3.26 \(\int \frac{1}{\sqrt{a \tanh ^2(x)}} \, dx\)

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

[Out]

(Log[Sinh[x]]*Tanh[x])/Sqrt[a*Tanh[x]^2]

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Rubi [A]  time = 0.0154868, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3658, 3475} \[ \frac{\tanh (x) \log (\sinh (x))}{\sqrt{a \tanh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Tanh[x]^2],x]

[Out]

(Log[Sinh[x]]*Tanh[x])/Sqrt[a*Tanh[x]^2]

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 3475

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a \tanh ^2(x)}} \, dx &=\frac{\tanh (x) \int \coth (x) \, dx}{\sqrt{a \tanh ^2(x)}}\\ &=\frac{\log (\sinh (x)) \tanh (x)}{\sqrt{a \tanh ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0077392, size = 16, normalized size = 1. \[ \frac{\tanh (x) \log (\sinh (x))}{\sqrt{a \tanh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Tanh[x]^2],x]

[Out]

(Log[Sinh[x]]*Tanh[x])/Sqrt[a*Tanh[x]^2]

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Maple [A]  time = 0.039, size = 29, normalized size = 1.8 \begin{align*} -{\frac{\tanh \left ( x \right ) \left ( \ln \left ( 1+\tanh \left ( x \right ) \right ) -2\,\ln \left ( \tanh \left ( x \right ) \right ) +\ln \left ( \tanh \left ( x \right ) -1 \right ) \right ) }{2}{\frac{1}{\sqrt{a \left ( \tanh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.59866, size = 42, normalized size = 2.62 \begin{align*} -\frac{x}{\sqrt{a}} - \frac{\log \left (e^{\left (-x\right )} + 1\right )}{\sqrt{a}} - \frac{\log \left (e^{\left (-x\right )} - 1\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/sqrt(a) - log(e^(-x) + 1)/sqrt(a) - log(e^(-x) - 1)/sqrt(a)

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Fricas [B]  time = 2.27023, size = 198, normalized size = 12.38 \begin{align*} -\frac{{\left (x e^{\left (2 \, x\right )} -{\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\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}}}{a e^{\left (2 \, x\right )} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \tanh ^{2}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*tanh(x)**2), x)

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Giac [B]  time = 1.19243, size = 50, normalized size = 3.12 \begin{align*} -\frac{x}{\sqrt{a} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right )} + \frac{\log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{\sqrt{a} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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