3.4.49 \(\int x^3 \text {sech}^2(a+b x) \tanh (a+b x) \, dx\) [349]

Optimal. Leaf size=83 \[ \frac {3 x^2}{2 b^2}-\frac {3 x \log \left (1+e^{2 (a+b x)}\right )}{b^3}-\frac {3 \text {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{2 b^4}-\frac {x^3 \text {sech}^2(a+b x)}{2 b}+\frac {3 x^2 \tanh (a+b x)}{2 b^2} \]

[Out]

3/2*x^2/b^2-3*x*ln(1+exp(2*b*x+2*a))/b^3-3/2*polylog(2,-exp(2*b*x+2*a))/b^4-1/2*x^3*sech(b*x+a)^2/b+3/2*x^2*ta
nh(b*x+a)/b^2

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Rubi [A]
time = 0.12, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5526, 4269, 3799, 2221, 2317, 2438} \begin {gather*} -\frac {3 \text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^4}-\frac {3 x \log \left (e^{2 (a+b x)}+1\right )}{b^3}+\frac {3 x^2 \tanh (a+b x)}{2 b^2}-\frac {x^3 \text {sech}^2(a+b x)}{2 b}+\frac {3 x^2}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 4269

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

Rule 5526

Int[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*Tanh[(a_.) + (b_.)*(x_)^(n_.)]^(q_.), x_Symbol] :> Simp[(-
x^(m - n + 1))*(Sech[a + b*x^n]^p/(b*n*p)), x] + Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Sech[a + b*x^n]^p, x]
, x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]

Rubi steps

\begin {align*} \int x^3 \text {sech}^2(a+b x) \tanh (a+b x) \, dx &=-\frac {x^3 \text {sech}^2(a+b x)}{2 b}+\frac {3 \int x^2 \text {sech}^2(a+b x) \, dx}{2 b}\\ &=-\frac {x^3 \text {sech}^2(a+b x)}{2 b}+\frac {3 x^2 \tanh (a+b x)}{2 b^2}-\frac {3 \int x \tanh (a+b x) \, dx}{b^2}\\ &=\frac {3 x^2}{2 b^2}-\frac {x^3 \text {sech}^2(a+b x)}{2 b}+\frac {3 x^2 \tanh (a+b x)}{2 b^2}-\frac {6 \int \frac {e^{2 (a+b x)} x}{1+e^{2 (a+b x)}} \, dx}{b^2}\\ &=\frac {3 x^2}{2 b^2}-\frac {3 x \log \left (1+e^{2 (a+b x)}\right )}{b^3}-\frac {x^3 \text {sech}^2(a+b x)}{2 b}+\frac {3 x^2 \tanh (a+b x)}{2 b^2}+\frac {3 \int \log \left (1+e^{2 (a+b x)}\right ) \, dx}{b^3}\\ &=\frac {3 x^2}{2 b^2}-\frac {3 x \log \left (1+e^{2 (a+b x)}\right )}{b^3}-\frac {x^3 \text {sech}^2(a+b x)}{2 b}+\frac {3 x^2 \tanh (a+b x)}{2 b^2}+\frac {3 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^4}\\ &=\frac {3 x^2}{2 b^2}-\frac {3 x \log \left (1+e^{2 (a+b x)}\right )}{b^3}-\frac {3 \text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^4}-\frac {x^3 \text {sech}^2(a+b x)}{2 b}+\frac {3 x^2 \tanh (a+b x)}{2 b^2}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 4.25, size = 176, normalized size = 2.12 \begin {gather*} -\frac {3 i b \pi x-3 i \pi \log \left (1+e^{2 b x}\right )+6 b x \log \left (1-e^{-2 \left (b x+\tanh ^{-1}(\coth (a))\right )}\right )+3 i \pi \log (\cosh (b x))+6 \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 )-3 \text {PolyLog}\left (2,e^{-2 \left (b x+\tanh ^{-1}(\coth (a))\right )}\right )+b^3 x^3 \text {sech}^2(a+b x)-3 b^2 x^2 \text {sech}(a) \text {sech}(a+b x) \sinh (b x)-3 b^2 e^{-\tanh ^{-1}(\coth (a))} x^2 \sqrt {-\text {csch}^2(a)} \tanh (a)}{2 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((3*I)*b*Pi*x - (3*I)*Pi*Log[1 + E^(2*b*x)] + 6*b*x*Log[1 - E^(-2*(b*x + ArcTanh[Coth[a]]))] + (3*I)*Pi*L
og[Cosh[b*x]] + 6*ArcTanh[Coth[a]]*(b*x + Log[1 - E^(-2*(b*x + ArcTanh[Coth[a]]))] - Log[I*Sinh[b*x + ArcTanh[
Coth[a]]]]) - 3*PolyLog[2, E^(-2*(b*x + ArcTanh[Coth[a]]))] + b^3*x^3*Sech[a + b*x]^2 - 3*b^2*x^2*Sech[a]*Sech
[a + b*x]*Sinh[b*x] - (3*b^2*x^2*Sqrt[-Csch[a]^2]*Tanh[a])/E^ArcTanh[Coth[a]])/b^4

