3.25 \(\int \sqrt {a \tanh ^2(x)} \, dx\)

Optimal. Leaf size=16 \[ \coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x)) \]

[Out]

coth(x)*ln(cosh(x))*(a*tanh(x)^2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3658, 3475} \[ \coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Tanh[x]^2],x]

[Out]

Coth[x]*Log[Cosh[x]]*Sqrt[a*Tanh[x]^2]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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
]])

Rubi steps

\begin {align*} \int \sqrt {a \tanh ^2(x)} \, dx &=\left (\coth (x) \sqrt {a \tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=\coth (x) \log (\cosh (x)) \sqrt {a \tanh ^2(x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Tanh[x]^2],x]

[Out]

Coth[x]*Log[Cosh[x]]*Sqrt[a*Tanh[x]^2]

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fricas [B]  time = 0.52, size = 72, normalized size = 4.50 \[ -\frac {{\left (x e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + x\right )} \sqrt {\frac {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{e^{\left (2 \, x\right )} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*e^(2*x) - (e^(2*x) + 1)*log(2*cosh(x)/(cosh(x) - sinh(x))) + x)*sqrt((a*e^(4*x) - 2*a*e^(2*x) + a)/(e^(4*x
) + 2*e^(2*x) + 1))/(e^(2*x) - 1)

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giac [B]  time = 2.90, size = 31, normalized size = 1.94 \[ -{\left (x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*sgn(e^(4*x) - 1) - log(e^(2*x) + 1)*sgn(e^(4*x) - 1))*sqrt(a)

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maple [A]  time = 0.08, size = 26, normalized size = 1.62 \[ -\frac {\sqrt {a \left (\tanh ^{2}\relax (x )\right )}\, \left (\ln \left (\tanh \relax (x )-1\right )+\ln \left (1+\tanh \relax (x )\right )\right )}{2 \tanh \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 19, normalized size = 1.19 \[ -\sqrt {a} x - \sqrt {a} \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a)*x - sqrt(a)*log(e^(-2*x) + 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \sqrt {a\,{\mathrm {tanh}\relax (x)}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*tanh(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \tanh ^{2}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*tanh(x)**2), x)

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