3.3.9 \(\int x^{3/2} \tanh ^{-1}(\sqrt {x}) \, dx\) [209]

Optimal. Leaf size=38 \[ \frac {x}{5}+\frac {x^2}{10}+\frac {2}{5} x^{5/2} \tanh ^{-1}\left (\sqrt {x}\right )+\frac {1}{5} \log (1-x) \]

[Out]

1/5*x+1/10*x^2+2/5*x^(5/2)*arctanh(x^(1/2))+1/5*ln(1-x)

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6037, 45} \begin {gather*} \frac {2}{5} x^{5/2} \tanh ^{-1}\left (\sqrt {x}\right )+\frac {x^2}{10}+\frac {x}{5}+\frac {1}{5} \log (1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*ArcTanh[Sqrt[x]],x]

[Out]

x/5 + x^2/10 + (2*x^(5/2)*ArcTanh[Sqrt[x]])/5 + Log[1 - x]/5

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6037

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTanh[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1
]

Rubi steps

\begin {align*} \int x^{3/2} \tanh ^{-1}\left (\sqrt {x}\right ) \, dx &=\frac {2}{5} x^{5/2} \tanh ^{-1}\left (\sqrt {x}\right )-\frac {1}{5} \int \frac {x^2}{1-x} \, dx\\ &=\frac {2}{5} x^{5/2} \tanh ^{-1}\left (\sqrt {x}\right )-\frac {1}{5} \int \left (-1+\frac {1}{1-x}-x\right ) \, dx\\ &=\frac {x}{5}+\frac {x^2}{10}+\frac {2}{5} x^{5/2} \tanh ^{-1}\left (\sqrt {x}\right )+\frac {1}{5} \log (1-x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.82 \begin {gather*} \frac {1}{10} \left (x (2+x)+4 x^{5/2} \tanh ^{-1}\left (\sqrt {x}\right )+2 \log (1-x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*ArcTanh[Sqrt[x]],x]

[Out]

(x*(2 + x) + 4*x^(5/2)*ArcTanh[Sqrt[x]] + 2*Log[1 - x])/10

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Maple [A]
time = 0.07, size = 35, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {2 x^{\frac {5}{2}} \arctanh \left (\sqrt {x}\right )}{5}+\frac {x^{2}}{10}+\frac {x}{5}+\frac {\ln \left (\sqrt {x}-1\right )}{5}+\frac {\ln \left (\sqrt {x}+1\right )}{5}\)	\(35\)
default	\(\frac {2 x^{\frac {5}{2}} \arctanh \left (\sqrt {x}\right )}{5}+\frac {x^{2}}{10}+\frac {x}{5}+\frac {\ln \left (\sqrt {x}-1\right )}{5}+\frac {\ln \left (\sqrt {x}+1\right )}{5}\)	\(35\)
meijerg	\(\frac {x \left (3 x +6\right )}{30}-\frac {x^{\frac {5}{2}} \left (\ln \left (1-\sqrt {x}\right )-\ln \left (\sqrt {x}+1\right )\right )}{5}+\frac {\ln \left (1-x \right )}{5}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*arctanh(x^(1/2)),x,method=_RETURNVERBOSE)

[Out]

2/5*x^(5/2)*arctanh(x^(1/2))+1/10*x^2+1/5*x+1/5*ln(x^(1/2)-1)+1/5*ln(x^(1/2)+1)

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Maxima [A]
time = 0.27, size = 24, normalized size = 0.63 \begin {gather*} \frac {2}{5} \, x^{\frac {5}{2}} \operatorname {artanh}\left (\sqrt {x}\right ) + \frac {1}{10} \, x^{2} + \frac {1}{5} \, x + \frac {1}{5} \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*arctanh(x^(1/2)),x, algorithm="maxima")

[Out]

2/5*x^(5/2)*arctanh(sqrt(x)) + 1/10*x^2 + 1/5*x + 1/5*log(x - 1)

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Fricas [A]
time = 0.38, size = 36, normalized size = 0.95 \begin {gather*} \frac {1}{5} \, x^{\frac {5}{2}} \log \left (-\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \frac {1}{10} \, x^{2} + \frac {1}{5} \, x + \frac {1}{5} \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*arctanh(x^(1/2)),x, algorithm="fricas")

