\(\int \frac {\sinh ^4(x)}{a+a \cosh (x)} \, dx\) [156]

Optimal result

Integrand size = 13, antiderivative size = 31 \[ \int \frac {\sinh ^4(x)}{a+a \cosh (x)} \, dx=\frac {x}{2 a}-\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\sinh ^3(x)}{3 a} \]

[Out]

1/2*x/a-1/2*cosh(x)*sinh(x)/a+1/3*sinh(x)^3/a

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2761, 2715, 8} \[ \int \frac {\sinh ^4(x)}{a+a \cosh (x)} \, dx=\frac {x}{2 a}+\frac {\sinh ^3(x)}{3 a}-\frac {\sinh (x) \cosh (x)}{2 a} \]

[In]

Int[Sinh[x]^4/(a + a*Cosh[x]),x]

[Out]

x/(2*a) - (Cosh[x]*Sinh[x])/(2*a) + Sinh[x]^3/(3*a)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2761

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*((g*Cos[e
+ f*x])^(p - 1)/(b*f*(p - 1))), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \frac {\sinh ^3(x)}{3 a}-\frac {\int \sinh ^2(x) \, dx}{a} \\ & = -\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\sinh ^3(x)}{3 a}+\frac {\int 1 \, dx}{2 a} \\ & = \frac {x}{2 a}-\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\sinh ^3(x)}{3 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {\sinh ^4(x)}{a+a \cosh (x)} \, dx=\frac {6 x-3 \sinh (x)-3 \sinh (2 x)+\sinh (3 x)}{12 a} \]

[In]

Integrate[Sinh[x]^4/(a + a*Cosh[x]),x]

[Out]

(6*x - 3*Sinh[x] - 3*Sinh[2*x] + Sinh[3*x])/(12*a)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(25)=50\).

Time = 3.72 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.94

method	result	size
risch	\(\frac {x}{2 a}+\frac {{\mathrm e}^{3 x}}{24 a}-\frac {{\mathrm e}^{2 x}}{8 a}-\frac {{\mathrm e}^{x}}{8 a}+\frac {{\mathrm e}^{-x}}{8 a}+\frac {{\mathrm e}^{-2 x}}{8 a}-\frac {{\mathrm e}^{-3 x}}{24 a}\)	\(60\)
default	\(\frac {-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}}{a}\)	\(85\)

[In]

int(sinh(x)^4/(a+a*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

1/2*x/a+1/24/a*exp(3*x)-1/8/a*exp(2*x)-1/8/a*exp(x)+1/8/a*exp(-x)+1/8/a*exp(-2*x)-1/24/a*exp(-3*x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {\sinh ^4(x)}{a+a \cosh (x)} \, dx=\frac {\sinh \left (x\right )^{3} + 3 \, {\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + 6 \, x}{12 \, a} \]

[In]

integrate(sinh(x)^4/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

1/12*(sinh(x)^3 + 3*(cosh(x)^2 - 2*cosh(x) - 1)*sinh(x) + 6*x)/a

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (22) = 44\).

Time = 0.43 (sec) , antiderivative size = 294, normalized size of antiderivative = 9.48 \[ \int \frac {\sinh ^4(x)}{a+a \cosh (x)} \, dx=\frac {3 x \tanh ^{6}{\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} - \frac {9 x \tanh ^{4}{\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} + \frac {9 x \tanh ^{2}{\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} - \frac {3 x}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} - \frac {6 \tanh ^{5}{\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} - \frac {16 \tanh ^{3}{\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} + \frac {6 \tanh {\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} \]

[In]

integrate(sinh(x)**4/(a+a*cosh(x)),x)

[Out]

3*x*tanh(x/2)**6/(6*a*tanh(x/2)**6 - 18*a*tanh(x/2)**4 + 18*a*tanh(x/2)**2 - 6*a) - 9*x*tanh(x/2)**4/(6*a*tanh
(x/2)**6 - 18*a*tanh(x/2)**4 + 18*a*tanh(x/2)**2 - 6*a) + 9*x*tanh(x/2)**2/(6*a*tanh(x/2)**6 - 18*a*tanh(x/2)*
*4 + 18*a*tanh(x/2)**2 - 6*a) - 3*x/(6*a*tanh(x/2)**6 - 18*a*tanh(x/2)**4 + 18*a*tanh(x/2)**2 - 6*a) - 6*tanh(
x/2)**5/(6*a*tanh(x/2)**6 - 18*a*tanh(x/2)**4 + 18*a*tanh(x/2)**2 - 6*a) - 16*tanh(x/2)**3/(6*a*tanh(x/2)**6 -
18*a*tanh(x/2)**4 + 18*a*tanh(x/2)**2 - 6*a) + 6*tanh(x/2)/(6*a*tanh(x/2)**6 - 18*a*tanh(x/2)**4 + 18*a*tanh(
x/2)**2 - 6*a)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).

Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {\sinh ^4(x)}{a+a \cosh (x)} \, dx=-\frac {{\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )}}{24 \, a} + \frac {x}{2 \, a} + \frac {3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )}}{24 \, a} \]

[In]

integrate(sinh(x)^4/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

-1/24*(3*e^(-x) + 3*e^(-2*x) - 1)*e^(3*x)/a + 1/2*x/a + 1/24*(3*e^(-x) + 3*e^(-2*x) - e^(-3*x))/a

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {\sinh ^4(x)}{a+a \cosh (x)} \, dx=\frac {{\left (3 \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 1\right )} e^{\left (-3 \, x\right )} + 12 \, x + e^{\left (3 \, x\right )} - 3 \, e^{\left (2 \, x\right )} - 3 \, e^{x}}{24 \, a} \]

[In]

integrate(sinh(x)^4/(a+a*cosh(x)),x, algorithm="giac")

[Out]

1/24*((3*e^(2*x) + 3*e^x - 1)*e^(-3*x) + 12*x + e^(3*x) - 3*e^(2*x) - 3*e^x)/a

Mupad [B] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int \frac {\sinh ^4(x)}{a+a \cosh (x)} \, dx=\frac {{\mathrm {e}}^{-x}}{8\,a}+\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}+\frac {x}{2\,a}-\frac {{\mathrm {e}}^x}{8\,a} \]

[In]

int(sinh(x)^4/(a + a*cosh(x)),x)

[Out]

exp(-x)/(8*a) + exp(-2*x)/(8*a) - exp(2*x)/(8*a) - exp(-3*x)/(24*a) + exp(3*x)/(24*a) + x/(2*a) - exp(x)/(8*a)