3.2.83 \(\int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx\) [183]

Optimal. Leaf size=170 \[ -\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {6 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5} \]

[Out]

-3*b^2/(b*x-arccoth(tanh(b*x+a)))^3/arccoth(tanh(b*x+a))^2+2*b/x/(b*x-arccoth(tanh(b*x+a)))^2/arccoth(tanh(b*x
+a))^2+1/2/x^2/(b*x-arccoth(tanh(b*x+a)))/arccoth(tanh(b*x+a))^2+6*b^2/(b*x-arccoth(tanh(b*x+a)))^4/arccoth(ta
nh(b*x+a))-6*b^2*ln(x)/(b*x-arccoth(tanh(b*x+a)))^5+6*b^2*ln(arccoth(tanh(b*x+a)))/(b*x-arccoth(tanh(b*x+a)))^
5

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Rubi [A]
time = 0.09, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2202, 2194, 2191, 2188, 29} \begin {gather*} \frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {6 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*ArcCoth[Tanh[a + b*x]]^3),x]

[Out]

(-3*b^2)/((b*x - ArcCoth[Tanh[a + b*x]])^3*ArcCoth[Tanh[a + b*x]]^2) + (2*b)/(x*(b*x - ArcCoth[Tanh[a + b*x]])
^2*ArcCoth[Tanh[a + b*x]]^2) + 1/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]])*ArcCoth[Tanh[a + b*x]]^2) + (6*b^2)/((b
*x - ArcCoth[Tanh[a + b*x]])^4*ArcCoth[Tanh[a + b*x]]) - (6*b^2*Log[x])/(b*x - ArcCoth[Tanh[a + b*x]])^5 + (6*
b^2*Log[ArcCoth[Tanh[a + b*x]]])/(b*x - ArcCoth[Tanh[a + b*x]])^5

Rule 29

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

Rule 2188

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Dist[1/c, Subst[Int[x^m, x], x, u], x]] /; FreeQ[m,
x] && PiecewiseLinearQ[u, x]

Rule 2191

Int[1/((u_)*(v_)), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Dist[b/(b*u - a*v), Int[1
/v, x], x] - Dist[a/(b*u - a*v), Int[1/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]

Rule 2194

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[v^(n + 1)/((n + 1)*
(b*u - a*v)), x] - Dist[a*((n + 1)/((n + 1)*(b*u - a*v))), Int[v^(n + 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; Pi
ecewiseLinearQ[u, v, x] && LtQ[n, -1]

Rule 2202

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(-u^(m + 1))*(
v^(n + 1)/((m + 1)*(b*u - a*v))), x] + Dist[b*((m + n + 2)/((m + 1)*(b*u - a*v))), Int[u^(m + 1)*v^n, x], x] /
; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \coth ^{-1}(\tanh (a+b x))^3} \, dx &=\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {(2 b) \int \frac {1}{x^2 \coth ^{-1}(\tanh (a+b x))^3} \, dx}{b x-\coth ^{-1}(\tanh (a+b x))}\\ &=\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}-\frac {\left (6 b^2\right ) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^3} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}\\ &=-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {\left (6 b^2\right ) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))^2} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )}\\ &=-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}+\frac {\left (6 b^2\right ) \int \frac {1}{x \coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}\\ &=-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {\left (6 b^2\right ) \int \frac {1}{x} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}+\frac {\left (6 b^3\right ) \int \frac {1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}\\ &=-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2}\\ &=-\frac {3 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \coth ^{-1}(\tanh (a+b x))^2}+\frac {2 b}{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \coth ^{-1}(\tanh (a+b x))^2}+\frac {1}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right ) \coth ^{-1}(\tanh (a+b x))^2}+\frac {6 b^2}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^4 \coth ^{-1}(\tanh (a+b x))}-\frac {6 b^2 \log (x)}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}+\frac {6 b^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 107, normalized size = 0.63 \begin {gather*} \frac {-b^4 x^4+8 b^3 x^3 \coth ^{-1}(\tanh (a+b x))-8 b x \coth ^{-1}(\tanh (a+b x))^3+\coth ^{-1}(\tanh (a+b x))^4-12 b^2 x^2 \coth ^{-1}(\tanh (a+b x))^2 \left (\log (x)-\log \left (\coth ^{-1}(\tanh (a+b x))\right )\right )}{2 x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^5 \coth ^{-1}(\tanh (a+b x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*ArcCoth[Tanh[a + b*x]]^3),x]

[Out]

