3.157 \(\int \frac{\sinh ^3(x)}{a+a \cosh (x)} \, dx\)

Optimal. Leaf size=19 \[ \frac{\cosh ^2(x)}{2 a}-\frac{\cosh (x)}{a} \]

[Out]

-(Cosh[x]/a) + Cosh[x]^2/(2*a)

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Rubi [A]  time = 0.0432605, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2667} \[ \frac{\cosh ^2(x)}{2 a}-\frac{\cosh (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^3/(a + a*Cosh[x]),x]

[Out]

-(Cosh[x]/a) + Cosh[x]^2/(2*a)

Rule 2667

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 + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

\begin{align*} \int \frac{\sinh ^3(x)}{a+a \cosh (x)} \, dx &=-\frac{\operatorname{Subst}(\int (a-x) \, dx,x,a \cosh (x))}{a^3}\\ &=-\frac{\cosh (x)}{a}+\frac{\cosh ^2(x)}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.0116991, size = 13, normalized size = 0.68 \[ \frac{2 \sinh ^4\left (\frac{x}{2}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^3/(a + a*Cosh[x]),x]

[Out]

(2*Sinh[x/2]^4)/a

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Maple [B]  time = 0.02, size = 47, normalized size = 2.5 \begin{align*} 8\,{\frac{1}{a} \left ( 1/16\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-2}-3/16\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-1}+1/16\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{-2}+3/16\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/a*(1/16/(tanh(1/2*x)+1)^2-3/16/(tanh(1/2*x)+1)+1/16/(tanh(1/2*x)-1)^2+3/16/(tanh(1/2*x)-1))

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Maxima [B]  time = 1.11178, size = 49, normalized size = 2.58 \begin{align*} -\frac{{\left (4 \, e^{\left (-x\right )} - 1\right )} e^{\left (2 \, x\right )}}{8 \, a} - \frac{4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^3/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

-1/8*(4*e^(-x) - 1)*e^(2*x)/a - 1/8*(4*e^(-x) - e^(-2*x))/a

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Fricas [A]  time = 1.88021, size = 58, normalized size = 3.05 \begin{align*} \frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^3/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

1/4*(cosh(x)^2 + sinh(x)^2 - 4*cosh(x))/a

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Sympy [B]  time = 1.106, size = 27, normalized size = 1.42 \begin{align*} \frac{2 \tanh ^{4}{\left (\frac{x}{2} \right )}}{a \tanh ^{4}{\left (\frac{x}{2} \right )} - 2 a \tanh ^{2}{\left (\frac{x}{2} \right )} + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**3/(a+a*cosh(x)),x)

[Out]

2*tanh(x/2)**4/(a*tanh(x/2)**4 - 2*a*tanh(x/2)**2 + a)

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Giac [A]  time = 1.24729, size = 36, normalized size = 1.89 \begin{align*} -\frac{{\left (4 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} - e^{\left (2 \, x\right )} + 4 \, e^{x}}{8 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^3/(a+a*cosh(x)),x, algorithm="giac")

[Out]

-1/8*((4*e^x - 1)*e^(-2*x) - e^(2*x) + 4*e^x)/a