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Maple [A]
time = 1.24, size = 121, normalized size = 1.46

method	result	size
risch	\(-\frac {x^{2} \left (2 b x \,{\mathrm e}^{2 b x +2 a}+3 \,{\mathrm e}^{2 b x +2 a}+3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}+\frac {3 x^{2}}{b^{2}}+\frac {6 a x}{b^{3}}+\frac {3 a^{2}}{b^{4}}-\frac {3 x \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b^{3}}-\frac {3 \polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{4}}-\frac {6 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.32, size = 110, normalized size = 1.33 \begin {gather*} -\frac {3 \, x^{2} + {\left (2 \, b x^{3} e^{\left (2 \, a\right )} + 3 \, x^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} + \frac {3 \, x^{2}}{b^{2}} - \frac {3 \, {\left (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 )\right )}}{2 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*x^2 + (2*b*x^3*e^(2*a) + 3*x^2*e^(2*a))*e^(2*b*x))/(b^2*e^(4*b*x + 4*a) + 2*b^2*e^(2*b*x + 2*a) + b^2) + 3
*x^2/b^2 - 3/2*(2*b*x*log(e^(2*b*x + 2*a) + 1) + dilog(-e^(2*b*x + 2*a)))/b^4

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Fricas [C] Result contains complex when optimal does not.
time = 0.38, size = 1113, normalized size = 13.41 \begin {gather*} \frac {3 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{4} + 12 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{4} - {\left (2 \, b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, a^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (2 \, b^{3} x^{3} - 3 \, b^{2} x^{2} - 18 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} + 6 \, a^{2}\right )} \sinh \left (b x + a\right )^{2} - 3 \, a^{2} - 3 \, {\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 ) - 3 \, {\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 ) + 3 \, {\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 ) + 3 \, {\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 ) - 3 \, {\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 ) - 3 \, {\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 (6 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{3} - {\left (2 \, b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, a^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{b^{4} \cosh \left (b x + a\right )^{4} + 4 \, b^{4} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b^{4} \sinh \left (b x + a\right )^{4} + 2 \, b^{4} \cosh \left (b x + a\right )^{2} + b^{4} + 2 \, {\left (3 \, b^{4} \cosh \left (b x + a\right )^{2} + b^{4}\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b^{4} \cosh \left (b x + a\right )^{3} + b^{4} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*(b^2*x^2 - a^2)*cosh(b*x + a)^4 + 12*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + 3*(b^2*x^2 - a^2)*sinh
(b*x + a)^4 - (2*b^3*x^3 - 3*b^2*x^2 + 6*a^2)*cosh(b*x + a)^2 - (2*b^3*x^3 - 3*b^2*x^2 - 18*(b^2*x^2 - a^2)*co
sh(b*x + a)^2 + 6*a^2)*sinh(b*x + a)^2 - 3*a^2 - 3*(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)) - 3*(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)) + 3*(a*cosh(b*x + a)^4 + 4*a*cos
h(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) + 3*(
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) - 3*((b*x + a)*cosh(b*x + a)^4 + 4*(b*x + a)*cosh(b*x + a)*sinh(b*x + a)^3 + (b*x + a)*s
inh(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) - 3*((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*(6*(b^2*x^2 - a^2)*cosh(b*x + a)^3 - (2*b^3*x^3 - 3*b^2*x^2 + 6*a^2)*cosh(b*x + a))*sinh(b*x + a))/(b^4*c
osh(b*x + a)^4 + 4*b^4*cosh(b*x + a)*sinh(b*x + a)^3 + b^4*sinh(b*x + a)^4 + 2*b^4*cosh(b*x + a)^2 + b^4 + 2*(
3*b^4*cosh(b*x + a)^2 + b^4)*sinh(b*x + a)^2 + 4*(b^4*cosh(b*x + a)^3 + b^4*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^{3} \sinh {\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**3*sech(b*x+a)**3*sinh(b*x+a),x)

[Out]

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

[Out]

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

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

[Out]

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

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