(-(b^4*x^4) + 8*b^3*x^3*ArcCoth[Tanh[a + b*x]] - 8*b*x*ArcCoth[Tanh[a + b*x]]^3 + ArcCoth[Tanh[a + b*x]]^4 - 1
2*b^2*x^2*ArcCoth[Tanh[a + b*x]]^2*(Log[x] - Log[ArcCoth[Tanh[a + b*x]]]))/(2*x^2*(b*x - ArcCoth[Tanh[a + b*x]
])^5*ArcCoth[Tanh[a + b*x]]^2)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/arccoth(tanh(b*x+a))^3,x)

[Out]

int(1/x^3/arccoth(tanh(b*x+a))^3,x)

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Maxima [C] Result contains complex when optimal does not.
time = 0.88, size = 331, normalized size = 1.95 \begin {gather*} \frac {192 \, b^{2} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}} - \frac {192 \, b^{2} \log \left (x\right )}{i \, \pi ^{5} - 10 \, \pi ^{4} a - 40 i \, \pi ^{3} a^{2} + 80 \, \pi ^{2} a^{3} + 80 i \, \pi a^{4} - 32 \, a^{5}} + \frac {4 \, {\left (96 \, b^{3} x^{3} - i \, \pi ^{3} + 6 \, \pi ^{2} a + 12 i \, \pi a^{2} - 8 \, a^{3} - 72 \, {\left (i \, \pi b^{2} - 2 \, a b^{2}\right )} x^{2} - 8 \, {\left (\pi ^{2} b + 4 i \, \pi a b - 4 \, a^{2} b\right )} x\right )}}{4 \, {\left (\pi ^{4} b^{2} + 8 i \, \pi ^{3} a b^{2} - 24 \, \pi ^{2} a^{2} b^{2} - 32 i \, \pi a^{3} b^{2} + 16 \, a^{4} b^{2}\right )} x^{4} - 4 \, {\left (i \, \pi ^{5} b - 10 \, \pi ^{4} a b - 40 i \, \pi ^{3} a^{2} b + 80 \, \pi ^{2} a^{3} b + 80 i \, \pi a^{4} b - 32 \, a^{5} b\right )} x^{3} - {\left (\pi ^{6} + 12 i \, \pi ^{5} a - 60 \, \pi ^{4} a^{2} - 160 i \, \pi ^{3} a^{3} + 240 \, \pi ^{2} a^{4} + 192 i \, \pi a^{5} - 64 \, a^{6}\right )} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/arccoth(tanh(b*x+a))^3,x, algorithm="maxima")

[Out]

