\(\int \frac {\sinh ^3(x)}{a+a \cosh (x)} \, dx\) [157]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \frac {\sinh ^3(x)}{a+a \cosh (x)} \, dx=-\frac {\cosh (x)}{a}+\frac {\cosh ^2(x)}{2 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2746} \[ \int \frac {\sinh ^3(x)}{a+a \cosh (x)} \, dx=\frac {\cosh ^2(x)}{2 a}-\frac {\cosh (x)}{a} \]

[In]

Int[Sinh[x]^3/(a + a*Cosh[x]),x]

[Out]

-(Cosh[x]/a) + Cosh[x]^2/(2*a)

Rule 2746

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*} \text {integral}& = -\frac {\text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {\sinh ^3(x)}{a+a \cosh (x)} \, dx=\frac {2 \sinh ^4\left (\frac {x}{2}\right )}{a} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {-\cosh \left (x \right )+\frac {\cosh \left (x \right )^{2}}{2}}{a}\)	\(16\)
default	\(\frac {-\cosh \left (x \right )+\frac {\cosh \left (x \right )^{2}}{2}}{a}\)	\(16\)
risch	\(\frac {{\mathrm e}^{2 x}}{8 a}-\frac {{\mathrm e}^{x}}{2 a}-\frac {{\mathrm e}^{-x}}{2 a}+\frac {{\mathrm e}^{-2 x}}{8 a}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {\sinh ^3(x)}{a+a \cosh (x)} \, dx=\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )}{4 \, a} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.58 \[ \int \frac {\sinh ^3(x)}{a+a \cosh (x)} \, dx=\frac {4 \tanh ^{2}{\left (\frac {x}{2} \right )}}{a \tanh ^{4}{\left (\frac {x}{2} \right )} - 2 a \tanh ^{2}{\left (\frac {x}{2} \right )} + a} - \frac {2}{a \tanh ^{4}{\left (\frac {x}{2} \right )} - 2 a \tanh ^{2}{\left (\frac {x}{2} \right )} + a} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {\sinh ^3(x)}{a+a \cosh (x)} \, dx=-\frac {{\left (4 \, e^{\left (-x\right )} - 1\right )} e^{\left (2 \, x\right )}}{8 \, a} - \frac {4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {\sinh ^3(x)}{a+a \cosh (x)} \, dx=-\frac {{\left (4 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} - e^{\left (2 \, x\right )} + 4 \, e^{x}}{8 \, a} \]

[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

Mupad [B] (verification not implemented)

Time = 1.64 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {\sinh ^3(x)}{a+a \cosh (x)} \, dx=\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-x}}{2\,a}+\frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^x}{2\,a} \]

[In]

int(sinh(x)^3/(a + a*cosh(x)),x)

[Out]

exp(-2*x)/(8*a) - exp(-x)/(2*a) + exp(2*x)/(8*a) - exp(x)/(2*a)