3.161 \(\int \frac {x}{\tanh ^{-1}(\tanh (a+b x))^{5/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac {4}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {2 x}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}} \]

[Out]

-2/3*x/b/arctanh(tanh(b*x+a))^(3/2)-4/3/b^2/arctanh(tanh(b*x+a))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2168, 2157, 30} \[ -\frac {4}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}-\frac {2 x}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x/ArcTanh[Tanh[a + b*x]]^(5/2),x]

[Out]

(-2*x)/(3*b*ArcTanh[Tanh[a + b*x]]^(3/2)) - 4/(3*b^2*Sqrt[ArcTanh[Tanh[a + b*x]]])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2157

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 2168

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 {x}{\tanh ^{-1}(\tanh (a+b x))^{5/2}} \, dx &=-\frac {2 x}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {2 \int \frac {1}{\tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx}{3 b}\\ &=-\frac {2 x}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{3 b^2}\\ &=-\frac {2 x}{3 b \tanh ^{-1}(\tanh (a+b x))^{3/2}}-\frac {4}{3 b^2 \sqrt {\tanh ^{-1}(\tanh (a+b x))}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 31, normalized size = 0.82 \[ -\frac {2 \left (2 \tanh ^{-1}(\tanh (a+b x))+b x\right )}{3 b^2 \tanh ^{-1}(\tanh (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/ArcTanh[Tanh[a + b*x]]^(5/2),x]

[Out]

(-2*(b*x + 2*ArcTanh[Tanh[a + b*x]]))/(3*b^2*ArcTanh[Tanh[a + b*x]]^(3/2))

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 41, normalized size = 1.08 \[ -\frac {2 \, {\left (3 \, b x + 2 \, a\right )} \sqrt {b x + a}}{3 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arctanh(tanh(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

-2/3*(3*b*x + 2*a)*sqrt(b*x + a)/(b^4*x^2 + 2*a*b^3*x + a^2*b^2)

________________________________________________________________________________________

giac [A]  time = 0.23, size = 20, normalized size = 0.53 \[ -\frac {2 \, {\left (3 \, b x + 2 \, a\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arctanh(tanh(b*x+a))^(5/2),x, algorithm="giac")

[Out]

-2/3*(3*b*x + 2*a)/((b*x + a)^(3/2)*b^2)

________________________________________________________________________________________

maple [A]  time = 0.15, size = 42, normalized size = 1.11 \[ \frac {-\frac {2}{\sqrt {\arctanh \left (\tanh \left (b x +a \right )\right )}}-\frac {2 \left (b x -\arctanh \left (\tanh \left (b x +a \right )\right )\right )}{3 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {3}{2}}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/arctanh(tanh(b*x+a))^(5/2),x)

[Out]

2/b^2*(-1/arctanh(tanh(b*x+a))^(1/2)-1/3*(b*x-arctanh(tanh(b*x+a)))/arctanh(tanh(b*x+a))^(3/2))

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 31, normalized size = 0.82 \[ -\frac {2 \, {\left (3 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}\right )}}{3 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/arctanh(tanh(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

-2/3*(3*b^2*x^2 + 5*a*b*x + 2*a^2)/((b*x + a)^(5/2)*b^2)

________________________________________________________________________________________

mupad [B]  time = 1.38, size = 152, normalized size = 4.00 \[ -\frac {8\,\sqrt {\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}-\frac {\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{2}}\,\left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+b\,x\right )}{3\,b^2\,{\left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/atanh(tanh(a + b*x))^(5/2),x)

[Out]

-(8*(log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))/2 - log(2/(exp(2*a)*exp(2*b*x) + 1))/2)^(1/2)*(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)) + b*x))/(3*b^2*(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)

________________________________________________________________________________________

sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/atanh(tanh(b*x+a))**(5/2),x)

[Out]

Exception raised: TypeError

________________________________________________________________________________________