192*b^2*log(-I*pi + 2*b*x + 2*a)/(I*pi^5 - 10*pi^4*a - 40*I*pi^3*a^2 + 80*pi^2*a^3 + 80*I*pi*a^4 - 32*a^5) - 1
92*b^2*log(x)/(I*pi^5 - 10*pi^4*a - 40*I*pi^3*a^2 + 80*pi^2*a^3 + 80*I*pi*a^4 - 32*a^5) + 4*(96*b^3*x^3 - I*pi
^3 + 6*pi^2*a + 12*I*pi*a^2 - 8*a^3 - 72*(I*pi*b^2 - 2*a*b^2)*x^2 - 8*(pi^2*b + 4*I*pi*a*b - 4*a^2*b)*x)/(4*(p
i^4*b^2 + 8*I*pi^3*a*b^2 - 24*pi^2*a^2*b^2 - 32*I*pi*a^3*b^2 + 16*a^4*b^2)*x^4 - 4*(I*pi^5*b - 10*pi^4*a*b - 4
0*I*pi^3*a^2*b + 80*pi^2*a^3*b + 80*I*pi*a^4*b - 32*a^5*b)*x^3 - (pi^6 + 12*I*pi^5*a - 60*pi^4*a^2 - 160*I*pi^
3*a^3 + 240*pi^2*a^4 + 192*I*pi*a^5 - 64*a^6)*x^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1316 vs. \(2 (168) = 336\).
time = 0.45, size = 1316, normalized size = 7.74 \begin {gather*} \frac {8 \, {\left (3 \, \pi ^{10} a + 44 \, \pi ^{8} a^{3} + 224 \, \pi ^{6} a^{5} + 384 \, \pi ^{4} a^{7} - 256 \, \pi ^{2} a^{9} - 1024 \, a^{11} + 192 \, {\left (\pi ^{6} b^{5} - 20 \, \pi ^{4} a^{2} b^{5} - 80 \, \pi ^{2} a^{4} b^{5} + 64 \, a^{6} b^{5}\right )} x^{5} + 96 \, {\left (11 \, \pi ^{6} a b^{4} - 140 \, \pi ^{4} a^{3} b^{4} - 624 \, \pi ^{2} a^{5} b^{4} + 448 \, a^{7} b^{4}\right )} x^{4} + 16 \, {\left (5 \, \pi ^{8} b^{3} + 16 \, \pi ^{6} a^{2} b^{3} - 1440 \, \pi ^{4} a^{4} b^{3} - 4864 \, \pi ^{2} a^{6} b^{3} + 3328 \, a^{8} b^{3}\right )} x^{3} + 4 \, {\left (65 \, \pi ^{8} a b^{2} - 272 \, \pi ^{6} a^{3} b^{2} - 4896 \, \pi ^{4} a^{5} b^{2} - 9472 \, \pi ^{2} a^{7} b^{2} + 6400 \, a^{9} b^{2}\right )} x^{2} + 2 \, {\left (3 \, \pi ^{10} b - 12 \, \pi ^{8} a^{2} b - 416 \, \pi ^{6} a^{4} b - 1920 \, \pi ^{4} a^{6} b - 2304 \, \pi ^{2} a^{8} b + 1024 \, a^{10} b\right )} x - 48 \, {\left (16 \, {\left (\pi ^{5} b^{6} - 40 \, \pi ^{3} a^{2} b^{6} + 80 \, \pi a^{4} b^{6}\right )} x^{6} + 64 \, {\left (\pi ^{5} a b^{5} - 40 \, \pi ^{3} a^{3} b^{5} + 80 \, \pi a^{5} b^{5}\right )} x^{5} + 8 \, {\left (\pi ^{7} b^{4} - 28 \, \pi ^{5} a^{2} b^{4} - 400 \, \pi ^{3} a^{4} b^{4} + 960 \, \pi a^{6} b^{4}\right )} x^{4} + 16 \, {\left (\pi ^{7} a b^{3} - 36 \, \pi ^{5} a^{3} b^{3} - 80 \, \pi ^{3} a^{5} b^{3} + 320 \, \pi a^{7} b^{3}\right )} x^{3} + {\left (\pi ^{9} b^{2} - 32 \, \pi ^{7} a^{2} b^{2} - 224 \, \pi ^{5} a^{4} b^{2} + 1280 \, \pi a^{8} b^{2}\right )} x^{2}\right )} \arctan \left (-\frac {2 \, b x + 2 \, a - \sqrt {4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) - 24 \, {\left (16 \, {\left (5 \, \pi ^{4} a b^{6} - 40 \, \pi ^{2} a^{3} b^{6} + 16 \, a^{5} b^{6}\right )} x^{6} + 64 \, {\left (5 \, \pi ^{4} a^{2} b^{5} - 40 \, \pi ^{2} a^{4} b^{5} + 16 \, a^{6} b^{5}\right )} x^{5} + 8 \, {\left (5 \, \pi ^{6} a b^{4} + 20 \, \pi ^{4} a^{3} b^{4} - 464 \, \pi ^{2} a^{5} b^{4} + 192 \, a^{7} b^{4}\right )} x^{4} + 16 \, {\left (5 \, \pi ^{6} a^{2} b^{3} - 20 \, \pi ^{4} a^{4} b^{3} - 144 \, \pi ^{2} a^{6} b^{3} + 64 \, a^{8} b^{3}\right )} x^{3} + {\left (5 \, \pi ^{8} a b^{2} - 224 \, \pi ^{4} a^{5} b^{2} - 512 \, \pi ^{2} a^{7} b^{2} + 256 \, a^{9} b^{2}\right )} x^{2}\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right ) + 48 \, {\left (16 \, {\left (5 \, \pi ^{4} a b^{6} - 40 \, \pi ^{2} a^{3} b^{6} + 16 \, a^{5} b^{6}\right )} x^{6} + 64 \, {\left (5 \, \pi ^{4} a^{2} b^{5} - 40 \, \pi ^{2} a^{4} b^{5} + 16 \, a^{6} b^{5}\right )} x^{5} + 8 \, {\left (5 \, \pi ^{6} a b^{4} + 20 \, \pi ^{4} a^{3} b^{4} - 464 \, \pi ^{2} a^{5} b^{4} + 192 \, a^{7} b^{4}\right )} x^{4} + 16 \, {\left (5 \, \pi ^{6} a^{2} b^{3} - 20 \, \pi ^{4} a^{4} b^{3} - 144 \, \pi ^{2} a^{6} b^{3} + 64 \, a^{8} b^{3}\right )} x^{3} + {\left (5 \, \pi ^{8} a b^{2} - 224 \, \pi ^{4} a^{5} b^{2} - 512 \, \pi ^{2} a^{7} b^{2} + 256 \, a^{9} b^{2}\right )} x^{2}\right )} \log \left (x\right )\right )}}{16 \, {\left (\pi ^{10} b^{4} + 20 \, \pi ^{8} a^{2} b^{4} + 160 \, \pi ^{6} a^{4} b^{4} + 640 \, \pi ^{4} a^{6} b^{4} + 1280 \, \pi ^{2} a^{8} b^{4} + 1024 \, a^{10} b^{4}\right )} x^{6} + 64 \, {\left (\pi ^{10} a b^{3} + 20 \, \pi ^{8} a^{3} b^{3} + 160 \, \pi ^{6} a^{5} b^{3} + 640 \, \pi ^{4} a^{7} b^{3} + 1280 \, \pi ^{2} a^{9} b^{3} + 1024 \, a^{11} b^{3}\right )} x^{5} + 8 \, {\left (\pi ^{12} b^{2} + 32 \, \pi ^{10} a^{2} b^{2} + 400 \, \pi ^{8} a^{4} b^{2} + 2560 \, \pi ^{6} a^{6} b^{2} + 8960 \, \pi ^{4} a^{8} b^{2} + 16384 \, \pi ^{2} a^{10} b^{2} + 12288 \, a^{12} b^{2}\right )} x^{4} + 16 \, {\left (\pi ^{12} a b + 24 \, \pi ^{10} a^{3} b + 240 \, \pi ^{8} a^{5} b + 1280 \, \pi ^{6} a^{7} b + 3840 \, \pi ^{4} a^{9} b + 6144 \, \pi ^{2} a^{11} b + 4096 \, a^{13} b\right )} x^{3} + {\left (\pi ^{14} + 28 \, \pi ^{12} a^{2} + 336 \, \pi ^{10} a^{4} + 2240 \, \pi ^{8} a^{6} + 8960 \, \pi ^{6} a^{8} + 21504 \, \pi ^{4} a^{10} + 28672 \, \pi ^{2} a^{12} + 16384 \, a^{14}\right )} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/arccoth(tanh(b*x+a))^3,x, algorithm="fricas")

