^{2 (a+b x)}\right )-2 \text {PolyLog}\left (3,e^{2 (a+b x)}\right )+2 b x \text {csch}(a) \text {sech}(a)+b^2 x^2 \text {sech}^2(a+b x)-2 b x \text {csch}(a) \text {csch}(a+b x) \sinh (b x)-2 b x \text {sech}(a) \text {sech}(a+b x) \sinh (b x)}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^2*Csch[a + b*x]^3*Sech[a + b*x]^3,x]
[Out]
-1/2*(4*ArcTanh[E^(2*(a + b*x))] + b^2*x^2*Csch[a + b*x]^2 + 4*b^2*x^2*Log[1 - E^(2*(a + b*x))] - 4*b^2*x^2*Lo
g[1 + E^(2*(a + b*x))] - 4*b*x*PolyLog[2, -E^(2*(a + b*x))] + 4*b*x*PolyLog[2, E^(2*(a + b*x))] + 2*PolyLog[3,
-E^(2*(a + b*x))] - 2*PolyLog[3, E^(2*(a + b*x))] + 2*b*x*Csch[a]*Sech[a] + b^2*x^2*Sech[a + b*x]^2 - 2*b*x*C
sch[a]*Csch[a + b*x]*Sinh[b*x] - 2*b*x*Sech[a]*Sech[a + b*x]*Sinh[b*x])/b^3
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(298\) vs.
\(2(144)=288\).
time = 2.05, size = 299, normalized size = 2.01
| | |
method |
result |
size |
| | |
risch |
\(-\frac {4 x \,{\mathrm e}^{2 b x +2 a} \left (b x \,{\mathrm e}^{4 b x +4 a}+{\mathrm e}^{4 b x +4 a}+b x -1\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {4 \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 x^{2} \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}+\frac {2 x \polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}-\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {4 x \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}-\frac {\ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b^{3}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b}-\frac {4 x \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\polylog \left (3, -{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}-\frac {2 a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}\) |
\(299\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*csch(b*x+a)^3*sech(b*x+a)^3,x,method=_RETURNVERBOSE)
[Out]
-4*x*exp(2*b*x+2*a)*(b*x*exp(4*b*x+4*a)+exp(4*b*x+4*a)+b*x-1)/b^2/(exp(2*b*x+2*a)+1)^2/(exp(2*b*x+2*a)-1)^2+4*
polylog(3,-exp(b*x+a))/b^3+4*polylog(3,exp(b*x+a))/b^3+2*x^2*ln(exp(2*b*x+2*a)+1)/b+2*x*polylog(2,-exp(2*b*x+2
*a))/b^2-2/b*ln(1-exp(b*x+a))*x^2-4*x*polylog(2,exp(b*x+a))/b^2+1/b^3*ln(exp(b*x+a)+1)+1/b^3*ln(exp(b*x+a)-1)-
1/b^3*ln(exp(2*b*x+2*a)+1)-2/b*ln(exp(b*x+a)+1)*x^2-4*x*polylog(2,-exp(b*x+a))/b^2-polylog(3,-exp(2*b*x+2*a))/
b^3+2/b^3*ln(1-exp(b*x+a))*a^2-2/b^3*a^2*ln(exp(b*x+a)-1)
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Maxima [A]
time = 0.28, size = 273, normalized size = 1.83 \begin {gather*} -\frac {4 \, {\left ({\left (b x^{2} e^{\left (6 \, a\right )} + x e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )} + {\left (b x^{2} e^{\left (2 \, a\right )} - x e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}\right )}}{b^{2} e^{\left (8 \, b x + 8 \, a\right )} - 2 \, b^{2} e^{\left (4 \, b x + 4 \, a\right )} + b^{2}} + \frac {2 \, b^{2} x^{2} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )})}{b^{3}} - \frac {2 \, {\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^{3}} - \frac {2 \, {\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^{3}} - \frac {\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{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*csch(b*x+a)^3*sech(b*x+a)^3,x, algorithm="maxima")
[Out]
-4*((b*x^2*e^(6*a) + x*e^(6*a))*e^(6*b*x) + (b*x^2*e^(2*a) - x*e^(2*a))*e^(2*b*x))/(b^2*e^(8*b*x + 8*a) - 2*b^
2*e^(4*b*x + 4*a) + b^2) + (2*b^2*x^2*log(e^(2*b*x + 2*a) + 1) + 2*b*x*dilog(-e^(2*b*x + 2*a)) - polylog(3, -e
^(2*b*x + 2*a)))/b^3 - 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 - 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^
(2*b*x + 2*a) + 1)/b^3 + log(e^(b*x + a) + 1)/b^3 + log(e^(b*x + a) - 1)/b^3
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Fricas [C] Result contains complex when optimal does not.