[Out]

1/5*x^(5/2)*log(-(x + 2*sqrt(x) + 1)/(x - 1)) + 1/10*x^2 + 1/5*x + 1/5*log(x - 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (29) = 58\).
time = 1.04, size = 121, normalized size = 3.18 \begin {gather*} \frac {4 x^{\frac {7}{2}} \operatorname {atanh}{\left (\sqrt {x} \right )}}{10 x - 10} - \frac {4 x^{\frac {5}{2}} \operatorname {atanh}{\left (\sqrt {x} \right )}}{10 x - 10} + \frac {x^{3}}{10 x - 10} + \frac {x^{2}}{10 x - 10} + \frac {4 x \log {\left (\sqrt {x} + 1 \right )}}{10 x - 10} - \frac {4 x \operatorname {atanh}{\left (\sqrt {x} \right )}}{10 x - 10} - \frac {4 \log {\left (\sqrt {x} + 1 \right )}}{10 x - 10} + \frac {4 \operatorname {atanh}{\left (\sqrt {x} \right )}}{10 x - 10} - \frac {2}{10 x - 10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*atanh(x**(1/2)),x)

[Out]

4*x**(7/2)*atanh(sqrt(x))/(10*x - 10) - 4*x**(5/2)*atanh(sqrt(x))/(10*x - 10) + x**3/(10*x - 10) + x**2/(10*x
- 10) + 4*x*log(sqrt(x) + 1)/(10*x - 10) - 4*x*atanh(sqrt(x))/(10*x - 10) - 4*log(sqrt(x) + 1)/(10*x - 10) + 4
*atanh(sqrt(x))/(10*x - 10) - 2/(10*x - 10)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (26) = 52\).
time = 0.41, size = 170, normalized size = 4.47 \begin {gather*} \frac {8 \, {\left (\frac {{\left (\sqrt {x} + 1\right )}^{3}}{{\left (\sqrt {x} - 1\right )}^{3}} - \frac {{\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + \frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}}{5 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{4}} + \frac {2 \, {\left (\frac {5 \, {\left (\sqrt {x} + 1\right )}^{4}}{{\left (\sqrt {x} - 1\right )}^{4}} + \frac {10 \, {\left (\sqrt {x} + 1\right )}^{2}}{{\left (\sqrt {x} - 1\right )}^{2}} + 1\right )} \log \left (-\frac {\sqrt {x} + 1}{\sqrt {x} - 1}\right )}{5 \, {\left (\frac {\sqrt {x} + 1}{\sqrt {x} - 1} - 1\right )}^{5}} + \frac {2}{5} \, \log \left (\frac {\sqrt {x} + 1}{{\left | \sqrt {x} - 1 \right |}}\right ) - \frac {2}{5} \, \log \left ({\left | -\frac {\sqrt {x} + 1}{\sqrt {x} - 1} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*arctanh(x^(1/2)),x, algorithm="giac")

[Out]

8/5*((sqrt(x) + 1)^3/(sqrt(x) - 1)^3 - (sqrt(x) + 1)^2/(sqrt(x) - 1)^2 + (sqrt(x) + 1)/(sqrt(x) - 1))/((sqrt(x
) + 1)/(sqrt(x) - 1) - 1)^4 + 2/5*(5*(sqrt(x) + 1)^4/(sqrt(x) - 1)^4 + 10*(sqrt(x) + 1)^2/(sqrt(x) - 1)^2 + 1)
*log(-(sqrt(x) + 1)/(sqrt(x) - 1))/((sqrt(x) + 1)/(sqrt(x) - 1) - 1)^5 + 2/5*log((sqrt(x) + 1)/abs(sqrt(x) - 1
)) - 2/5*log(abs(-(sqrt(x) + 1)/(sqrt(x) - 1) + 1))

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Mupad [B]
time = 0.86, size = 24, normalized size = 0.63 \begin {gather*} \frac {x}{5}+\frac {\ln \left (x-1\right )}{5}+\frac {2\,x^{5/2}\,\mathrm {atanh}\left (\sqrt {x}\right )}{5}+\frac {x^2}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*atanh(x^(1/2)),x)

[Out]

x/5 + log(x - 1)/5 + (2*x^(5/2)*atanh(x^(1/2)))/5 + x^2/10

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