3.5.40 \(\int x^2 \cosh (a+b x) \coth ^2(a+b x) \, dx\) [440]

Optimal. Leaf size=95 \[ -\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {2 \text {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {2 \sinh (a+b x)}{b^3}+\frac {x^2 \sinh (a+b x)}{b} \]

[Out]

-4*x*arctanh(exp(b*x+a))/b^2-2*x*cosh(b*x+a)/b^2-x^2*csch(b*x+a)/b-2*polylog(2,-exp(b*x+a))/b^3+2*polylog(2,ex
p(b*x+a))/b^3+2*sinh(b*x+a)/b^3+x^2*sinh(b*x+a)/b

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Rubi [A]
time = 0.10, antiderivative size = 95, 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 = {5558, 3377, 2717, 5527, 4267, 2317, 2438} \begin {gather*} -\frac {2 \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {2 \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {2 \sinh (a+b x)}{b^3}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {x^2 \sinh (a+b x)}{b}-\frac {x^2 \text {csch}(a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Cosh[a + b*x]*Coth[a + b*x]^2,x]

[Out]

(-4*x*ArcTanh[E^(a + b*x)])/b^2 - (2*x*Cosh[a + b*x])/b^2 - (x^2*Csch[a + b*x])/b - (2*PolyLog[2, -E^(a + b*x)
])/b^3 + (2*PolyLog[2, E^(a + b*x)])/b^3 + (2*Sinh[a + b*x])/b^3 + (x^2*Sinh[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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5527

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

Rule 5558

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[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 \cosh (a+b x) \coth ^2(a+b x) \, dx &=\int x^2 \cosh (a+b x) \, dx+\int x^2 \coth (a+b x) \text {csch}(a+b x) \, dx\\ &=-\frac {x^2 \text {csch}(a+b x)}{b}+\frac {x^2 \sinh (a+b x)}{b}+\frac {2 \int x \text {csch}(a+b x) \, dx}{b}-\frac {2 \int x \sinh (a+b x) \, dx}{b}\\ &=-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}+\frac {x^2 \sinh (a+b x)}{b}+\frac {2 \int \cosh (a+b x) \, dx}{b^2}-\frac {2 \int \log \left (1-e^{a+b x}\right ) \, dx}{b^2}+\frac {2 \int \log \left (1+e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}+\frac {2 \sinh (a+b x)}{b^3}+\frac {x^2 \sinh (a+b x)}{b}-\frac {2 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {2 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac {4 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \cosh (a+b x)}{b^2}-\frac {x^2 \text {csch}(a+b x)}{b}-\frac {2 \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {2 \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {2 \sinh (a+b x)}{b^3}+\frac {x^2 \sinh (a+b x)}{b}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(230\) vs. \(2(95)=190\).
time = 0.31, size = 230, normalized size = 2.42 \begin {gather*} \frac {\text {csch}\left (\frac {1}{2} (a+b x)\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \left (-2-3 b^2 x^2+2 \cosh (2 (a+b x))+b^2 x^2 \cosh (2 (a+b x))+4 a \log \left (1-e^{-a-b x}\right ) \sinh (a+b x)+4 b x \log \left (1-e^{-a-b x}\right ) \sinh (a+b x)-4 a \log \left (1+e^{-a-b x}\right ) \sinh (a+b x)-4 b x \log \left (1+e^{-a-b x}\right ) \sinh (a+b x)-4 a \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right ) \sinh (a+b x)+4 \text {PolyLog}\left (2,-e^{-a-b x}\right ) \sinh (a+b x)-4 \text {PolyLog}\left (2,e^{-a-b x}\right ) \sinh (a+b x)-2 b x \sinh (2 (a+b x))\right )}{4 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cosh[a + b*x]*Coth[a + b*x]^2,x]

[Out]

(Csch[(a + b*x)/2]*Sech[(a + b*x)/2]*(-2 - 3*b^2*x^2 + 2*Cosh[2*(a + b*x)] + b^2*x^2*Cosh[2*(a + b*x)] + 4*a*L
og[1 - E^(-a - b*x)]*Sinh[a + b*x] + 4*b*x*Log[1 - E^(-a - b*x)]*Sinh[a + b*x] - 4*a*Log[1 + E^(-a - b*x)]*Sin
h[a + b*x] - 4*b*x*Log[1 + E^(-a - b*x)]*Sinh[a + b*x] - 4*a*Log[Tanh[(a + b*x)/2]]*Sinh[a + b*x] + 4*PolyLog[
2, -E^(-a - b*x)]*Sinh[a + b*x] - 4*PolyLog[2, E^(-a - b*x)]*Sinh[a + b*x] - 2*b*x*Sinh[2*(a + b*x)]))/(4*b^3)