time = 0.46, size = 4779, normalized size = 32.07 \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)^3*sech(b*x+a)^3,x, algorithm="fricas")
[Out]
-(4*(b^2*x^2 + b*x)*cosh(b*x + a)^6 + 80*(b^2*x^2 + b*x)*cosh(b*x + a)^3*sinh(b*x + a)^3 + 60*(b^2*x^2 + b*x)*
cosh(b*x + a)^2*sinh(b*x + a)^4 + 24*(b^2*x^2 + b*x)*cosh(b*x + a)*sinh(b*x + a)^5 + 4*(b^2*x^2 + b*x)*sinh(b*
x + a)^6 + 4*(b^2*x^2 - b*x)*cosh(b*x + a)^2 + 4*(15*(b^2*x^2 + b*x)*cosh(b*x + a)^4 + b^2*x^2 - b*x)*sinh(b*x
+ a)^2 + 4*(b*x*cosh(b*x + a)^8 + 56*b*x*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*b*x*cosh(b*x + a)^2*sinh(b*x +
a)^6 + 8*b*x*cosh(b*x + a)*sinh(b*x + a)^7 + b*x*sinh(b*x + a)^8 - 2*b*x*cosh(b*x + a)^4 + 2*(35*b*x*cosh(b*x
+ a)^4 - b*x)*sinh(b*x + a)^4 + 8*(7*b*x*cosh(b*x + a)^5 - b*x*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*b*x*cosh(
b*x + a)^6 - 3*b*x*cosh(b*x + a)^2)*sinh(b*x + a)^2 + b*x + 8*(b*x*cosh(b*x + a)^7 - b*x*cosh(b*x + a)^3)*sinh
(b*x + a))*dilog(cosh(b*x + a) + sinh(b*x + a)) - 4*(b*x*cosh(b*x + a)^8 + 56*b*x*cosh(b*x + a)^3*sinh(b*x + a
)^5 + 28*b*x*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*b*x*cosh(b*x + a)*sinh(b*x + a)^7 + b*x*sinh(b*x + a)^8 - 2*b
*x*cosh(b*x + a)^4 + 2*(35*b*x*cosh(b*x + a)^4 - b*x)*sinh(b*x + a)^4 + 8*(7*b*x*cosh(b*x + a)^5 - b*x*cosh(b*
x + a))*sinh(b*x + a)^3 + 4*(7*b*x*cosh(b*x + a)^6 - 3*b*x*cosh(b*x + a)^2)*sinh(b*x + a)^2 + b*x + 8*(b*x*cos
h(b*x + a)^7 - b*x*cosh(b*x + a)^3)*sinh(b*x + a))*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 4*(b*x*cosh(b*x
+ a)^8 + 56*b*x*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*b*x*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*b*x*cosh(b*x + a)
*sinh(b*x + a)^7 + b*x*sinh(b*x + a)^8 - 2*b*x*cosh(b*x + a)^4 + 2*(35*b*x*cosh(b*x + a)^4 - b*x)*sinh(b*x + a
)^4 + 8*(7*b*x*cosh(b*x + a)^5 - b*x*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*b*x*cosh(b*x + a)^6 - 3*b*x*cosh(b*
x + a)^2)*sinh(b*x + a)^2 + b*x + 8*(b*x*cosh(b*x + a)^7 - b*x*cosh(b*x + a)^3)*sinh(b*x + a))*dilog(-I*cosh(b
*x + a) - I*sinh(b*x + a)) + 4*(b*x*cosh(b*x + a)^8 + 56*b*x*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*b*x*cosh(b*x
+ a)^2*sinh(b*x + a)^6 + 8*b*x*cosh(b*x + a)*sinh(b*x + a)^7 + b*x*sinh(b*x + a)^8 - 2*b*x*cosh(b*x + a)^4 +
2*(35*b*x*cosh(b*x + a)^4 - b*x)*sinh(b*x + a)^4 + 8*(7*b*x*cosh(b*x + a)^5 - b*x*cosh(b*x + a))*sinh(b*x + a)
^3 + 4*(7*b*x*cosh(b*x + a)^6 - 3*b*x*cosh(b*x + a)^2)*sinh(b*x + a)^2 + b*x + 8*(b*x*cosh(b*x + a)^7 - b*x*co
sh(b*x + a)^3)*sinh(b*x + a))*dilog(-cosh(b*x + a) - sinh(b*x + a)) + ((2*b^2*x^2 - 1)*cosh(b*x + a)^8 + 56*(2
*b^2*x^2 - 1)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(2*b^2*x^2 - 1)*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*(2*b^2*
x^2 - 1)*cosh(b*x + a)*sinh(b*x + a)^7 + (2*b^2*x^2 - 1)*sinh(b*x + a)^8 - 2*(2*b^2*x^2 - 1)*cosh(b*x + a)^4 +
2*(35*(2*b^2*x^2 - 1)*cosh(b*x + a)^4 - 2*b^2*x^2 + 1)*sinh(b*x + a)^4 + 2*b^2*x^2 + 8*(7*(2*b^2*x^2 - 1)*cos
