3.3.21 \(\int x^2 (a+b \tanh ^{-1}(c x^{3/2}))^2 \, dx\) [221]

Optimal. Leaf size=101 \[ \frac {2 a b x^{3/2}}{3 c}+\frac {2 b^2 x^{3/2} \tanh ^{-1}\left (c x^{3/2}\right )}{3 c}-\frac {\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{3 c^2}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2+\frac {b^2 \log \left (1-c^2 x^3\right )}{3 c^2} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6039, 6037, 6127, 6021, 266, 6095} \begin {gather*} -\frac {\left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2}{3 c^2}+\frac {2 a b x^{3/2}}{3 c}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2+\frac {b^2 \log \left (1-c^2 x^3\right )}{3 c^2}+\frac {2 b^2 x^{3/2} \tanh ^{-1}\left (c x^{3/2}\right )}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTanh[c*x^(3/2)])^2,x]

[Out]

(2*a*b*x^(3/2))/(3*c) + (2*b^2*x^(3/2)*ArcTanh[c*x^(3/2)])/(3*c) - (a + b*ArcTanh[c*x^(3/2)])^2/(3*c^2) + (x^3
*(a + b*ArcTanh[c*x^(3/2)])^2)/3 + (b^2*Log[1 - c^2*x^3])/(3*c^2)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 6021

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

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
]

Rule 6039

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m
+ 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[S
implify[(m + 1)/n]]

Rule 6095

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]

Rule 6127

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])
^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rubi steps

\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \, dx &=\int x^2 \left (a+b \tanh ^{-1}\left (c x^{3/2}\right )\right )^2 \, dx\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 122, normalized size = 1.21 \begin {gather*} \frac {2 a b c x^{3/2}+a^2 c^2 x^3+2 b c x^{3/2} \left (b+a c x^{3/2}\right ) \tanh ^{-1}\left (c x^{3/2}\right )+b^2 \left (-1+c^2 x^3\right ) \tanh ^{-1}\left (c x^{3/2}\right )^2+b (a+b) \log \left (1-c x^{3/2}\right )-a b \log \left (1+c x^{3/2}\right )+b^2 \log \left (1+c x^{3/2}\right )}{3 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*ArcTanh[c*x^(3/2)])^2,x]

[Out]

(2*a*b*c*x^(3/2) + a^2*c^2*x^3 + 2*b*c*x^(3/2)*(b + a*c*x^(3/2))*ArcTanh[c*x^(3/2)] + b^2*(-1 + c^2*x^3)*ArcTa
nh[c*x^(3/2)]^2 + b*(a + b)*Log[1 - c*x^(3/2)] - a*b*Log[1 + c*x^(3/2)] + b^2*Log[1 + c*x^(3/2)])/(3*c^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(257\) vs. \(2(81)=162\).
time = 0.27, size = 258, normalized size = 2.55

