3.170 \(\int \sqrt {x} \tanh ^{-1}(\tanh (a+b x)) \, dx\)

Optimal. Leaf size=27 \[ \frac {2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))-\frac {4}{15} b x^{5/2} \]

[Out]

-4/15*b*x^(5/2)+2/3*x^(3/2)*arctanh(tanh(b*x+a))

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 30} \[ \frac {2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))-\frac {4}{15} b x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*ArcTanh[Tanh[a + b*x]],x]

[Out]

(-4*b*x^(5/2))/15 + (2*x^(3/2)*ArcTanh[Tanh[a + b*x]])/3

Rule 30

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

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 \sqrt {x} \tanh ^{-1}(\tanh (a+b x)) \, dx &=\frac {2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))-\frac {1}{3} (2 b) \int x^{3/2} \, dx\\ &=-\frac {4}{15} b x^{5/2}+\frac {2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 23, normalized size = 0.85 \[ \frac {2}{15} x^{3/2} \left (5 \tanh ^{-1}(\tanh (a+b x))-2 b x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*ArcTanh[Tanh[a + b*x]],x]

[Out]

(2*x^(3/2)*(-2*b*x + 5*ArcTanh[Tanh[a + b*x]]))/15

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fricas [A]  time = 0.57, size = 16, normalized size = 0.59 \[ \frac {2}{15} \, {\left (3 \, b x^{2} + 5 \, a x\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*b*x^2 + 5*a*x)*sqrt(x)

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giac [A]  time = 1.50, size = 13, normalized size = 0.48 \[ \frac {2}{5} \, b x^{\frac {5}{2}} + \frac {2}{3} \, a x^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*x^(5/2) + 2/3*a*x^(3/2)

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maple [A]  time = 0.24, size = 20, normalized size = 0.74 \[ -\frac {4 b \,x^{\frac {5}{2}}}{15}+\frac {2 x^{\frac {3}{2}} \arctanh \left (\tanh \left (b x +a \right )\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*b*x^(5/2)+2/3*x^(3/2)*arctanh(tanh(b*x+a))

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maxima [A]  time = 0.33, size = 19, normalized size = 0.70 \[ -\frac {4}{15} \, b x^{\frac {5}{2}} + \frac {2}{3} \, x^{\frac {3}{2}} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*b*x^(5/2) + 2/3*x^(3/2)*arctanh(tanh(b*x + a))

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mupad [B]  time = 1.09, size = 57, normalized size = 2.11 \[ \frac {x^{3/2}\,\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{3}-\frac {4\,b\,x^{5/2}}{15}-\frac {x^{3/2}\,\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*atanh(tanh(a + b*x)),x)

[Out]

(x^(3/2)*log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)))/3 - (4*b*x^(5/2))/15 - (x^(3/2)*log(1/(exp(2*a)
*exp(2*b*x) + 1)))/3

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(tanh(b*x+a))*x**(1/2),x)

[Out]

Integral(sqrt(x)*atanh(tanh(a + b*x)), x)

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