3.3.77 \(\int \frac {\cosh ^4(x)}{a+a \sinh ^2(x)} \, dx\) [277]

Optimal. Leaf size=20 \[ \frac {x}{2 a}+\frac {\cosh (x) \sinh (x)}{2 a} \]

[Out]

1/2*x/a+1/2*cosh(x)*sinh(x)/a

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Rubi [A]
time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3254, 2715, 8} \begin {gather*} \frac {x}{2 a}+\frac {\sinh (x) \cosh (x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^4/(a + a*Sinh[x]^2),x]

[Out]

x/(2*a) + (Cosh[x]*Sinh[x])/(2*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 3254

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\cosh ^4(x)}{a+a \sinh ^2(x)} \, dx &=\frac {\int \cosh ^2(x) \, dx}{a}\\ &=\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\int 1 \, dx}{2 a}\\ &=\frac {x}{2 a}+\frac {\cosh (x) \sinh (x)}{2 a}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 18, normalized size = 0.90 \begin {gather*} \frac {\frac {x}{2}+\frac {1}{4} \sinh (2 x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^4/(a + a*Sinh[x]^2),x]

[Out]

(x/2 + Sinh[2*x]/4)/a

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(16)=32\).
time = 0.43, size = 65, normalized size = 3.25

method	result	size
risch	\(\frac {x}{2 a}+\frac {{\mathrm e}^{2 x}}{8 a}-\frac {{\mathrm e}^{-2 x}}{8 a}\)	\(26\)
default	\(\frac {-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2}{4 \tanh \left (\frac {x}{2}\right )+4}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {2}{4 \tanh \left (\frac {x}{2}\right )-4}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}}{a}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^4/(a+a*sinh(x)^2),x,method=_RETURNVERBOSE)

[Out]

2/a*(-1/4/(tanh(1/2*x)+1)^2+1/4/(tanh(1/2*x)+1)+1/4*ln(tanh(1/2*x)+1)+1/4/(tanh(1/2*x)-1)^2+1/4/(tanh(1/2*x)-1
)-1/4*ln(tanh(1/2*x)-1))

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Maxima [A]
time = 0.26, size = 25, normalized size = 1.25 \begin {gather*} \frac {x}{2 \, a} + \frac {e^{\left (2 \, x\right )}}{8 \, a} - \frac {e^{\left (-2 \, x\right )}}{8 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^4/(a+a*sinh(x)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 12, normalized size = 0.60 \begin {gather*} \frac {\cosh \left (x\right ) \sinh \left (x\right ) + x}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^4/(a+a*sinh(x)^2),x, algorithm="fricas")

[Out]

1/2*(cosh(x)*sinh(x) + x)/a

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (14) = 28\).
time = 1.56, size = 153, normalized size = 7.65 \begin {gather*} \frac {x \tanh ^{4}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} - \frac {2 x \tanh ^{2}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {x}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {2 \tanh ^{3}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {2 \tanh {\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**4/(a+a*sinh(x)**2),x)

[Out]

x*tanh(x/2)**4/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)**2 + 2*a) - 2*x*tanh(x/2)**2/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2
)**2 + 2*a) + x/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)**2 + 2*a) + 2*tanh(x/2)**3/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)
**2 + 2*a) + 2*tanh(x/2)/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)**2 + 2*a)

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Giac [A]
time = 0.42, size = 28, normalized size = 1.40 \begin {gather*} -\frac {{\left (2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} - 4 \, x - e^{\left (2 \, x\right )}}{8 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^4/(a+a*sinh(x)^2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.33, size = 25, normalized size = 1.25 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}+\frac {x}{2\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^4/(a + a*sinh(x)^2),x)

[Out]

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

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