method	result	size
derivativedivides	\(\frac {\frac {c^{2} x^{3} a^{2}}{3}+\frac {b^{2} c^{2} x^{3} \arctanh \left (c \,x^{\frac {3}{2}}\right )^{2}}{3}+\frac {2 b^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right ) c \,x^{\frac {3}{2}}}{3}+\frac {b^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}-1\right )}{3}-\frac {b^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3}-\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}-1\right ) \ln \left (\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )}{6}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}-1\right )^{2}}{12}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}-1\right )}{3}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}+1\right )^{2}}{12}-\frac {b^{2} \ln \left (-\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{6}+\frac {b^{2} \ln \left (-\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right ) \ln \left (\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )}{6}+\frac {2 a b \,c^{2} x^{3} \arctanh \left (c \,x^{\frac {3}{2}}\right )}{3}+\frac {2 c \,x^{\frac {3}{2}} a b}{3}+\frac {a b \ln \left (c \,x^{\frac {3}{2}}-1\right )}{3}-\frac {a b \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3}}{c^{2}}\)	\(258\)
default	\(\frac {\frac {c^{2} x^{3} a^{2}}{3}+\frac {b^{2} c^{2} x^{3} \arctanh \left (c \,x^{\frac {3}{2}}\right )^{2}}{3}+\frac {2 b^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right ) c \,x^{\frac {3}{2}}}{3}+\frac {b^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}-1\right )}{3}-\frac {b^{2} \arctanh \left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3}-\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}-1\right ) \ln \left (\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )}{6}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}-1\right )^{2}}{12}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}-1\right )}{3}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3}+\frac {b^{2} \ln \left (c \,x^{\frac {3}{2}}+1\right )^{2}}{12}-\frac {b^{2} \ln \left (-\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{6}+\frac {b^{2} \ln \left (-\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right ) \ln \left (\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )}{6}+\frac {2 a b \,c^{2} x^{3} \arctanh \left (c \,x^{\frac {3}{2}}\right )}{3}+\frac {2 c \,x^{\frac {3}{2}} a b}{3}+\frac {a b \ln \left (c \,x^{\frac {3}{2}}-1\right )}{3}-\frac {a b \ln \left (c \,x^{\frac {3}{2}}+1\right )}{3}}{c^{2}}\)	\(258\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arctanh(c*x^(3/2)))^2,x,method=_RETURNVERBOSE)

[Out]

2/3/c^2*(1/2*c^2*x^3*a^2+1/2*b^2*c^2*x^3*arctanh(c*x^(3/2))^2+b^2*arctanh(c*x^(3/2))*c*x^(3/2)+1/2*b^2*arctanh
(c*x^(3/2))*ln(c*x^(3/2)-1)-1/2*b^2*arctanh(c*x^(3/2))*ln(c*x^(3/2)+1)-1/4*b^2*ln(c*x^(3/2)-1)*ln(1/2*c*x^(3/2
)+1/2)+1/8*b^2*ln(c*x^(3/2)-1)^2+1/2*b^2*ln(c*x^(3/2)-1)+1/2*b^2*ln(c*x^(3/2)+1)+1/8*b^2*ln(c*x^(3/2)+1)^2-1/4
*b^2*ln(-1/2*c*x^(3/2)+1/2)*ln(c*x^(3/2)+1)+1/4*b^2*ln(-1/2*c*x^(3/2)+1/2)*ln(1/2*c*x^(3/2)+1/2)+a*b*c^2*x^3*a
rctanh(c*x^(3/2))+c*x^(3/2)*a*b+1/2*a*b*ln(c*x^(3/2)-1)-1/2*a*b*ln(c*x^(3/2)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (81) = 162\).
time = 0.26, size = 186, normalized size = 1.84 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )^{2} + \frac {1}{3} \, a^{2} x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + c {\left (\frac {2 \, x^{\frac {3}{2}}}{c^{2}} - \frac {\log \left (c x^{\frac {3}{2}} + 1\right )}{c^{3}} + \frac {\log \left (c x^{\frac {3}{2}} - 1\right )}{c^{3}}\right )}\right )} a b + \frac {1}{12} \, {\left (4 \, c {\left (\frac {2 \, x^{\frac {3}{2}}}{c^{2}} - \frac {\log \left (c x^{\frac {3}{2}} + 1\right )}{c^{3}} + \frac {\log \left (c x^{\frac {3}{2}} - 1\right )}{c^{3}}\right )} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) - \frac {2 \, {\left (\log \left (c x^{\frac {3}{2}} - 1\right ) - 2\right )} \log \left (c x^{\frac {3}{2}} + 1\right ) - \log \left (c x^{\frac {3}{2}} + 1\right )^{2} - \log \left (c x^{\frac {3}{2}} - 1\right )^{2} - 4 \, \log \left (c x^{\frac {3}{2}} - 1\right )}{c^{2}}\right )} b^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c*x^(3/2)))^2,x, algorithm="maxima")

[Out]

