3.483 \(\int x \text {csch}(a+b x) \text {sech}^3(a+b x) \, dx\)
Optimal. Leaf size=95 \[ -\frac {\text {Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^2}+\frac {\text {Li}_2\left (e^{2 a+2 b x}\right )}{2 b^2}-\frac {\tanh (a+b x)}{2 b^2}-\frac {2 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {x \tanh ^2(a+b x)}{2 b}+\frac {x}{2 b} \]
[Out]
1/2*x/b-2*x*arctanh(exp(2*b*x+2*a))/b-1/2*polylog(2,-exp(2*b*x+2*a))/b^2+1/2*polylog(2,exp(2*b*x+2*a))/b^2-1/2
*tanh(b*x+a)/b^2-1/2*x*tanh(b*x+a)^2/b
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Rubi [A] time = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used
= {2620, 14, 5462, 2548, 12, 4182, 2279, 2391, 3473, 8} \[ -\frac {\text {PolyLog}\left (2,-e^{2 a+2 b x}\right )}{2 b^2}+\frac {\text {PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b^2}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}-\frac {2 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}+\frac {x}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[x*Csch[a + b*x]*Sech[a + b*x]^3,x]
[Out]
x/(2*b) - (2*x*ArcTanh[E^(2*a + 2*b*x)])/b - PolyLog[2, -E^(2*a + 2*b*x)]/(2*b^2) + PolyLog[2, E^(2*a + 2*b*x)
]/(2*b^2) - Tanh[a + b*x]/(2*b^2) - (x*Tanh[a + b*x]^2)/(2*b)
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
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 2279
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 2391
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 2548
Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]
Rule 2620
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Rule 3473
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Rule 4182
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 5462
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]
Rubi steps
\begin {align*} \int x \text {csch}(a+b x) \text {sech}^3(a+b x) \, dx &=\frac {x \log (\tanh (a+b x))}{b}-\frac {x \tanh ^2(a+b x)}{2 b}-\int \left (\frac {\log (\tanh (a+b x))}{b}-\frac {\tanh ^2(a+b x)}{2 b}\right ) \, dx\\ &=\frac {x \log (\tanh (a+b x))}{b}-\frac {x \tanh ^2(a+b x)}{2 b}+\frac {\int \tanh ^2(a+b x) \, dx}{2 b}-\frac {\int \log (\tanh (a+b x)) \, dx}{b}\\ &=-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}+\frac {\int 1 \, dx}{2 b}+\frac {\int 2 b x \text {csch}(2 a+2 b x) \, dx}{b}\\ &=\frac {x}{2 b}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}+2 \int x \text {csch}(2 a+2 b x) \, dx\\ &=\frac {x}{2 b}-\frac {2 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}-\frac {\int \log \left (1-e^{2 a+2 b x}\right ) \, dx}{b}+\frac {\int \log \left (1+e^{2 a+2 b x}\right ) \, dx}{b}\\ &=\frac {x}{2 b}-\frac {2 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}-\frac {\operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{2 b^2}+\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 a+2 b x}\right )}{2 b^2}\\ &=\frac {x}{2 b}-\frac {2 x \tanh ^{-1}\left (e^{2 a+2 b x}\right )}{b}-\frac {\text {Li}_2\left (-e^{2 a+2 b x}\right )}{2 b^2}+\frac {\text {Li}_2\left (e^{2 a+2 b x}\right )}{2 b^2}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 139, normalized size = 1.46 \[ \frac {\text {Li}_2\left (-e^{-2 (a+b x)}\right )-\text {Li}_2\left (e^{-2 (a+b x)}\right )+2 a \log \left (1-e^{-2 (a+b x)}\right )+2 b x \log \left (1-e^{-2 (a+b x)}\right )-2 a \log \left (e^{-2 (a+b x)}+1\right )-2 b x \log \left (e^{-2 (a+b x)}+1\right )-\tanh (a+b x)+b x \text {sech}^2(a+b x)-2 a \log (\sinh (a+b x))+2 a \log (\cosh (a+b x))}{2 b^2} \]
Antiderivative was successfully verified.
