3.218 \(\int \frac{x \sinh (x)}{(a+b \cosh (x))^2} \, dx\)

Optimal. Leaf size=60 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b \sqrt{a-b} \sqrt{a+b}}-\frac{x}{b (a+b \cosh (x))} \]

[Out]

(2*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]) - x/(b*(a + b*Cosh[x]))

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Rubi [A]  time = 0.0568916, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5465, 2659, 208} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b \sqrt{a-b} \sqrt{a+b}}-\frac{x}{b (a+b \cosh (x))} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sinh[x])/(a + b*Cosh[x])^2,x]

[Out]

(2*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]) - x/(b*(a + b*Cosh[x]))

Rule 5465

Int[(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)], x_Symbo
l] :> Simp[((e + f*x)^m*(a + b*Cosh[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e +
f*x)^(m - 1)*(a + b*Cosh[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n,
-1]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{x \sinh (x)}{(a+b \cosh (x))^2} \, dx &=-\frac{x}{b (a+b \cosh (x))}+\frac{\int \frac{1}{a+b \cosh (x)} \, dx}{b}\\ &=-\frac{x}{b (a+b \cosh (x))}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b \sqrt{a+b}}-\frac{x}{b (a+b \cosh (x))}\\ \end{align*}

Mathematica [A]  time = 0.124802, size = 59, normalized size = 0.98 \[ -\frac{2 \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{b \sqrt{b^2-a^2}}-\frac{x}{b (a+b \cosh (x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Sinh[x])/(a + b*Cosh[x])^2,x]

[Out]

(-2*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/(b*Sqrt[-a^2 + b^2]) - x/(b*(a + b*Cosh[x]))

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Maple [B]  time = 0.069, size = 138, normalized size = 2.3 \begin{align*} -2\,{\frac{x{{\rm e}^{x}}}{b \left ( b{{\rm e}^{2\,x}}+2\,a{{\rm e}^{x}}+b \right ) }}+{\frac{1}{b}\ln \left ({{\rm e}^{x}}+{\frac{1}{b} \left ( a\sqrt{{a}^{2}-{b}^{2}}-{a}^{2}+{b}^{2} \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}}-{\frac{1}{b}\ln \left ({{\rm e}^{x}}+{\frac{1}{b} \left ( a\sqrt{{a}^{2}-{b}^{2}}+{a}^{2}-{b}^{2} \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.93979, size = 1202, normalized size = 20.03 \begin{align*} \left [-\frac{2 \,{\left (a^{2} - b^{2}\right )} x \cosh \left (x\right ) + 2 \,{\left (a^{2} - b^{2}\right )} x \sinh \left (x\right ) -{\left (b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b\right )} \sqrt{a^{2} - b^{2}} \log \left (\frac{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right )}{a^{2} b^{2} - b^{4} +{\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{2} +{\left (a^{2} b^{2} - b^{4}\right )} \sinh \left (x\right )^{2} + 2 \,{\left (a^{3} b - a b^{3}\right )} \cosh \left (x\right ) + 2 \,{\left (a^{3} b - a b^{3} +{\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}, -\frac{2 \,{\left ({\left (a^{2} - b^{2}\right )} x \cosh \left (x\right ) +{\left (a^{2} - b^{2}\right )} x \sinh \left (x\right ) +{\left (b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b\right )} \sqrt{-a^{2} + b^{2}} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right )\right )}}{a^{2} b^{2} - b^{4} +{\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{2} +{\left (a^{2} b^{2} - b^{4}\right )} \sinh \left (x\right )^{2} + 2 \,{\left (a^{3} b - a b^{3}\right )} \cosh \left (x\right ) + 2 \,{\left (a^{3} b - a b^{3} +{\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(2*(a^2 - b^2)*x*cosh(x) + 2*(a^2 - b^2)*x*sinh(x) - (b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x)
+ a)*sinh(x) + b)*sqrt(a^2 - b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 - b^2 + 2*(b^2*c
osh(x) + a*b)*sinh(x) - 2*sqrt(a^2 - b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x
) + 2*(b*cosh(x) + a)*sinh(x) + b)))/(a^2*b^2 - b^4 + (a^2*b^2 - b^4)*cosh(x)^2 + (a^2*b^2 - b^4)*sinh(x)^2 +
2*(a^3*b - a*b^3)*cosh(x) + 2*(a^3*b - a*b^3 + (a^2*b^2 - b^4)*cosh(x))*sinh(x)), -2*((a^2 - b^2)*x*cosh(x) +
(a^2 - b^2)*x*sinh(x) + (b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) + b)*sqrt(-a^2 +
b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a)/(a^2 - b^2)))/(a^2*b^2 - b^4 + (a^2*b^2 - b^4)*cosh(
x)^2 + (a^2*b^2 - b^4)*sinh(x)^2 + 2*(a^3*b - a*b^3)*cosh(x) + 2*(a^3*b - a*b^3 + (a^2*b^2 - b^4)*cosh(x))*sin
h(x))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sinh \left (x\right )}{{\left (b \cosh \left (x\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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