3.4.43 \(\int x^2 \text {sech}(a+b x) \tanh (a+b x) \, dx\) [343]

Optimal. Leaf size=69 \[ \frac {4 x \text {ArcTan}\left (e^{a+b x}\right )}{b^2}-\frac {2 i \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}-\frac {x^2 \text {sech}(a+b x)}{b} \]

[Out]

4*x*arctan(exp(b*x+a))/b^2-2*I*polylog(2,-I*exp(b*x+a))/b^3+2*I*polylog(2,I*exp(b*x+a))/b^3-x^2*sech(b*x+a)/b

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Rubi [A]
time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5526, 4265, 2317, 2438} \begin {gather*} \frac {4 x \text {ArcTan}\left (e^{a+b x}\right )}{b^2}-\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {x^2 \text {sech}(a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x*ArcTan[E^(a + b*x)])/b^2 - ((2*I)*PolyLog[2, (-I)*E^(a + b*x)])/b^3 + ((2*I)*PolyLog[2, I*E^(a + b*x)])/b
^3 - (x^2*Sech[a + b*x])/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 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^2 \text {sech}(a+b x) \tanh (a+b x) \, dx &=-\frac {x^2 \text {sech}(a+b x)}{b}+\frac {2 \int x \text {sech}(a+b x) \, dx}{b}\\ &=\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {sech}(a+b x)}{b}-\frac {(2 i) \int \log \left (1-i e^{a+b x}\right ) \, dx}{b^2}+\frac {(2 i) \int \log \left (1+i e^{a+b x}\right ) \, dx}{b^2}\\ &=\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {sech}(a+b x)}{b}-\frac {(2 i) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {(2 i) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {x^2 \text {sech}(a+b x)}{b}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 125, normalized size = 1.81 \begin {gather*} -\frac {(-2 i a+\pi -2 i b x) \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )-(-2 i a+\pi ) \log \left (\cot \left (\frac {1}{4} (2 i a+\pi +2 i b x)\right )\right )+2 i \left (\text {PolyLog}\left (2,-i e^{a+b x}\right )-\text {PolyLog}\left (2,i e^{a+b x}\right )\right )+b^2 x^2 \text {sech}(a+b x)}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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. 153 vs. \(2 (62 ) = 124\).
time = 1.77, size = 154, normalized size = 2.23

method	result	size
risch	\(-\frac {2 x^{2} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )}-\frac {2 i \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 i \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 i \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 i \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{b^{3}}-\frac {2 i \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 i \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {4 a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x^2*exp(b*x+a)/b/(exp(2*b*x+2*a)+1)-2*I/b^2*ln(1+I*exp(b*x+a))*x-2*I/b^3*ln(1+I*exp(b*x+a))*a+2*I/b^2*ln(1-
I*exp(b*x+a))*x+2*I/b^3*ln(1-I*exp(b*x+a))*a-2*I/b^3*dilog(1+I*exp(b*x+a))+2*I/b^3*dilog(1-I*exp(b*x+a))-4/b^3
*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^2*sech(b*x+a)^2*sinh(b*x+a),x, algorithm="maxima")

[Out]

-2*x^2*e^(b*x + a)/(b*e^(2*b*x + 2*a) + b) + 4*integrate(x*e^(b*x + a)/(b*e^(2*b*x + 2*a) + b), 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. 468 vs. \(2 (56) = 112\).
time = 0.41, size = 468, normalized size = 6.78 \begin {gather*} -\frac {2 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right ) + {\left (-i \, \cosh \left (b x + a\right )^{2} - 2 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )^{2} - 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 )^{2} + 2 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )^{2} + 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 )^{2} + 2 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, a \sinh \left (b x + a\right )^{2} + 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 )^{2} - 2 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, a \sinh \left (b x + a\right )^{2} - 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 )^{2} + 2 \, {\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (i \, b x + i \, a\right )} \sinh \left (b x + a\right )^{2} + i \, b x + 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 )^{2} + 2 \, {\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (-i \, b x - i \, a\right )} \sinh \left (b x + a\right )^{2} - i \, b x - i \, a\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right )\right )}}{b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} + b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sech(b*x+a)^2*sinh(b*x+a),x, algorithm="fricas")

[Out]

-2*(b^2*x^2*cosh(b*x + a) + b^2*x^2*sinh(b*x + a) + (-I*cosh(b*x + a)^2 - 2*I*cosh(b*x + a)*sinh(b*x + a) - I*
sinh(b*x + a)^2 - I)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) + (I*cosh(b*x + a)^2 + 2*I*cosh(b*x + a)*sinh(b*
x + a) + I*sinh(b*x + a)^2 + I)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + (I*a*cosh(b*x + a)^2 + 2*I*a*cosh(
b*x + a)*sinh(b*x + a) + I*a*sinh(b*x + a)^2 + I*a)*log(cosh(b*x + a) + sinh(b*x + a) + I) + (-I*a*cosh(b*x +
a)^2 - 2*I*a*cosh(b*x + a)*sinh(b*x + a) - I*a*sinh(b*x + a)^2 - I*a)*log(cosh(b*x + a) + sinh(b*x + a) - I) +
((I*b*x + I*a)*cosh(b*x + a)^2 + 2*(I*b*x + I*a)*cosh(b*x + a)*sinh(b*x + a) + (I*b*x + I*a)*sinh(b*x + a)^2
+ I*b*x + I*a)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) + ((-I*b*x - I*a)*cosh(b*x + a)^2 + 2*(-I*b*x - I*a)
*cosh(b*x + a)*sinh(b*x + a) + (-I*b*x - I*a)*sinh(b*x + a)^2 - I*b*x - I*a)*log(-I*cosh(b*x + a) - I*sinh(b*x
+ a) + 1))/(b^3*cosh(b*x + a)^2 + 2*b^3*cosh(b*x + a)*sinh(b*x + a) + b^3*sinh(b*x + a)^2 + b^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sinh {\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*sech(b*x+a)**2*sinh(b*x+a),x)

[Out]

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

[Out]

integrate(x^2*sech(b*x + a)^2*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^2\,\mathrm {sinh}\left (a+b\,x\right )}{{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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