3.3.4 \(\int x (a+b \tanh ^{-1}(c \sqrt {x}))^3 \, dx\) [204]

Optimal. Leaf size=234 \[ \frac {b^3 \sqrt {x}}{2 c^3}-\frac {b^3 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4}+\frac {b^2 x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}+\frac {2 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{c^4}+\frac {3 b \sqrt {x} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 c^3}+\frac {b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{2 c^4}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {4 b^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^4}-\frac {2 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{c^4} \]

[Out]

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

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Rubi [A]
time = 0.42, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6039, 6037, 6127, 327, 212, 6131, 6055, 2449, 2352, 6021, 6095} \begin {gather*} -\frac {4 b^2 \log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^4}+\frac {b^2 x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}+\frac {2 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{c^4}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{2 c^4}+\frac {3 b \sqrt {x} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 c^3}+\frac {b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 c}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {2 b^3 \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{c^4}-\frac {b^3 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4}+\frac {b^3 \sqrt {x}}{2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcTanh[c*Sqrt[x]])^3,x]

[Out]

(b^3*Sqrt[x])/(2*c^3) - (b^3*ArcTanh[c*Sqrt[x]])/(2*c^4) + (b^2*x*(a + b*ArcTanh[c*Sqrt[x]]))/(2*c^2) + (2*b*(
a + b*ArcTanh[c*Sqrt[x]])^2)/c^4 + (3*b*Sqrt[x]*(a + b*ArcTanh[c*Sqrt[x]])^2)/(2*c^3) + (b*x^(3/2)*(a + b*ArcT
anh[c*Sqrt[x]])^2)/(2*c) - (a + b*ArcTanh[c*Sqrt[x]])^3/(2*c^4) + (x^2*(a + b*ArcTanh[c*Sqrt[x]])^3)/2 - (4*b^
2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x])])/c^4 - (2*b^3*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/c^4

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 327

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

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 6055

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

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]

Rule 6131

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

Rubi steps

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

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Mathematica [A]
time = 0.38, size = 285, normalized size = 1.22 \begin {gather*} \frac {-2 a b^2+6 a^2 b c \sqrt {x}+2 b^3 c \sqrt {x}+2 a b^2 c^2 x+2 a^2 b c^3 x^{3/2}+2 a^3 c^4 x^2+2 b^2 \left (b \left (-4+3 c \sqrt {x}+c^3 x^{3/2}\right )+3 a \left (-1+c^4 x^2\right )\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^2+2 b^3 \left (-1+c^4 x^2\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^3+2 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (3 a^2 c^4 x^2+b^2 \left (-1+c^2 x\right )+2 a b c \sqrt {x} \left (3+c^2 x\right )-8 b^2 \log \left (1+e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+3 a^2 b \log \left (1-c \sqrt {x}\right )-3 a^2 b \log \left (1+c \sqrt {x}\right )+8 a b^2 \log \left (1-c^2 x\right )+8 b^3 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )}{4 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*b^2 + 6*a^2*b*c*Sqrt[x] + 2*b^3*c*Sqrt[x] + 2*a*b^2*c^2*x + 2*a^2*b*c^3*x^(3/2) + 2*a^3*c^4*x^2 + 2*b^2*
(b*(-4 + 3*c*Sqrt[x] + c^3*x^(3/2)) + 3*a*(-1 + c^4*x^2))*ArcTanh[c*Sqrt[x]]^2 + 2*b^3*(-1 + c^4*x^2)*ArcTanh[
c*Sqrt[x]]^3 + 2*b*ArcTanh[c*Sqrt[x]]*(3*a^2*c^4*x^2 + b^2*(-1 + c^2*x) + 2*a*b*c*Sqrt[x]*(3 + c^2*x) - 8*b^2*
Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 3*a^2*b*Log[1 - c*Sqrt[x]] - 3*a^2*b*Log[1 + c*Sqrt[x]] + 8*a*b^2*Log[1
- c^2*x] + 8*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(4*c^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 7.33, size = 1250, normalized size = 5.34

