3.3.79 \(\int \frac {\tanh ^{-1}(\coth (a+b x))}{x^2} \, dx\) [279]

Optimal. Leaf size=17 \[ -\frac {\tanh ^{-1}(\coth (a+b x))}{x}+b \log (x) \]

[Out]

-arctanh(coth(b*x+a))/x+b*ln(x)

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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2199, 29} \begin {gather*} b \log (x)-\frac {\tanh ^{-1}(\coth (a+b x))}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Coth[a + b*x]]/x^2,x]

[Out]

-(ArcTanh[Coth[a + b*x]]/x) + b*Log[x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2199

Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[u^(m + 1)*(v^
n/(a*(m + 1))), x] - Dist[b*(n/(a*(m + 1))), Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m
, n}, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0] &&  !(ILtQ[m + n, -2] && (Fra
ctionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] &&  !IntegerQ[m]
) || (ILtQ[m, 0] &&  !IntegerQ[n]))

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.06 \begin {gather*} b-\frac {\tanh ^{-1}(\coth (a+b x))}{x}+b \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Coth[a + b*x]]/x^2,x]

[Out]

b - ArcTanh[Coth[a + b*x]]/x + b*Log[x]

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Maple [A]
time = 0.09, size = 18, normalized size = 1.06

method	result	size
default	\(-\frac {\arctanh \left (\coth \left (b x +a \right )\right )}{x}+b \ln \left (x \right )\)	\(18\)
risch	\(-\frac {\ln \left ({\mathrm e}^{b x +a}\right )}{x}+\frac {-2 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{2}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )+i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right )^{3}+2 i \pi +i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 b x +2 a}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{3}-i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 b x +2 a}}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}-2 i \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right )^{2}+2 i \pi \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 b x +2 a}-1}\right )^{3}+4 \ln \left (x \right ) x b}{4 x}\)	\(339\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(coth(b*x+a))/x^2,x,method=_RETURNVERBOSE)

[Out]

-arctanh(coth(b*x+a))/x+b*ln(x)

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Maxima [A]
time = 0.30, size = 17, normalized size = 1.00 \begin {gather*} b \log \left (x\right ) - \frac {\operatorname {artanh}\left (\coth \left (b x + a\right )\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(coth(b*x+a))/x^2,x, algorithm="maxima")

[Out]

b*log(x) - arctanh(coth(b*x + a))/x

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Fricas [A]
time = 0.41, size = 13, normalized size = 0.76 \begin {gather*} \frac {b x \log \left (x\right ) - a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(coth(b*x+a))/x^2,x, algorithm="fricas")

[Out]

(b*x*log(x) - a)/x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (14) = 28\).
time = 3.96, size = 68, normalized size = 4.00 \begin {gather*} \begin {cases} - \frac {\operatorname {atanh}{\left (\coth {\left (b x + \log {\left (- e^{- b x} \right )} \right )} \right )}}{x} & \text {for}\: a = \log {\left (- e^{- b x} \right )} \\- \frac {\operatorname {atanh}{\left (\coth {\left (b x + \log {\left (e^{- b x} \right )} \right )} \right )}}{x} & \text {for}\: a = \log {\left (e^{- b x} \right )} \\b \log {\left (x \right )} - \frac {\operatorname {atanh}{\left (\frac {1}{\tanh {\left (a + b x \right )}} \right )}}{x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(coth(b*x+a))/x**2,x)

[Out]

Piecewise((-atanh(coth(b*x + log(-exp(-b*x))))/x, Eq(a, log(-exp(-b*x)))), (-atanh(coth(b*x + log(exp(-b*x))))
/x, Eq(a, log(exp(-b*x)))), (b*log(x) - atanh(1/tanh(a + b*x))/x, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (17) = 34\).
time = 0.40, size = 70, normalized size = 4.12 \begin {gather*} b \log \left ({\left | x \right |}\right ) - \frac {\log \left (-\frac {\frac {e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} + 1}{\frac {e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - 1}\right )}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(coth(b*x+a))/x^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.08, size = 17, normalized size = 1.00 \begin {gather*} b\,\ln \left (x\right )-\frac {\mathrm {atanh}\left (\mathrm {coth}\left (a+b\,x\right )\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(coth(a + b*x))/x^2,x)

[Out]

b*log(x) - atanh(coth(a + b*x))/x

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