3.4.72 \(\int x \text {sech}(a+b x) \tanh ^2(a+b x) \, dx\) [372]

Optimal. Leaf size=91 \[ \frac {x \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {i \text {PolyLog}\left (2,-i e^{a+b x}\right )}{2 b^2}+\frac {i \text {PolyLog}\left (2,i e^{a+b x}\right )}{2 b^2}-\frac {\text {sech}(a+b x)}{2 b^2}-\frac {x \text {sech}(a+b x) \tanh (a+b x)}{2 b} \]

[Out]

x*arctan(exp(b*x+a))/b-1/2*I*polylog(2,-I*exp(b*x+a))/b^2+1/2*I*polylog(2,I*exp(b*x+a))/b^2-1/2*sech(b*x+a)/b^
2-1/2*x*sech(b*x+a)*tanh(b*x+a)/b

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Rubi [A]
time = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5563, 4265, 2317, 2438, 4270} \begin {gather*} \frac {x \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {i \text {Li}_2\left (-i e^{a+b x}\right )}{2 b^2}+\frac {i \text {Li}_2\left (i e^{a+b x}\right )}{2 b^2}-\frac {\text {sech}(a+b x)}{2 b^2}-\frac {x \tanh (a+b x) \text {sech}(a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*ArcTan[E^(a + b*x)])/b - ((I/2)*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + ((I/2)*PolyLog[2, I*E^(a + b*x)])/b^2 -
Sech[a + b*x]/(2*b^2) - (x*Sech[a + b*x]*Tanh[a + b*x])/(2*b)

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 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 4270

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

Rule 5563

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]*Tanh[(a_.) + (b_.)*(x_)]^(p_), x_Symbol] :> Int[(c + d
*x)^m*Sech[a + b*x]*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sech[a + b*x]^3*Tanh[a + b*x]^(p - 2), x] /; F
reeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.55, size = 93, normalized size = 1.02 \begin {gather*} -\frac {-2 b x \text {ArcTan}(\cosh (a+b x)+\sinh (a+b x))+i \text {PolyLog}(2,-i (\cosh (a+b x)+\sinh (a+b x)))-i \text {PolyLog}(2,i (\cosh (a+b x)+\sinh (a+b x)))+\text {sech}(a+b x)+b x \text {sech}(a+b x) \tanh (a+b x)}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(-2*b*x*ArcTan[Cosh[a + b*x] + Sinh[a + b*x]] + I*PolyLog[2, (-I)*(Cosh[a + b*x] + Sinh[a + b*x])] - I*Po
lyLog[2, I*(Cosh[a + b*x] + Sinh[a + b*x])] + Sech[a + b*x] + b*x*Sech[a + b*x]*Tanh[a + b*x])/b^2

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (76 ) = 152\).
time = 1.36, size = 178, normalized size = 1.96

method	result	size
risch	\(-\frac {{\mathrm e}^{b x +a} \left (b x \,{\mathrm e}^{2 b x +2 a}-b x +{\mathrm e}^{2 b x +2 a}+1\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}-\frac {i \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{2 b}-\frac {i \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}+\frac {i \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{2 b}+\frac {i \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{2 b^{2}}-\frac {i \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {i \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\)	\(178\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sech(b*x+a)^3*sinh(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-exp(b*x+a)*(b*x*exp(2*b*x+2*a)-b*x+exp(2*b*x+2*a)+1)/b^2/(exp(2*b*x+2*a)+1)^2-1/2*I/b*ln(1+I*exp(b*x+a))*x-1/
2*I/b^2*ln(1+I*exp(b*x+a))*a+1/2*I/b*ln(1-I*exp(b*x+a))*x+1/2*I/b^2*ln(1-I*exp(b*x+a))*a-1/2*I/b^2*dilog(1+I*e
xp(b*x+a))+1/2*I/b^2*dilog(1-I*exp(b*x+a))-1/b^2*a*arctan(exp(b*x+a))