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1250\)
default	\(\text {Expression too large to display}\)	\(1250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arctanh(c*x^(1/2)))^3,x,method=_RETURNVERBOSE)

[Out]

2/c^4*(3/16*I*b^3*arctanh(c*x^(1/2))^2*Pi*csgn(I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))^2*csgn(I*(1+c*x^(1/2))^2/(c^2
*x-1))-1/4*b^3+1/4*a^2*b*c^3*x^(3/2)+3/4*a^2*b*c*x^(1/2)+3/4*a*b^2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)-3/4*a*b^
2*arctanh(c*x^(1/2))*ln(1+c*x^(1/2))-3/8*a*b^2*ln(c*x^(1/2)-1)*ln(1/2*c*x^(1/2)+1/2)-3/8*a*b^2*ln(-1/2*c*x^(1/
2)+1/2)*ln(1+c*x^(1/2))+3/8*a*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1/2*c*x^(1/2)+1/2)+1/4*b^3*arctanh(c*x^(1/2))*c^2*
x+1/4*b^3*arctanh(c*x^(1/2))^2*c^3*x^(3/2)+3/4*b^3*arctanh(c*x^(1/2))^2*c*x^(1/2)+1/4*b^3*c*x^(1/2)+3/16*a*b^2
*ln(c*x^(1/2)-1)^2+3/16*a*b^2*ln(1+c*x^(1/2))^2+a*b^2*ln(c*x^(1/2)-1)+a*b^2*ln(1+c*x^(1/2))+3/8*a^2*b*ln(c*x^(
1/2)-1)-3/8*a^2*b*ln(1+c*x^(1/2))-2*b^3*arctanh(c*x^(1/2))*ln(1+I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-2*b^3*arctan
h(c*x^(1/2))*ln(1-I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+3/8*b^3*arctanh(c*x^(1/2))^2*ln(c*x^(1/2)-1)-3/8*b^3*arcta
nh(c*x^(1/2))^2*ln(1+c*x^(1/2))+3/4*b^3*arctanh(c*x^(1/2))^2*ln((1+c*x^(1/2))/(-c^2*x+1)^(1/2))+1/4*a*b^2*x*c^
2+3/16*I*b^3*arctanh(c*x^(1/2))^2*Pi*csgn(I/(1+(1+c*x^(1/2))^2/(-c^2*x+1)))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1)/(
1+(1+c*x^(1/2))^2/(-c^2*x+1)))^2-3/16*I*b^3*arctanh(c*x^(1/2))^2*Pi*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))*csgn(I*(
1+c*x^(1/2))^2/(c^2*x-1)/(1+(1+c*x^(1/2))^2/(-c^2*x+1)))^2+3/8*I*b^3*arctanh(c*x^(1/2))^2*Pi*csgn(I*(1+c*x^(1/
2))/(-c^2*x+1)^(1/2))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))^2+1/2*a*b^2*arctanh(c*x^(1/2))*c^3*x^(3/2)+3/2*a*b^2*a
rctanh(c*x^(1/2))*c*x^(1/2)-3/16*I*b^3*arctanh(c*x^(1/2))^2*Pi*csgn(I/(1+(1+c*x^(1/2))^2/(-c^2*x+1)))*csgn(I*(
1+c*x^(1/2))^2/(c^2*x-1))*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1)/(1+(1+c*x^(1/2))^2/(-c^2*x+1)))+3/8*I*b^3*arctanh(c
*x^(1/2))^2*Pi*csgn(I/(1+(1+c*x^(1/2))^2/(-c^2*x+1)))^2-3/8*I*b^3*arctanh(c*x^(1/2))^2*Pi*csgn(I/(1+(1+c*x^(1/
2))^2/(-c^2*x+1)))^3+3/4*a^2*b*c^4*x^2*arctanh(c*x^(1/2))+3/4*a*b^2*c^4*x^2*arctanh(c*x^(1/2))^2+3/16*I*b^3*ar
ctanh(c*x^(1/2))^2*Pi*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))^3+3/16*I*b^3*arctanh(c*x^(1/2))^2*Pi*csgn(I*(1+c*x^(1/
