3.4.93 \(\int x \tanh ^3(a+b x) \, dx\) [393]

Optimal. Leaf size=82 \[ \frac {x}{2 b}-\frac {x^2}{2}+\frac {x \log \left (1+e^{2 (a+b x)}\right )}{b}+\frac {\text {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{2 b^2}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b} \]

[Out]

1/2*x/b-1/2*x^2+x*ln(1+exp(2*b*x+2*a))/b+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 = 0.09, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3801, 3554, 8, 3799, 2221, 2317, 2438} \begin {gather*} \frac {\text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}-\frac {\tanh (a+b x)}{2 b^2}+\frac {x \log \left (e^{2 (a+b x)}+1\right )}{b}-\frac {x \tanh ^2(a+b x)}{2 b}+\frac {x}{2 b}-\frac {x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Tanh[a + b*x]^3,x]

[Out]

x/(2*b) - x^2/2 + (x*Log[1 + E^(2*(a + b*x))])/b + PolyLog[2, -E^(2*(a + 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 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 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 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 3554

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 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 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rubi steps

\begin {align*} \int x \tanh ^3(a+b x) \, dx &=-\frac {x \tanh ^2(a+b x)}{2 b}+\frac {\int \tanh ^2(a+b x) \, dx}{2 b}+\int x \tanh (a+b x) \, dx\\ &=-\frac {x^2}{2}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}+2 \int \frac {e^{2 (a+b x)} x}{1+e^{2 (a+b x)}} \, dx+\frac {\int 1 \, dx}{2 b}\\ &=\frac {x}{2 b}-\frac {x^2}{2}+\frac {x \log \left (1+e^{2 (a+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+b x)}\right ) \, dx}{b}\\ &=\frac {x}{2 b}-\frac {x^2}{2}+\frac {x \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {\tanh (a+b x)}{2 b^2}-\frac {x \tanh ^2(a+b x)}{2 b}-\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^2}\\ &=\frac {x}{2 b}-\frac {x^2}{2}+\frac {x \log \left (1+e^{2 (a+b x)}\right )}{b}+\frac {\text {Li}_2\left (-e^{2 (a+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 [C] Result contains complex when optimal does not.
time = 5.83, size = 175, normalized size = 2.13 \begin {gather*} \frac {i b \pi x-i \pi \log \left (1+e^{2 b x}\right )+2 b x \log \left (1-e^{-2 \left (b x+\tanh ^{-1}(\coth (a))\right )}\right )+i \pi \log (\cosh (b x))+2 \tanh ^{-1}(\coth (a)) \left (b x+\log \left (1-e^{-2 \left (b x+\tanh ^{-1}(\coth (a))\right )}\right )-\log \left (i \sinh \left (b x+\tanh ^{-1}(\coth (a))\right )\right )\right )-\text {PolyLog}\left (2,e^{-2 \left (b x+\tanh ^{-1}(\coth (a))\right )}\right )+b x \text {sech}^2(a+b x)-\text {sech}(a) \text {sech}(a+b x) \sinh (b x)+b^2 x^2 \tanh (a)-b^2 e^{-\tanh ^{-1}(\coth (a))} x^2 \sqrt {-\text {csch}^2(a)} \tanh (a)}{2 b^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x*Tanh[a + b*x]^3,x]

[Out]

(I*b*Pi*x - I*Pi*Log[1 + E^(2*b*x)] + 2*b*x*Log[1 - E^(-2*(b*x + ArcTanh[Coth[a]]))] + I*Pi*Log[Cosh[b*x]] + 2
*ArcTanh[Coth[a]]*(b*x + Log[1 - E^(-2*(b*x + ArcTanh[Coth[a]]))] - Log[I*Sinh[b*x + ArcTanh[Coth[a]]]]) - Pol
yLog[2, E^(-2*(b*x + ArcTanh[Coth[a]]))] + b*x*Sech[a + b*x]^2 - Sech[a]*Sech[a + b*x]*Sinh[b*x] + b^2*x^2*Tan
h[a] - (b^2*x^2*Sqrt[-Csch[a]^2]*Tanh[a])/E^ArcTanh[Coth[a]])/(2*b^2)

