3.4.85 \(\int x^2 \sinh (a+b x) \tanh ^2(a+b x) \, dx\) [385]

Optimal. Leaf size=104 \[ -\frac {4 x \text {ArcTan}\left (e^{a+b x}\right )}{b^2}+\frac {2 \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b}+\frac {2 i \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}-\frac {2 i \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}+\frac {x^2 \text {sech}(a+b x)}{b}-\frac {2 x \sinh (a+b x)}{b^2} \]

[Out]

-4*x*arctan(exp(b*x+a))/b^2+2*cosh(b*x+a)/b^3+x^2*cosh(b*x+a)/b+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-2*x*sinh(b*x+a)/b^2

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Rubi [A]
time = 0.09, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5557, 3377, 2718, 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 {2 \cosh (a+b x)}{b^3}-\frac {2 x \sinh (a+b x)}{b^2}+\frac {x^2 \cosh (a+b x)}{b}+\frac {x^2 \text {sech}(a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*x*ArcTan[E^(a + b*x)])/b^2 + (2*Cosh[a + b*x])/b^3 + (x^2*Cosh[a + b*x])/b + ((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 - (2*x*Sinh[a + b*x])/b^2

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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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

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]

Rule 5557

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

Rubi steps

\begin {align*} \int x^2 \sinh (a+b x) \tanh ^2(a+b x) \, dx &=\int x^2 \sinh (a+b x) \, dx-\int x^2 \text {sech}(a+b x) \tanh (a+b x) \, dx\\ &=\frac {x^2 \cosh (a+b x)}{b}+\frac {x^2 \text {sech}(a+b x)}{b}-\frac {2 \int x \cosh (a+b x) \, dx}{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 \cosh (a+b x)}{b}+\frac {x^2 \text {sech}(a+b x)}{b}-\frac {2 x \sinh (a+b x)}{b^2}+\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 {2 \int \sinh (a+b x) \, dx}{b^2}\\ &=-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {2 \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b}+\frac {x^2 \text {sech}(a+b x)}{b}-\frac {2 x \sinh (a+b x)}{b^2}+\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 \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b}+\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}-\frac {2 x \sinh (a+b x)}{b^2}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 172, normalized size = 1.65 \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)+\cosh (b x) \left (\left (2+b^2 x^2\right ) \cosh (a)-2 b x \sinh (a)\right )+\left (-2 b x \cosh (a)+\left (2+b^2 x^2\right ) \sinh (a)\right ) \sinh (b x)}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sinh[a + b*x]*Tanh[a + b*x]^2,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] + Cosh[b*x]*((2 + b^2*x^2)*Cosh[a] - 2*b*x*Sinh[a]) + (-2*b*x*Cosh[a] + (2 + b^2*x^2)*Sinh[a])*Sinh[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. 204 vs. \(2 (97 ) = 194\).
time = 2.32, size = 205, normalized size = 1.97

method	result	size
risch	\(\frac {\left (b^{2} x^{2}-2 b x +2\right ) {\mathrm e}^{b x +a}}{2 b^{3}}+\frac {\left (b^{2} x^{2}+2 b x +2\right ) {\mathrm e}^{-b x -a}}{2 b^{3}}+\frac {2 x^{2} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )}+\frac {2 i \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 i \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{b^{3}}-\frac {2 i \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 i \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 i \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 i \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*x^2-2*b*x+2)/b^3*exp(b*x+a)+1/2*(b^2*x^2+2*b*x+2)/b^3*exp(-b*x-a)+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)^3,x, algorithm="maxima")

[Out]