[Out]

8*(3*pi^10*a + 44*pi^8*a^3 + 224*pi^6*a^5 + 384*pi^4*a^7 - 256*pi^2*a^9 - 1024*a^11 + 192*(pi^6*b^5 - 20*pi^4*
a^2*b^5 - 80*pi^2*a^4*b^5 + 64*a^6*b^5)*x^5 + 96*(11*pi^6*a*b^4 - 140*pi^4*a^3*b^4 - 624*pi^2*a^5*b^4 + 448*a^
7*b^4)*x^4 + 16*(5*pi^8*b^3 + 16*pi^6*a^2*b^3 - 1440*pi^4*a^4*b^3 - 4864*pi^2*a^6*b^3 + 3328*a^8*b^3)*x^3 + 4*
(65*pi^8*a*b^2 - 272*pi^6*a^3*b^2 - 4896*pi^4*a^5*b^2 - 9472*pi^2*a^7*b^2 + 6400*a^9*b^2)*x^2 + 2*(3*pi^10*b -
12*pi^8*a^2*b - 416*pi^6*a^4*b - 1920*pi^4*a^6*b - 2304*pi^2*a^8*b + 1024*a^10*b)*x - 48*(16*(pi^5*b^6 - 40*p
i^3*a^2*b^6 + 80*pi*a^4*b^6)*x^6 + 64*(pi^5*a*b^5 - 40*pi^3*a^3*b^5 + 80*pi*a^5*b^5)*x^5 + 8*(pi^7*b^4 - 28*pi
^5*a^2*b^4 - 400*pi^3*a^4*b^4 + 960*pi*a^6*b^4)*x^4 + 16*(pi^7*a*b^3 - 36*pi^5*a^3*b^3 - 80*pi^3*a^5*b^3 + 320
*pi*a^7*b^3)*x^3 + (pi^9*b^2 - 32*pi^7*a^2*b^2 - 224*pi^5*a^4*b^2 + 1280*pi*a^8*b^2)*x^2)*arctan(-(2*b*x + 2*a
- sqrt(4*b^2*x^2 + 8*a*b*x + pi^2 + 4*a^2))/pi) - 24*(16*(5*pi^4*a*b^6 - 40*pi^2*a^3*b^6 + 16*a^5*b^6)*x^6 +
64*(5*pi^4*a^2*b^5 - 40*pi^2*a^4*b^5 + 16*a^6*b^5)*x^5 + 8*(5*pi^6*a*b^4 + 20*pi^4*a^3*b^4 - 464*pi^2*a^5*b^4
+ 192*a^7*b^4)*x^4 + 16*(5*pi^6*a^2*b^3 - 20*pi^4*a^4*b^3 - 144*pi^2*a^6*b^3 + 64*a^8*b^3)*x^3 + (5*pi^8*a*b^2
- 224*pi^4*a^5*b^2 - 512*pi^2*a^7*b^2 + 256*a^9*b^2)*x^2)*log(4*b^2*x^2 + 8*a*b*x + pi^2 + 4*a^2) + 48*(16*(5
*pi^4*a*b^6 - 40*pi^2*a^3*b^6 + 16*a^5*b^6)*x^6 + 64*(5*pi^4*a^2*b^5 - 40*pi^2*a^4*b^5 + 16*a^6*b^5)*x^5 + 8*(
5*pi^6*a*b^4 + 20*pi^4*a^3*b^4 - 464*pi^2*a^5*b^4 + 192*a^7*b^4)*x^4 + 16*(5*pi^6*a^2*b^3 - 20*pi^4*a^4*b^3 -
144*pi^2*a^6*b^3 + 64*a^8*b^3)*x^3 + (5*pi^8*a*b^2 - 224*pi^4*a^5*b^2 - 512*pi^2*a^7*b^2 + 256*a^9*b^2)*x^2)*l
og(x))/(16*(pi^10*b^4 + 20*pi^8*a^2*b^4 + 160*pi^6*a^4*b^4 + 640*pi^4*a^6*b^4 + 1280*pi^2*a^8*b^4 + 1024*a^10*
b^4)*x^6 + 64*(pi^10*a*b^3 + 20*pi^8*a^3*b^3 + 160*pi^6*a^5*b^3 + 640*pi^4*a^7*b^3 + 1280*pi^2*a^9*b^3 + 1024*
a^11*b^3)*x^5 + 8*(pi^12*b^2 + 32*pi^10*a^2*b^2 + 400*pi^8*a^4*b^2 + 2560*pi^6*a^6*b^2 + 8960*pi^4*a^8*b^2 + 1
6384*pi^2*a^10*b^2 + 12288*a^12*b^2)*x^4 + 16*(pi^12*a*b + 24*pi^10*a^3*b + 240*pi^8*a^5*b + 1280*pi^6*a^7*b +
3840*pi^4*a^9*b + 6144*pi^2*a^11*b + 4096*a^13*b)*x^3 + (pi^14 + 28*pi^12*a^2 + 336*pi^10*a^4 + 2240*pi^8*a^6
+ 8960*pi^6*a^8 + 21504*pi^4*a^10 + 28672*pi^2*a^12 + 16384*a^14)*x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \operatorname {acoth}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/acoth(tanh(b*x+a))**3,x)

[Out]

Integral(1/(x**3*acoth(tanh(a + b*x))**3), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.43, size = 343, normalized size = 2.02 \begin {gather*} \frac {192 i \, b^{2} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{\pi ^{5} - 10 i \, \pi ^{4} a - 40 \, \pi ^{3} a^{2} + 80 i \, \pi ^{2} a^{3} + 80 \, \pi a^{4} - 32 i \, a^{5}} - \frac {192 i \, b^{2} \log \left (x\right )}{\pi ^{5} - 10 i \, \pi ^{4} a - 40 \, \pi ^{3} a^{2} + 80 i \, \pi ^{2} a^{3} + 80 \, \pi a^{4} - 32 i \, a^{5}} - \frac {4 \, {\left (i \, \pi - 12 \, b x + 2 \, a\right )}}{\pi ^{4} x^{2} - 8 i \, \pi ^{3} a x^{2} - 24 \, \pi ^{2} a^{2} x^{2} + 32 i \, \pi a^{3} x^{2} + 16 \, a^{4} x^{2}} + \frac {16 \, {\left (12 \, b^{3} x + 7 i \, \pi b^{2} + 14 \, a b^{2}\right )}}{4 \, \pi ^{4} b^{2} x^{2} - 32 i \, \pi ^{3} a b^{2} x^{2} - 96 \, \pi ^{2} a^{2} b^{2} x^{2} + 128 i \, \pi a^{3} b^{2} x^{2} + 64 \, a^{4} b^{2} x^{2} + 4 i \, \pi ^{5} b x + 40 \, \pi ^{4} a b x - 160 i \, \pi ^{3} a^{2} b x - 320 \, \pi ^{2} a^{3} b x + 320 i \, \pi a^{4} b x + 128 \, a^{5} b x - \pi ^{6} + 12 i \, \pi ^{5} a + 60 \, \pi ^{4} a^{2} - 160 i \, \pi ^{3} a^{3} - 240 \, \pi ^{2} a^{4} + 192 i \, \pi a^{5} + 64 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/arccoth(tanh(b*x+a))^3,x, algorithm="giac")