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Maple [A]
time = 2.48, size = 111, normalized size = 1.35

method	result	size
risch	\(-\frac {x^{2}}{2}+\frac {2 b x \,{\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 b x +2 a}+1}{b^{2} \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}-\frac {2 a x}{b}-\frac {a^{2}}{b^{2}}+\frac {x \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}+\frac {\polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 131, normalized size = 1.60 \begin {gather*} -x^{2} + \frac {b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} + b^{2} x^{2} + 2 \, {\left (b^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 2}{2 \, {\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} + \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 + 1/2*(b^2*x^2*e^(4*b*x + 4*a) + b^2*x^2 + 2*(b^2*x^2*e^(2*a) + 2*b*x*e^(2*a) + e^(2*a))*e^(2*b*x) + 2)/(
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

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Fricas [C] Result contains complex when optimal does not.
time = 0.40, size = 1106, normalized size = 13.49 \begin {gather*} -\frac {{\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} - 2 \, a^{2}\right )} \sinh \left (b x + a\right )^{4} + b^{2} x^{2} + 2 \, {\left (b^{2} x^{2} - 2 \, a^{2} - 2 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} x^{2} + 3 \, {\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, a^{2} - 2 \, b x - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, a^{2} - 2 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 2 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 2 \, {\left (a \cosh \left (b x + a\right )^{4} + 4 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + a \sinh \left (b x + a\right )^{4} + 2 \, a \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, a \cosh \left (b x + a\right )^{2} + a\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (a \cosh \left (b x + a\right )^{3} + a \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + 2 \, {\left (a \cosh \left (b x + a\right )^{4} + 4 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + a \sinh \left (b x + a\right )^{4} + 2 \, a \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, a \cosh \left (b x + a\right )^{2} + a\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (a \cosh \left (b x + a\right )^{3} + a \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - 2 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b x + a\right )} \sinh \left (b x + a\right )^{4} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + b x + a\right )} \sinh \left (b x + a\right )^{2} + b x + 4 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{3} + {\left (b x + a\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + a\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b x + a\right )} \sinh \left (b x + a\right )^{4} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + b x + a\right )} \sinh \left (b x + a\right )^{2} + b x + 4 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{3} + {\left (b x + a\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + a\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) + 4 \, {\left ({\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} - 2 \, a^{2} - 2 \, b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 2}{2 \, {\left (b^{2} \cosh \left (b x + a\right )^{4} + 4 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{2} \sinh \left (b x + a\right )^{4} + 2 \, b^{2} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} + b^{2}\right )} \sinh \left (b x + a\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (b x + a\right )^{3} + b^{2} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((b^2*x^2 - 2*a^2)*cosh(b*x + a)^4 + 4*(b^2*x^2 - 2*a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 - 2*a^2
)*sinh(b*x + a)^4 + b^2*x^2 + 2*(b^2*x^2 - 2*a^2 - 2*b*x - 1)*cosh(b*x + a)^2 + 2*(b^2*x^2 + 3*(b^2*x^2 - 2*a^
2)*cosh(b*x + a)^2 - 2*a^2 - 2*b*x - 1)*sinh(b*x + a)^2 - 2*a^2 - 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(I*cosh(b*x + a) + I*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(-I*cosh(b*x + a) - I*sinh(b*x + a)) + 2*(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) + 2*(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) - 2*((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) - 2*((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*s
inh(b*x + a) + 1) + 4*((b^2*x^2 - 2*a^2)*cosh(b*x + a)^3 + (b^2*x^2 - 2*a^2 - 2*b*x - 1)*cosh(b*x + a))*sinh(b
*x + a) - 2)/(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))*
sinh(b*x + a))

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

[Out]

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

[Out]

integrate(x*sech(b*x + a)^3*sinh(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\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\mathrm {cosh}\left (a+b\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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