1/2*((b^2*x^2*e^(4*a) - 2*b*x*e^(4*a) + 2*e^(4*a))*e^(3*b*x) + 2*(3*b^2*x^2*e^(2*a) + 2*e^(2*a))*e^(b*x) + (b^
2*x^2 + 2*b*x + 2)*e^(-b*x))/(b^3*e^(2*b*x + 3*a) + b^3*e^a) - 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. 879 vs. \(2 (91) = 182\).
time = 0.43, size = 879, normalized size = 8.45 \begin {gather*} \frac {{\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \sinh \left (b x + a\right )^{4} + b^{2} x^{2} + 2 \, {\left (3 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b^{2} x^{2} + 3 \, {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right )^{2} + 2 \, b x - 4 \, {\left (i \, \cosh \left (b x + a\right )^{3} + 3 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + i \, \sinh \left (b x + a\right )^{3} + {\left (3 i \, \cosh \left (b x + a\right )^{2} + i\right )} \sinh \left (b x + a\right ) + i \, \cosh \left (b x + a\right )\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 4 \, {\left (-i \, \cosh \left (b x + a\right )^{3} - 3 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - i \, \sinh \left (b x + a\right )^{3} + {\left (-3 i \, \cosh \left (b x + a\right )^{2} - i\right )} \sinh \left (b x + a\right ) - i \, \cosh \left (b x + a\right )\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) - 4 \, {\left (-i \, a \cosh \left (b x + a\right )^{3} - 3 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - i \, a \sinh \left (b x + a\right )^{3} - i \, a \cosh \left (b x + a\right ) + {\left (-3 i \, a \cosh \left (b x + a\right )^{2} - i \, a\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) - 4 \, {\left (i \, a \cosh \left (b x + a\right )^{3} + 3 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + i \, a \sinh \left (b x + a\right )^{3} + i \, a \cosh \left (b x + a\right ) + {\left (3 i \, a \cosh \left (b x + a\right )^{2} + i \, a\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - 4 \, {\left ({\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + {\left (-i \, b x - i \, a\right )} \sinh \left (b x + a\right )^{3} + {\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right ) + {\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 )\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - 4 \, {\left ({\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + {\left (i \, b x + i \, a\right )} \sinh \left (b x + a\right )^{3} + {\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right ) + {\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 )\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 \, b x + 2\right )} \cosh \left (b x + a\right )^{3} + {\left (3 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 2}{2 \, {\left (b^{3} \cosh \left (b x + a\right )^{3} + 3 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{3} \sinh \left (b x + a\right )^{3} + b^{3} \cosh \left (b x + a\right ) + {\left (3 \, b^{3} \cosh \left (b x + a\right )^{2} + b^{3}\right )} \sinh \left (b x + a\right )\right )}} \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)^3,x, algorithm="fricas")

[Out]

1/2*((b^2*x^2 - 2*b*x + 2)*cosh(b*x + a)^4 + 4*(b^2*x^2 - 2*b*x + 2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2
- 2*b*x + 2)*sinh(b*x + a)^4 + b^2*x^2 + 2*(3*b^2*x^2 + 2)*cosh(b*x + a)^2 + 2*(3*b^2*x^2 + 3*(b^2*x^2 - 2*b*x
+ 2)*cosh(b*x + a)^2 + 2)*sinh(b*x + a)^2 + 2*b*x - 4*(I*cosh(b*x + a)^3 + 3*I*cosh(b*x + a)*sinh(b*x + a)^2
+ I*sinh(b*x + a)^3 + (3*I*cosh(b*x + a)^2 + I)*sinh(b*x + a) + I*cosh(b*x + a))*dilog(I*cosh(b*x + a) + I*sin
h(b*x + a)) - 4*(-I*cosh(b*x + a)^3 - 3*I*cosh(b*x + a)*sinh(b*x + a)^2 - I*sinh(b*x + a)^3 + (-3*I*cosh(b*x +
a)^2 - I)*sinh(b*x + a) - I*cosh(b*x + a))*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) - 4*(-I*a*cosh(b*x + a)^
3 - 3*I*a*cosh(b*x + a)*sinh(b*x + a)^2 - I*a*sinh(b*x + a)^3 - I*a*cosh(b*x + a) + (-3*I*a*cosh(b*x + a)^2 -
I*a)*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + I) - 4*(I*a*cosh(b*x + a)^3 + 3*I*a*cosh(b*x + a)*sinh
(b*x + a)^2 + I*a*sinh(b*x + a)^3 + I*a*cosh(b*x + a) + (3*I*a*cosh(b*x + a)^2 + I*a)*sinh(b*x + a))*log(cosh(
b*x + a) + sinh(b*x + a) - I) - 4*((-I*b*x - I*a)*cosh(b*x + a)^3 + 3*(-I*b*x - I*a)*cosh(b*x + a)*sinh(b*x +
a)^2 + (-I*b*x - I*a)*sinh(b*x + a)^3 + (-I*b*x - I*a)*cosh(b*x + a) + (3*(-I*b*x - I*a)*cosh(b*x + a)^2 - I*b
*x - I*a)*sinh(b*x + a))*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) - 4*((I*b*x + I*a)*cosh(b*x + a)^3 + 3*(I*
b*x + I*a)*cosh(b*x + a)*sinh(b*x + a)^2 + (I*b*x + I*a)*sinh(b*x + a)^3 + (I*b*x + I*a)*cosh(b*x + a) + (3*(I
*b*x + I*a)*cosh(b*x + a)^2 + I*b*x + I*a)*sinh(b*x + a))*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) + 4*((b^
2*x^2 - 2*b*x + 2)*cosh(b*x + a)^3 + (3*b^2*x^2 + 2)*cosh(b*x + a))*sinh(b*x + a) + 2)/(b^3*cosh(b*x + a)^3 +
3*b^3*cosh(b*x + a)*sinh(b*x + a)^2 + b^3*sinh(b*x + a)^3 + b^3*cosh(b*x + a) + (3*b^3*cosh(b*x + a)^2 + b^3)*
sinh(b*x + a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sinh ^{3}{\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)**3,x)

[Out]

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

[Out]

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

[Out]

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

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