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Maple [A]
time = 2.30, size = 185, normalized size = 1.95

method	result	size
risch	\(\frac {\left (b^{2} x^{2}-2 b x +2\right ) {\mathrm e}^{b x +a}}{2 b^{3}}-\frac {\left (b^{2} x^{2}+2 b x +2\right ) {\mathrm e}^{-b x -a}}{2 b^{3}}-\frac {2 x^{2} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{2}}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{3}}-\frac {2 \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cosh(b*x+a)^3*csch(b*x+a)^2,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/b^2*ln(1-exp(b*x+a))*x+2/b^3*ln(1-exp(b*x+a))*a+2*polylog(2,exp(b*x+a))/b^3-2/b^2*ln(exp(b*x+a)+1)*x-2/b^
3*ln(exp(b*x+a)+1)*a-2*polylog(2,-exp(b*x+a))/b^3+4/b^3*a*arctanh(exp(b*x+a))

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Maxima [A]
time = 0.35, size = 157, normalized size = 1.65 \begin {gather*} \frac {{\left (b^{2} x^{2} e^{\left (4 \, a\right )} - 2 \, b x e^{\left (4 \, a\right )} + 2 \, e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} - 2 \, {\left (3 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x\right )}}{2 \, {\left (b^{3} e^{\left (2 \, b x + 3 \, a\right )} - b^{3} e^{a}\right )}} - \frac {2 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{3}} + \frac {2 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(b*x+a)^3*csch(b*x+a)^2,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) - 2*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a
)))/b^3 + 2*(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (90) = 180\).
time = 0.40, size = 731, normalized size = 7.69 \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 (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 4 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 4 \, {\left (b x \cosh \left (b x + a\right )^{3} + 3 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b x \sinh \left (b x + a\right )^{3} - b x \cosh \left (b x + a\right ) + {\left (3 \, b x \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 4 \, {\left (a \cosh \left (b x + a\right )^{3} + 3 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + a \sinh \left (b x + a\right )^{3} - a \cosh \left (b x + a\right ) + {\left (3 \, a \cosh \left (b x + a\right )^{2} - a\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 4 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + {\left (b x + a\right )} \sinh \left (b x + a\right )^{3} - {\left (b x + a\right )} \cosh \left (b x + a\right ) + {\left (3 \, {\left (b x + a\right )} \cosh \left (b x + a\right )^{2} - b x - a\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \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*cosh(b*x+a)^3*csch(b*x+a)^2,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*(cosh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 + si
nh(b*x + a)^3 + (3*cosh(b*x + a)^2 - 1)*sinh(b*x + a) - cosh(b*x + a))*dilog(cosh(b*x + a) + sinh(b*x + a)) -
4*(cosh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b*x + a)^3 + (3*cosh(b*x + a)^2 - 1)*sinh(b*x + a)
- cosh(b*x + a))*dilog(-cosh(b*x + a) - sinh(b*x + a)) - 4*(b*x*cosh(b*x + a)^3 + 3*b*x*cosh(b*x + a)*sinh(b*
x + a)^2 + b*x*sinh(b*x + a)^3 - b*x*cosh(b*x + a) + (3*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a))*log(cosh(b*x
+ a) + sinh(b*x + a) + 1) - 4*(a*cosh(b*x + a)^3 + 3*a*cosh(b*x + a)*sinh(b*x + a)^2 + a*sinh(b*x + a)^3 - a*
cosh(b*x + a) + (3*a*cosh(b*x + a)^2 - a)*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 4*((b*x + a)
*cosh(b*x + a)^3 + 3*(b*x + a)*cosh(b*x + a)*sinh(b*x + a)^2 + (b*x + a)*sinh(b*x + a)^3 - (b*x + a)*cosh(b*x
+ a) + (3*(b*x + a)*cosh(b*x + a)^2 - b*x - a)*sinh(b*x + a))*log(-cosh(b*x + a) - 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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cosh(b*x+a)**3*csch(b*x+a)**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

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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*cosh(b*x+a)^3*csch(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(x^2*cosh(b*x + a)^3*csch(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^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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