[In]
Integrate[x*Csch[a + b*x]*Sech[a + b*x]^3,x]
[Out]
(2*a*Log[1 - E^(-2*(a + b*x))] + 2*b*x*Log[1 - E^(-2*(a + b*x))] - 2*a*Log[1 + E^(-2*(a + b*x))] - 2*b*x*Log[1
+ E^(-2*(a + b*x))] + 2*a*Log[Cosh[a + b*x]] - 2*a*Log[Sinh[a + b*x]] + PolyLog[2, -E^(-2*(a + b*x))] - PolyL
og[2, E^(-2*(a + b*x))] + b*x*Sech[a + b*x]^2 - Tanh[a + b*x])/(2*b^2)
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fricas [C] time = 0.47, size = 1543, normalized size = 16.24 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*csch(b*x+a)*sech(b*x+a)^3,x, algorithm="fricas")
[Out]
((2*b*x + 1)*cosh(b*x + a)^2 + 2*(2*b*x + 1)*cosh(b*x + a)*sinh(b*x + a) + (2*b*x + 1)*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)*dilog(cosh(b*x + a) + sinh(b*x + a)
) - (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)*dilog(I*cosh(b*x + a) + I*
sinh(b*x + a)) - (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)*dilog(-I*cosh
(b*x + a) - I*sinh(b*x + a)) + (cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cos
h(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)
*dilog(-cosh(b*x + a) - sinh(b*x + a)) + (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))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + (a*cosh(b*x + a)^4 + 4*a*c
osh(b*x + a)*sinh(b*x + a)^3 + a*sinh(b*x + a)^4 + 2*a*cosh(b*x + a)^2 + 2*(3*a*cosh(b*x + a)^2 + a)*sinh(b*x
+ a)^2 + 4*(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) + I) + (
a*cosh(b*x + a)^4 + 4*a*cosh(b*x + a)*sinh(b*x + a)^3 + a*sinh(b*x + a)^4 + 2*a*cosh(b*x + a)^2 + 2*(3*a*cosh(
b*x + a)^2 + a)*sinh(b*x + a)^2 + 4*(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) - I) - (a*cosh(b*x + a)^4 + 4*a*cosh(b*x + a)*sinh(b*x + a)^3 + a*sinh(b*x + a)^4 + 2*a*cosh(
b*x + a)^2 + 2*(3*a*cosh(b*x + a)^2 + a)*sinh(b*x + a)^2 + 4*(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) - ((b*x + a)*cosh(b*x + a)^4 + 4*(b*x + a)*cosh(b*x + a)*sinh(b
*x + a)^3 + (b*x + a)*sinh(b*x + a)^4 + 2*(b*x + a)*cosh(b*x + a)^2 + 2*(3*(b*x + a)*cosh(b*x + a)^2 + b*x + a
)*sinh(b*x + a)^2 + b*x + 4*((b*x + a)*cosh(b*x + a)^3 + (b*x + a)*cosh(b*x + a))*sinh(b*x + a) + a)*log(I*cos
h(b*x + a) + I*sinh(b*x + a) + 1) - ((b*x + a)*cosh(b*x + a)^4 + 4*(b*x + a)*cosh(b*x + a)*sinh(b*x + a)^3 + (
b*x + a)*sinh(b*x + a)^4 + 2*(b*x + a)*cosh(b*x + a)^2 + 2*(3*(b*x + a)*cosh(b*x + a)^2 + b*x + a)*sinh(b*x +
a)^2 + b*x + 4*((b*x + a)*cosh(b*x + a)^3 + (b*x + a)*cosh(b*x + a))*sinh(b*x + a) + a)*log(-I*cosh(b*x + a) -
I*sinh(b*x + a) + 1) + ((b*x + a)*cosh(b*x + a)^4 + 4*(b*x + a)*cosh(b*x + a)*sinh(b*x + a)^3 + (b*x + a)*sin
h(b*x + a)^4 + 2*(b*x + a)*cosh(b*x + a)^2 + 2*(3*(b*x + a)*cosh(b*x + a)^2 + b*x + a)*sinh(b*x + a)^2 + b*x +
4*((b*x + a)*cosh(b*x + a)^3 + (b*x + a)*cosh(b*x + a))*sinh(b*x + a) + a)*log(-cosh(b*x + a) - sinh(b*x + a)
+ 1) + 1)/(b^2*cosh(b*x + a)^4 + 4*b^2*cosh(b*x + a)*sinh(b*x + a)^3 + b^2*sinh(b*x + a)^4 + 2*b^2*cosh(b*x +
a)^2 + 2*(3*b^2*cosh(b*x + a)^2 + b^2)*sinh(b*x + a)^2 + b^2 + 4*(b^2*cosh(b*x + a)^3 + b^2*cosh(b*x + a))*si
nh(b*x + a))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {csch}\left (b x + a\right ) \operatorname {sech}\left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*csch(b*x+a)*sech(b*x+a)^3,x, algorithm="giac")
[Out]
integrate(x*csch(b*x + a)*sech(b*x + a)^3, x)
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maple [B] time = 0.42, size = 166, normalized size = 1.75 \[ \frac {2 b x \,{\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 b x +2 a}+1}{b^{2} \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}-\frac {x \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}-\frac {\polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*csch(b*x+a)*sech(b*x+a)^3,x)
[Out]
(2*b*x*exp(2*b*x+2*a)+exp(2*b*x+2*a)+1)/b^2/(1+exp(2*b*x+2*a))^2-x*ln(1+exp(2*b*x+2*a))/b-1/2*polylog(2,-exp(2
*b*x+2*a))/b^2+1/b*ln(1-exp(b*x+a))*x+1/b^2*ln(1-exp(b*x+a))*a+polylog(2,exp(b*x+a))/b^2+1/b*ln(1+exp(b*x+a))*
x+polylog(2,-exp(b*x+a))/b^2-1/b^2*a*ln(exp(b*x+a)-1)
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maxima [A] time = 0.36, size = 142, normalized size = 1.49 \[ \frac {{\left (2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 1}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} - \frac {2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right )}{2 \, b^{2}} + \frac {b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{2}} + \frac {b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*csch(b*x+a)*sech(b*x+a)^3,x, algorithm="maxima")
[Out]
((2*b*x*e^(2*a) + e^(2*a))*e^(2*b*x) + 1)/(b^2*e^(4*b*x + 4*a) + 2*b^2*e^(2*b*x + 2*a) + b^2) - 1/2*(2*b*x*log
(e^(2*b*x + 2*a) + 1) + dilog(-e^(2*b*x + 2*a)))/b^2 + (b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^2 +
(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^2
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\mathrm {cosh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(cosh(a + b*x)^3*sinh(a + b*x)),x)
[Out]
int(x/(cosh(a + b*x)^3*sinh(a + b*x)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {csch}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*csch(b*x+a)*sech(b*x+a)**3,x)
[Out]
Integral(x*csch(a + b*x)*sech(a + b*x)**3, x)
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