1/3*b^2*x^3*arctanh(c*x^(3/2))^2 + 1/3*a^2*x^3 + 1/3*(2*x^3*arctanh(c*x^(3/2)) + c*(2*x^(3/2)/c^2 - log(c*x^(3
/2) + 1)/c^3 + log(c*x^(3/2) - 1)/c^3))*a*b + 1/12*(4*c*(2*x^(3/2)/c^2 - log(c*x^(3/2) + 1)/c^3 + log(c*x^(3/2
) - 1)/c^3)*arctanh(c*x^(3/2)) - (2*(log(c*x^(3/2) - 1) - 2)*log(c*x^(3/2) + 1) - log(c*x^(3/2) + 1)^2 - log(c
*x^(3/2) - 1)^2 - 4*log(c*x^(3/2) - 1))/c^2)*b^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (81) = 162\).
time = 0.38, size = 179, normalized size = 1.77 \begin {gather*} \frac {4 \, a^{2} c^{2} x^{3} + 8 \, a b c x^{\frac {3}{2}} + {\left (b^{2} c^{2} x^{3} - b^{2}\right )} \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right )^{2} + 4 \, {\left (a b c^{2} - a b + b^{2}\right )} \log \left (c x^{\frac {3}{2}} + 1\right ) - 4 \, {\left (a b c^{2} - a b - b^{2}\right )} \log \left (c x^{\frac {3}{2}} - 1\right ) + 4 \, {\left (a b c^{2} x^{3} + b^{2} c x^{\frac {3}{2}} - a b c^{2}\right )} \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right )}{12 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c*x^(3/2)))^2,x, algorithm="fricas")

[Out]

1/12*(4*a^2*c^2*x^3 + 8*a*b*c*x^(3/2) + (b^2*c^2*x^3 - b^2)*log(-(c^2*x^3 + 2*c*x^(3/2) + 1)/(c^2*x^3 - 1))^2
+ 4*(a*b*c^2 - a*b + b^2)*log(c*x^(3/2) + 1) - 4*(a*b*c^2 - a*b - b^2)*log(c*x^(3/2) - 1) + 4*(a*b*c^2*x^3 + b
^2*c*x^(3/2) - a*b*c^2)*log(-(c^2*x^3 + 2*c*x^(3/2) + 1)/(c^2*x^3 - 1)))/c^2

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*atanh(c*x**(3/2)))**2,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c*x^(3/2)))^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^(3/2)) + a)^2*x^2, x)

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Mupad [B]
time = 1.31, size = 105, normalized size = 1.04 \begin {gather*} \frac {c\,\left (\frac {2\,b^2\,x^{3/2}\,\mathrm {atanh}\left (c\,x^{3/2}\right )}{3}+\frac {2\,a\,b\,x^{3/2}}{3}\right )-\frac {b^2\,{\mathrm {atanh}\left (c\,x^{3/2}\right )}^2}{3}+\frac {b^2\,\ln \left (c^2\,x^3-1\right )}{3}-\frac {2\,a\,b\,\mathrm {atanh}\left (c\,x^{3/2}\right )}{3}}{c^2}+\frac {a^2\,x^3}{3}+\frac {b^2\,x^3\,{\mathrm {atanh}\left (c\,x^{3/2}\right )}^2}{3}+\frac {2\,a\,b\,x^3\,\mathrm {atanh}\left (c\,x^{3/2}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*atanh(c*x^(3/2)))^2,x)

[Out]

(c*((2*b^2*x^(3/2)*atanh(c*x^(3/2)))/3 + (2*a*b*x^(3/2))/3) - (b^2*atanh(c*x^(3/2))^2)/3 + (b^2*log(c^2*x^3 -
1))/3 - (2*a*b*atanh(c*x^(3/2)))/3)/c^2 + (a^2*x^3)/3 + (b^2*x^3*atanh(c*x^(3/2))^2)/3 + (2*a*b*x^3*atanh(c*x^
(3/2)))/3

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