[Out]

192*I*b^2*log(I*pi + 2*b*x + 2*a)/(pi^5 - 10*I*pi^4*a - 40*pi^3*a^2 + 80*I*pi^2*a^3 + 80*pi*a^4 - 32*I*a^5) -
192*I*b^2*log(x)/(pi^5 - 10*I*pi^4*a - 40*pi^3*a^2 + 80*I*pi^2*a^3 + 80*pi*a^4 - 32*I*a^5) - 4*(I*pi - 12*b*x
+ 2*a)/(pi^4*x^2 - 8*I*pi^3*a*x^2 - 24*pi^2*a^2*x^2 + 32*I*pi*a^3*x^2 + 16*a^4*x^2) + 16*(12*b^3*x + 7*I*pi*b^
2 + 14*a*b^2)/(4*pi^4*b^2*x^2 - 32*I*pi^3*a*b^2*x^2 - 96*pi^2*a^2*b^2*x^2 + 128*I*pi*a^3*b^2*x^2 + 64*a^4*b^2*
x^2 + 4*I*pi^5*b*x + 40*pi^4*a*b*x - 160*I*pi^3*a^2*b*x - 320*pi^2*a^3*b*x + 320*I*pi*a^4*b*x + 128*a^5*b*x -
pi^6 + 12*I*pi^5*a + 60*pi^4*a^2 - 160*I*pi^3*a^3 - 240*pi^2*a^4 + 192*I*pi*a^5 + 64*a^6)

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Mupad [B]
time = 8.12, size = 1251, normalized size = 7.36 \begin {gather*} \frac {\frac {4}{\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x}+\frac {32\,b\,x}{{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2}-\frac {288\,b^2\,x^2}{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+4\,a^2\right )}+\frac {384\,b^3\,x^3}{{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+4\,a^2\right )}}{x^3\,\left (8\,a\,b-4\,b\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\right )+x^2\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+4\,a^2\right )+4\,b^2\,x^4}-\frac {384\,b^2\,\mathrm {atanh}\left (\frac {4\,b\,x\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )+4\,a^2\right )}{{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3}-\frac {{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^5+40\,a^2\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3-80\,a^3\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2-32\,a^5-10\,a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^4+80\,a^4\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}{{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^5}\right )}{{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*acoth(tanh(a + b*x))^3),x)

[Out]

(4/(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x) + (32*
b*x)/(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^2 -
(288*b^2*x^2)/((log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2
*b*x)*((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b
*x)^2 - 4*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1))
+ 2*b*x) + 4*a^2)) + (384*b^3*x^3)/((log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)
*exp(2*b*x) - 1)) + 2*b*x)^2*((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)
*exp(2*b*x) - 1)) + 2*b*x)^2 - 4*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp
(2*a)*exp(2*b*x) - 1)) + 2*b*x) + 4*a^2)))/(x^3*(8*a*b - 4*b*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(
2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)) + x^2*((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)
*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^2 - 4*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp
(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x) + 4*a^2) + 4*b^2*x^4) - (384*b^2*atanh((4*
b*x*((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x
)^2 - 4*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) +
2*b*x) + 4*a^2))/(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) +
2*b*x)^3 - ((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1))
+ 2*b*x)^5 + 40*a^2*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*exp(2*b*
x) - 1)) + 2*b*x)^3 - 80*a^3*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + log(-2/(exp(2*a)*
exp(2*b*x) - 1)) + 2*b*x)^2 - 32*a^5 - 10*a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + lo
g(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^4 + 80*a^4*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) -
1)) + log(-2/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x))/(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b
*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^5))/(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/
(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^5

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