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]

ln(sinh(x))*tanh(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} \[ \frac {\tanh (x) \log (\sinh (x))}{\sqrt {a \tanh ^2(x)}} \]

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 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 \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*}

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

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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fricas [B]  time = 0.45, size = 76, normalized size = 4.75 \[ -\frac {{\left (x e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + x\right )} \sqrt {\frac {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{a e^{\left (2 \, x\right )} - a} \]

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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giac [A]  time = 0.14, size = 1, normalized size = 0.06 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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maple [A]  time = 0.09, size = 29, normalized size = 1.81 \[ -\frac {\tanh \relax (x ) \left (\ln \left (\tanh \relax (x )-1\right )+\ln \left (1+\tanh \relax (x )\right )-2 \ln \left (\tanh \relax (x )\right )\right )}{2 \sqrt {a \left (\tanh ^{2}\relax (x )\right )}} \]

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

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maxima [B]  time = 0.45, size = 31, normalized size = 1.94 \[ -\frac {x}{\sqrt {a}} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{\sqrt {a}} - \frac {\log \left (e^{\left (-x\right )} - 1\right )}{\sqrt {a}} \]

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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mupad [B]  time = 1.20, size = 14, normalized size = 0.88 \[ \frac {\mathrm {atanh}\left (\frac {\mathrm {tanh}\relax (x)}{\sqrt {{\mathrm {tanh}\relax (x)}^2}}\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atanh(tanh(x)/(tanh(x)^2)^(1/2))/a^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \tanh ^{2}{\relax (x )}}}\, dx \]

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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