3.5.41 \(\int x \cosh (a+b x) \coth ^2(a+b x) \, dx\) [441]

Optimal. Leaf size=47 \[ -\frac {\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac {\cosh (a+b x)}{b^2}-\frac {x \text {csch}(a+b x)}{b}+\frac {x \sinh (a+b x)}{b} \]

[Out]

-arctanh(cosh(b*x+a))/b^2-cosh(b*x+a)/b^2-x*csch(b*x+a)/b+x*sinh(b*x+a)/b

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Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5558, 3377, 2718, 5527, 3855} \begin {gather*} -\frac {\cosh (a+b x)}{b^2}-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^2}+\frac {x \sinh (a+b x)}{b}-\frac {x \text {csch}(a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[Cosh[a + b*x]]/b^2) - Cosh[a + b*x]/b^2 - (x*Csch[a + b*x])/b + (x*Sinh[a + b*x])/b

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 3855

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

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 \cosh (a+b x) \coth ^2(a+b x) \, dx &=\int x \cosh (a+b x) \, dx+\int x \coth (a+b x) \text {csch}(a+b x) \, dx\\ &=-\frac {x \text {csch}(a+b x)}{b}+\frac {x \sinh (a+b x)}{b}+\frac {\int \text {csch}(a+b x) \, dx}{b}-\frac {\int \sinh (a+b x) \, dx}{b}\\ &=-\frac {\tanh ^{-1}(\cosh (a+b x))}{b^2}-\frac {\cosh (a+b x)}{b^2}-\frac {x \text {csch}(a+b x)}{b}+\frac {x \sinh (a+b x)}{b}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 66, normalized size = 1.40 \begin {gather*} \frac {-2 \cosh (a+b x)-b x \coth \left (\frac {1}{2} (a+b x)\right )+2 \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )+2 b x \sinh (a+b x)+b x \tanh \left (\frac {1}{2} (a+b x)\right )}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cosh[a + b*x] - b*x*Coth[(a + b*x)/2] + 2*Log[Tanh[(a + b*x)/2]] + 2*b*x*Sinh[a + b*x] + b*x*Tanh[(a + b*x
)/2])/(2*b^2)

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Maple [A]
time = 2.16, size = 89, normalized size = 1.89

method	result	size
risch	\(\frac {\left (b x -1\right ) {\mathrm e}^{b x +a}}{2 b^{2}}-\frac {\left (b x +1\right ) {\mathrm e}^{-b x -a}}{2 b^{2}}-\frac {2 \,{\mathrm e}^{b x +a} x}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{2}}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(b*x+a)^3*csch(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/2*(b*x-1)/b^2*exp(b*x+a)-1/2*(b*x+1)/b^2*exp(-b*x-a)-2*exp(b*x+a)*x/b/(exp(2*b*x+2*a)-1)+1/b^2*ln(exp(b*x+a)
-1)-1/b^2*ln(exp(b*x+a)+1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (47) = 94\).
time = 0.34, size = 109, normalized size = 2.32 \begin {gather*} -\frac {6 \, b x e^{\left (b x + 2 \, a\right )} - {\left (b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} - {\left (b x + 1\right )} e^{\left (-b x\right )}}{2 \, {\left (b^{2} e^{\left (2 \, b x + 3 \, a\right )} - b^{2} e^{a}\right )}} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(6*b*x*e^(b*x + 2*a) - (b*x*e^(4*a) - e^(4*a))*e^(3*b*x) - (b*x + 1)*e^(-b*x))/(b^2*e^(2*b*x + 3*a) - b^2
*e^a) - log((e^(b*x + a) + 1)*e^(-a))/b^2 + log((e^(b*x + a) - 1)*e^(-a))/b^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (47) = 94\).
time = 0.43, size = 367, normalized size = 7.81 \begin {gather*} \frac {{\left (b x - 1\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b x - 1\right )} \sinh \left (b x + a\right )^{4} - 6 \, b x \cosh \left (b x + a\right )^{2} + 6 \, {\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} + b x - 2 \, {\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 )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\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 )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 4 \, {\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{3} - 3 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{2 \, {\left (b^{2} \cosh \left (b x + a\right )^{3} + 3 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{2} \sinh \left (b x + a\right )^{3} - b^{2} \cosh \left (b x + a\right ) + {\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} - b^{2}\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((b*x - 1)*cosh(b*x + a)^4 + 4*(b*x - 1)*cosh(b*x + a)*sinh(b*x + a)^3 + (b*x - 1)*sinh(b*x + a)^4 - 6*b*x
*cosh(b*x + a)^2 + 6*((b*x - 1)*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^2 + b*x - 2*(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))*log(cosh(b*x
+ a) + sinh(b*x + a) + 1) + 2*(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))*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 4*((b*x - 1)*cosh(b*x +
a)^3 - 3*b*x*cosh(b*x + a))*sinh(b*x + a) + 1)/(b^2*cosh(b*x + a)^3 + 3*b^2*cosh(b*x + a)*sinh(b*x + a)^2 + b
^2*sinh(b*x + a)^3 - b^2*cosh(b*x + a) + (3*b^2*cosh(b*x + a)^2 - b^2)*sinh(b*x + a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \cosh ^{3}{\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (47) = 94\).
time = 0.44, size = 144, normalized size = 3.06 \begin {gather*} \frac {b x e^{\left (4 \, b x + 4 \, a\right )} - 6 \, b x e^{\left (2 \, b x + 2 \, a\right )} + b x - 2 \, e^{\left (3 \, b x + 3 \, a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, e^{\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, e^{\left (3 \, b x + 3 \, a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - 2 \, e^{\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - e^{\left (4 \, b x + 4 \, a\right )} + 1}{2 \, {\left (b^{2} e^{\left (3 \, b x + 3 \, a\right )} - b^{2} e^{\left (b x + a\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x*e^(4*b*x + 4*a) - 6*b*x*e^(2*b*x + 2*a) + b*x - 2*e^(3*b*x + 3*a)*log(e^(b*x + a) + 1) + 2*e^(b*x + a
)*log(e^(b*x + a) + 1) + 2*e^(3*b*x + 3*a)*log(e^(b*x + a) - 1) - 2*e^(b*x + a)*log(e^(b*x + a) - 1) - e^(4*b*
x + 4*a) + 1)/(b^2*e^(3*b*x + 3*a) - b^2*e^(b*x + a))

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Mupad [B]
time = 0.09, size = 95, normalized size = 2.02 \begin {gather*} {\mathrm {e}}^{a+b\,x}\,\left (\frac {x}{2\,b}-\frac {1}{2\,b^2}\right )-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^4}}{b^2}\right )}{\sqrt {-b^4}}-{\mathrm {e}}^{-a-b\,x}\,\left (\frac {x}{2\,b}+\frac {1}{2\,b^2}\right )-\frac {2\,x\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(a + b*x)*(x/(2*b) - 1/(2*b^2)) - (2*atan((exp(b*x)*exp(a)*(-b^4)^(1/2))/b^2))/(-b^4)^(1/2) - exp(- a - b*x
)*(x/(2*b) + 1/(2*b^2)) - (2*x*exp(a + b*x))/(b*(exp(2*a + 2*b*x) - 1))

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