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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*sech(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="maxima")

[Out]

-((b*x*e^(3*a) + e^(3*a))*e^(3*b*x) - (b*x*e^a - e^a)*e^(b*x))/(b^2*e^(4*b*x + 4*a) + 2*b^2*e^(2*b*x + 2*a) +
b^2) + 2*integrate(1/2*x*e^(b*x + a)/(e^(2*b*x + 2*a) + 1), x)

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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. 1064 vs. \(2 (70) = 140\).
time = 0.38, size = 1064, normalized size = 11.69 \begin {gather*} -\frac {2 \, {\left (b x + 1\right )} \cosh \left (b x + a\right )^{3} + 6 \, {\left (b x + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 2 \, {\left (b x + 1\right )} \sinh \left (b x + a\right )^{3} - 2 \, {\left (b x - 1\right )} \cosh \left (b x + a\right ) - {\left (i \, \cosh \left (b x + a\right )^{4} + 4 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + i \, \sinh \left (b x + a\right )^{4} - 2 \, {\left (-3 i \, \cosh \left (b x + a\right )^{2} - i\right )} \sinh \left (b x + a\right )^{2} + 2 i \, \cosh \left (b x + a\right )^{2} - 4 \, {\left (-i \, \cosh \left (b x + a\right )^{3} - i \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + i\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - {\left (-i \, \cosh \left (b x + a\right )^{4} - 4 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - i \, \sinh \left (b x + a\right )^{4} - 2 \, {\left (3 i \, \cosh \left (b x + a\right )^{2} + i\right )} \sinh \left (b x + a\right )^{2} - 2 i \, \cosh \left (b x + a\right )^{2} - 4 \, {\left (i \, \cosh \left (b x + a\right )^{3} + i \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - i\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) - {\left (-i \, a \cosh \left (b x + a\right )^{4} - 4 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - i \, a \sinh \left (b x + a\right )^{4} - 2 i \, a \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 i \, a \cosh \left (b x + a\right )^{2} + i \, a\right )} \sinh \left (b x + a\right )^{2} - 4 \, {\left (i \, a \cosh \left (b x + a\right )^{3} + i \, a \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - i \, a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) - {\left (i \, a \cosh \left (b x + a\right )^{4} + 4 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + i \, a \sinh \left (b x + a\right )^{4} + 2 i \, a \cosh \left (b x + a\right )^{2} - 2 \, {\left (-3 i \, a \cosh \left (b x + a\right )^{2} - i \, a\right )} \sinh \left (b x + a\right )^{2} - 4 \, {\left (-i \, a \cosh \left (b x + a\right )^{3} - i \, a \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + i \, a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - {\left ({\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right )^{4} - 4 \, {\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (-i \, b x - i \, a\right )} \sinh \left (b x + a\right )^{4} - 2 \, {\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 \, {\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right )^{2} + i \, b x + i \, a\right )} \sinh \left (b x + a\right )^{2} - i \, b x - 4 \, {\left ({\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right )^{3} + {\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - i \, a\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - {\left ({\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right )^{4} - 4 \, {\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (i \, b x + i \, a\right )} \sinh \left (b x + a\right )^{4} - 2 \, {\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (3 \, {\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right )^{2} - i \, b x - i \, a\right )} \sinh \left (b x + a\right )^{2} + i \, b x - 4 \, {\left ({\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right )^{3} + {\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + i \, a\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left (3 \, {\left (b x + 1\right )} \cosh \left (b x + a\right )^{2} - b x + 1\right )} \sinh \left (b x + a\right )}{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)^2,x, algorithm="fricas")

[Out]

