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

Optimal. Leaf size=25 \[ \frac{\log (a+b \sinh (x))}{b}-\frac{x \cosh (x)}{a+b \sinh (x)} \]

[Out]

Log[a + b*Sinh[x]]/b - (x*Cosh[x])/(a + b*Sinh[x])

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Rubi [A]  time = 0.057565, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5636, 2668, 31} \[ \frac{\log (a+b \sinh (x))}{b}-\frac{x \cosh (x)}{a+b \sinh (x)} \]

Antiderivative was successfully verified.

[In]

Int[(x*(b - a*Sinh[x]))/(a + b*Sinh[x])^2,x]

[Out]

Log[a + b*Sinh[x]]/b - (x*Cosh[x])/(a + b*Sinh[x])

Rule 5636

Int[(((e_.) + (f_.)*(x_))*((A_) + (B_.)*Sinh[(c_.) + (d_.)*(x_)]))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)])^2,
x_Symbol] :> Simp[(B*(e + f*x)*Cosh[c + d*x])/(a*d*(a + b*Sinh[c + d*x])), x] - Dist[(B*f)/(a*d), Int[Cosh[c +
d*x]/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && EqQ[a*A + b*B, 0]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.168827, size = 25, normalized size = 1. \[ \frac{\log (a+b \sinh (x))}{b}-\frac{x \cosh (x)}{a+b \sinh (x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*Sinh[x]]/b - (x*Cosh[x])/(a + b*Sinh[x])

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Maple [B]  time = 0.175, size = 58, normalized size = 2.3 \begin{align*} -2\,{\frac{x}{b}}+2\,{\frac{x \left ( a{{\rm e}^{x}}-b \right ) }{b \left ( b{{\rm e}^{2\,x}}+2\,a{{\rm e}^{x}}-b \right ) }}+{\frac{1}{b}\ln \left ({{\rm e}^{2\,x}}+2\,{\frac{a{{\rm e}^{x}}}{b}}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.4049, size = 396, normalized size = 15.84 \begin{align*} -\frac{2 \, b x \cosh \left (x\right )^{2} + 2 \, b x \sinh \left (x\right )^{2} + 2 \, a x \cosh \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 )} \log \left (\frac{2 \,{\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (2 \, b x \cosh \left (x\right ) + a x\right )} \sinh \left (x\right )}{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.19389, size = 130, normalized size = 5.2 \begin{align*} -\frac{2 \, b x e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )} \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right ) - 2 \, a e^{x} \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right ) + 2 \, b x + b \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right )}{b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} - b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*x*e^(2*x) - b*e^(2*x)*log(-b*e^(2*x) - 2*a*e^x + b) - 2*a*e^x*log(-b*e^(2*x) - 2*a*e^x + b) + 2*b*x + b*
log(-b*e^(2*x) - 2*a*e^x + b))/(b^2*e^(2*x) + 2*a*b*e^x - b^2)