3.1.38 \(\int \sqrt {a \tanh ^4(x)} \, dx\) [38]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3739, 3554, 8} \begin {gather*} x \coth ^2(x) \sqrt {a \tanh ^4(x)}-\coth (x) \sqrt {a \tanh ^4(x)} \end {gather*}

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 8

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

Rule 3554

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 3739

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
]])

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*}

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Mathematica [A]
time = 0.01, size = 19, normalized size = 0.61 \begin {gather*} \coth (x) (-1+x \coth (x)) \sqrt {a \tanh ^4(x)} \end {gather*}

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 = 0.72, size = 32, normalized size = 1.03

method	result	size
derivativedivides	\(-\frac {\sqrt {a \left (\tanh ^{4}\left (x \right )\right )}\, \left (2 \tanh \left (x \right )+\ln \left (\tanh \left (x \right )-1\right )-\ln \left (1+\tanh \left (x \right )\right )\right )}{2 \tanh \left (x \right )^{2}}\)	\(32\)
default	\(-\frac {\sqrt {a \left (\tanh ^{4}\left (x \right )\right )}\, \left (2 \tanh \left (x \right )+\ln \left (\tanh \left (x \right )-1\right )-\ln \left (1+\tanh \left (x \right )\right )\right )}{2 \tanh \left (x \right )^{2}}\)	\(32\)
risch	\(\frac {\sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{4}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2} x}{\left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {2 \sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{4}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.48, size = 19, normalized size = 0.61 \begin {gather*} \sqrt {a} x - \frac {2 \, \sqrt {a}}{e^{\left (-2 \, x\right )} + 1} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (27) = 54\).
time = 0.37, size = 213, normalized size = 6.87 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \tanh ^{4}{\left (x \right )}}\, dx \end {gather*}

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 = 0.42, size = 16, normalized size = 0.52 \begin {gather*} \sqrt {a} {\left (x + \frac {2}{e^{\left (2 \, x\right )} + 1}\right )} \end {gather*}

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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {a\,{\mathrm {tanh}\left (x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*tanh(x)^4)^(1/2), x)

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