-1/2*(2*(b*x + 1)*cosh(b*x + a)^3 + 6*(b*x + 1)*cosh(b*x + a)*sinh(b*x + a)^2 + 2*(b*x + 1)*sinh(b*x + a)^3 -
2*(b*x - 1)*cosh(b*x + a) - (I*cosh(b*x + a)^4 + 4*I*cosh(b*x + a)*sinh(b*x + a)^3 + I*sinh(b*x + a)^4 - 2*(-3
*I*cosh(b*x + a)^2 - I)*sinh(b*x + a)^2 + 2*I*cosh(b*x + a)^2 - 4*(-I*cosh(b*x + a)^3 - I*cosh(b*x + a))*sinh(
b*x + a) + I)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - (-I*cosh(b*x + a)^4 - 4*I*cosh(b*x + a)*sinh(b*x + a)
^3 - I*sinh(b*x + a)^4 - 2*(3*I*cosh(b*x + a)^2 + I)*sinh(b*x + a)^2 - 2*I*cosh(b*x + a)^2 - 4*(I*cosh(b*x + a
)^3 + I*cosh(b*x + a))*sinh(b*x + a) - I)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) - (-I*a*cosh(b*x + a)^4 -
4*I*a*cosh(b*x + a)*sinh(b*x + a)^3 - I*a*sinh(b*x + a)^4 - 2*I*a*cosh(b*x + a)^2 - 2*(3*I*a*cosh(b*x + a)^2 +
I*a)*sinh(b*x + a)^2 - 4*(I*a*cosh(b*x + a)^3 + I*a*cosh(b*x + a))*sinh(b*x + a) - I*a)*log(cosh(b*x + a) + s
inh(b*x + a) + I) - (I*a*cosh(b*x + a)^4 + 4*I*a*cosh(b*x + a)*sinh(b*x + a)^3 + I*a*sinh(b*x + a)^4 + 2*I*a*c
osh(b*x + a)^2 - 2*(-3*I*a*cosh(b*x + a)^2 - I*a)*sinh(b*x + a)^2 - 4*(-I*a*cosh(b*x + a)^3 - I*a*cosh(b*x + a
))*sinh(b*x + a) + I*a)*log(cosh(b*x + a) + sinh(b*x + a) - I) - ((-I*b*x - I*a)*cosh(b*x + a)^4 - 4*(I*b*x +
I*a)*cosh(b*x + a)*sinh(b*x + a)^3 + (-I*b*x - I*a)*sinh(b*x + a)^4 - 2*(I*b*x + I*a)*cosh(b*x + a)^2 - 2*(3*(
I*b*x + I*a)*cosh(b*x + a)^2 + I*b*x + I*a)*sinh(b*x + a)^2 - I*b*x - 4*((I*b*x + I*a)*cosh(b*x + a)^3 + (I*b*
x + I*a)*cosh(b*x + a))*sinh(b*x + a) - I*a)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) - ((I*b*x + I*a)*cosh(
b*x + a)^4 - 4*(-I*b*x - I*a)*cosh(b*x + a)*sinh(b*x + a)^3 + (I*b*x + I*a)*sinh(b*x + a)^4 - 2*(-I*b*x - I*a)
*cosh(b*x + a)^2 - 2*(3*(-I*b*x - I*a)*cosh(b*x + a)^2 - I*b*x - I*a)*sinh(b*x + a)^2 + I*b*x - 4*((-I*b*x - I
*a)*cosh(b*x + a)^3 + (-I*b*x - I*a)*cosh(b*x + a))*sinh(b*x + a) + I*a)*log(-I*cosh(b*x + a) - I*sinh(b*x + a
) + 1) + 2*(3*(b*x + 1)*cosh(b*x + a)^2 - b*x + 1)*sinh(b*x + a))/(b^2*cosh(b*x + a)^4 + 4*b^2*cosh(b*x + a)*s
inh(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 ^{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*sech(b*x+a)**3*sinh(b*x+a)**2,x)

[Out]

Integral(x*sinh(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*sech(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(x*sech(b*x + a)^3*sinh(b*x + a)^2, 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 )}^2}{{\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)^2)/cosh(a + b*x)^3,x)

[Out]

int((x*sinh(a + b*x)^2)/cosh(a + b*x)^3, x)

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