3.38 \(\int \sqrt{a \tanh ^4(x)} \, dx\)

Optimal. Leaf size=31 \[ x \coth ^2(x) \sqrt{a \tanh ^4(x)}-\coth (x) \sqrt{a \tanh ^4(x)} \]

[Out]

-(Coth[x]*Sqrt[a*Tanh[x]^4]) + x*Coth[x]^2*Sqrt[a*Tanh[x]^4]

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Rubi [A]  time = 0.0137952, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3658, 3473, 8} \[ x \coth ^2(x) \sqrt{a \tanh ^4(x)}-\coth (x) \sqrt{a \tanh ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Tanh[x]^4],x]

[Out]

-(Coth[x]*Sqrt[a*Tanh[x]^4]) + x*Coth[x]^2*Sqrt[a*Tanh[x]^4]

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

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

Rubi steps

\begin{align*} \int \sqrt{a \tanh ^4(x)} \, dx &=\left (\coth ^2(x) \sqrt{a \tanh ^4(x)}\right ) \int \tanh ^2(x) \, dx\\ &=-\coth (x) \sqrt{a \tanh ^4(x)}+\left (\coth ^2(x) \sqrt{a \tanh ^4(x)}\right ) \int 1 \, dx\\ &=-\coth (x) \sqrt{a \tanh ^4(x)}+x \coth ^2(x) \sqrt{a \tanh ^4(x)}\\ \end{align*}

Mathematica [A]  time = 0.0168269, size = 19, normalized size = 0.61 \[ \coth (x) (x \coth (x)-1) \sqrt{a \tanh ^4(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Tanh[x]^4],x]

[Out]

Coth[x]*(-1 + x*Coth[x])*Sqrt[a*Tanh[x]^4]

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Maple [A]  time = 0.032, size = 32, normalized size = 1. \begin{align*} -{\frac{2\,\tanh \left ( x \right ) +\ln \left ( \tanh \left ( x \right ) -1 \right ) -\ln \left ( 1+\tanh \left ( x \right ) \right ) }{2\, \left ( \tanh \left ( x \right ) \right ) ^{2}}\sqrt{a \left ( \tanh \left ( x \right ) \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*tanh(x)^4)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.60266, size = 26, normalized size = 0.84 \begin{align*} \sqrt{a} x - \frac{2 \, \sqrt{a}}{e^{\left (-2 \, x\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^4)^(1/2),x, algorithm="maxima")

[Out]

sqrt(a)*x - 2*sqrt(a)/(e^(-2*x) + 1)

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Fricas [B]  time = 2.32083, size = 647, normalized size = 20.87 \begin{align*} \frac{{\left (x \cosh \left (x\right )^{2} +{\left (x e^{\left (4 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + x\right )} \sinh \left (x\right )^{2} +{\left (x \cosh \left (x\right )^{2} + x + 2\right )} e^{\left (4 \, x\right )} + 2 \,{\left (x \cosh \left (x\right )^{2} + x + 2\right )} e^{\left (2 \, x\right )} + 2 \,{\left (x \cosh \left (x\right ) e^{\left (4 \, x\right )} + 2 \, x \cosh \left (x\right ) e^{\left (2 \, x\right )} + x \cosh \left (x\right )\right )} \sinh \left (x\right ) + x + 2\right )} \sqrt{\frac{a e^{\left (8 \, x\right )} - 4 \, a e^{\left (6 \, x\right )} + 6 \, a e^{\left (4 \, x\right )} - 4 \, a e^{\left (2 \, x\right )} + a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}}}{{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (4 \, x\right )} - 2 \,{\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \,{\left (\cosh \left (x\right ) e^{\left (4 \, x\right )} - 2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^4)^(1/2),x, algorithm="fricas")

[Out]

(x*cosh(x)^2 + (x*e^(4*x) + 2*x*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^2 + x + 2)*e^(4*x) + 2*(x*cosh(x)^2 + x +
2)*e^(2*x) + 2*(x*cosh(x)*e^(4*x) + 2*x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x) + x + 2)*sqrt((a*e^(8*x) - 4*a*e^
(6*x) + 6*a*e^(4*x) - 4*a*e^(2*x) + a)/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))/((e^(4*x) - 2*e^(2*x
) + 1)*sinh(x)^2 + cosh(x)^2 + (cosh(x)^2 + 1)*e^(4*x) - 2*(cosh(x)^2 + 1)*e^(2*x) + 2*(cosh(x)*e^(4*x) - 2*co
sh(x)*e^(2*x) + cosh(x))*sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \tanh ^{4}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)**4)**(1/2),x)

[Out]

Integral(sqrt(a*tanh(x)**4), x)

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Giac [A]  time = 1.19429, size = 22, normalized size = 0.71 \begin{align*} \sqrt{a}{\left (x + \frac{2}{e^{\left (2 \, x\right )} + 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*tanh(x)^4)^(1/2),x, algorithm="giac")

[Out]

sqrt(a)*(x + 2/(e^(2*x) + 1))