h(b*x + a)^5 - (2*b^2*x^2 - 1)*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*(2*b^2*x^2 - 1)*cosh(b*x + a)^6 - 3*(2*b^
2*x^2 - 1)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 8*((2*b^2*x^2 - 1)*cosh(b*x + a)^7 - (2*b^2*x^2 - 1)*cosh(b*x +
a)^3)*sinh(b*x + a) - 1)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - ((2*a^2 - 1)*cosh(b*x + a)^8 + 56*(2*a^2 - 1
)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(2*a^2 - 1)*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*(2*a^2 - 1)*cosh(b*x +
a)*sinh(b*x + a)^7 + (2*a^2 - 1)*sinh(b*x + a)^8 - 2*(2*a^2 - 1)*cosh(b*x + a)^4 + 2*(35*(2*a^2 - 1)*cosh(b*x
+ a)^4 - 2*a^2 + 1)*sinh(b*x + a)^4 + 8*(7*(2*a^2 - 1)*cosh(b*x + a)^5 - (2*a^2 - 1)*cosh(b*x + a))*sinh(b*x +
a)^3 + 4*(7*(2*a^2 - 1)*cosh(b*x + a)^6 - 3*(2*a^2 - 1)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 2*a^2 + 8*((2*a^2
- 1)*cosh(b*x + a)^7 - (2*a^2 - 1)*cosh(b*x + a)^3)*sinh(b*x + a) - 1)*log(cosh(b*x + a) + sinh(b*x + a) + I)
- ((2*a^2 - 1)*cosh(b*x + a)^8 + 56*(2*a^2 - 1)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(2*a^2 - 1)*cosh(b*x + a)
^2*sinh(b*x + a)^6 + 8*(2*a^2 - 1)*cosh(b*x + a)*sinh(b*x + a)^7 + (2*a^2 - 1)*sinh(b*x + a)^8 - 2*(2*a^2 - 1)
*cosh(b*x + a)^4 + 2*(35*(2*a^2 - 1)*cosh(b*x + a)^4 - 2*a^2 + 1)*sinh(b*x + a)^4 + 8*(7*(2*a^2 - 1)*cosh(b*x
+ a)^5 - (2*a^2 - 1)*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*(2*a^2 - 1)*cosh(b*x + a)^6 - 3*(2*a^2 - 1)*cosh(b*
x + a)^2)*sinh(b*x + a)^2 + 2*a^2 + 8*((2*a^2 - 1)*cosh(b*x + a)^7 - (2*a^2 - 1)*cosh(b*x + a)^3)*sinh(b*x + a
) - 1)*log(cosh(b*x + a) + sinh(b*x + a) - I) + ((2*a^2 - 1)*cosh(b*x + a)^8 + 56*(2*a^2 - 1)*cosh(b*x + a)^3*
sinh(b*x + a)^5 + 28*(2*a^2 - 1)*cosh(b*x + a)^2*sinh(b*x + a)^6 + 8*(2*a^2 - 1)*cosh(b*x + a)*sinh(b*x + a)^7
+ (2*a^2 - 1)*sinh(b*x + a)^8 - 2*(2*a^2 - 1)*cosh(b*x + a)^4 + 2*(35*(2*a^2 - 1)*cosh(b*x + a)^4 - 2*a^2 + 1
)*sinh(b*x + a)^4 + 8*(7*(2*a^2 - 1)*cosh(b*x + a)^5 - (2*a^2 - 1)*cosh(b*x + a))*sinh(b*x + a)^3 + 4*(7*(2*a^
2 - 1)*cosh(b*x + a)^6 - 3*(2*a^2 - 1)*cosh(b*x + a)^2)*sinh(b*x + a)^2 + 2*a^2 + 8*((2*a^2 - 1)*cosh(b*x + a)
^7 - (2*a^2 - 1)*cosh(b*x + a)^3)*sinh(b*x + a) - 1)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - 2*((b^2*x^2 - a^
2)*cosh(b*x + a)^8 + 56*(b^2*x^2 - a^2)*cosh(b*x + a)^3*sinh(b*x + a)^5 + 28*(b^2*x^2 - a^2)*cosh(b*x + a)^2*s
inh(b*x + a)^6 + 8*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a)^7 + (b^2*x^2 - a^2)*sinh(b*x + a)^8 - 2*(b^2*x^
2 - a^2)*cosh(b*x + a)^4 + 2*(35*(b^2*x^2 - a^2...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \operatorname {csch}^{3}{\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)**3*sech(b*x+a)**3,x)
[Out]
Integral(x**2*csch(a + b*x)**3*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)^3*sech(b*x+a)^3,x, algorithm="giac")
[Out]
integrate(x^2*csch(b*x + a)^3*sech(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 )}^3\,{\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)^3*sinh(a + b*x)^3),x)
[Out]
int(x^2/(cosh(a + b*x)^3*sinh(a + b*x)^3), x)
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