2))^2/(c^2*x-1)/(1+(1+c*x^(1/2))^2/(-c^2*x+1)))^3+1/4*b^3*c^4*x^2*arctanh(c*x^(1/2))^3-3/8*I*b^3*arctanh(c*x^(
1/2))^2*Pi+1/4*c^4*x^2*a^3+b^3*arctanh(c*x^(1/2))^2-1/4*b^3*arctanh(c*x^(1/2))^3-2*b^3*dilog(1-I*(1+c*x^(1/2))
/(-c^2*x+1)^(1/2))-2*b^3*dilog(1+I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-1/4*b^3*arctanh(c*x^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1184 vs. \(2 (191) = 382\).
time = 0.69, size = 1184, normalized size = 5.06 \begin {gather*} \frac {1}{2} \, a^{3} x^{2} - \frac {1}{32} \, a b^{2} c {\left (\frac {3 \, c^{3} x^{2} + 10 \, c x - 2 \, {\left (3 \, c^{3} x^{2} + 4 \, c^{2} x^{\frac {3}{2}} + 6 \, c x + 12 \, \sqrt {x}\right )} \log \left (c \sqrt {x} + 1\right )}{c^{4}} - \frac {14 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}} - \frac {14 \, \log \left (c \sqrt {x} - 1\right )}{c^{5}}\right )} - \frac {1}{16} \, {\left (12 \, x^{2} \log \left (c \sqrt {x} + 1\right ) - c {\left (\frac {3 \, c^{3} x^{2} - 4 \, c^{2} x^{\frac {3}{2}} + 6 \, c x - 12 \, \sqrt {x}}{c^{4}} + \frac {12 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}}\right )}\right )} a b^{2} \log \left (-c \sqrt {x} + 1\right ) + \frac {1}{16} \, {\left (12 \, x^{2} \log \left (c \sqrt {x} + 1\right ) - c {\left (\frac {3 \, c^{3} x^{2} - 4 \, c^{2} x^{\frac {3}{2}} + 6 \, c x - 12 \, \sqrt {x}}{c^{4}} + \frac {12 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}}\right )}\right )} a^{2} b - \frac {1}{16} \, {\left (12 \, x^{2} \log \left (-c \sqrt {x} + 1\right ) - c {\left (\frac {3 \, c^{3} x^{2} + 4 \, c^{2} x^{\frac {3}{2}} + 6 \, c x + 12 \, \sqrt {x}}{c^{4}} + \frac {12 \, \log \left (c \sqrt {x} - 1\right )}{c^{5}}\right )}\right )} a^{2} b + \frac {{\left (9 \, {\left (8 \, \log \left (-c \sqrt {x} + 1\right )^{2} - 4 \, \log \left (-c \sqrt {x} + 1\right ) + 1\right )} {\left (c \sqrt {x} - 1\right )}^{4} + 32 \, {\left (9 \, \log \left (-c \sqrt {x} + 1\right )^{2} - 6 \, \log \left (-c \sqrt {x} + 1\right ) + 2\right )} {\left (c \sqrt {x} - 1\right )}^{3} + 216 \, {\left (2 \, \log \left (-c \sqrt {x} + 1\right )^{2} - 2 \, \log \left (-c \sqrt {x} + 1\right ) + 1\right )} {\left (c \sqrt {x} - 1\right )}^{2} + 288 \, {\left (\log \left (-c \sqrt {x} + 1\right )^{2} - 2 \, \log \left (-c \sqrt {x} + 1\right ) + 2\right )} {\left (c \sqrt {x} - 1\right )}\right )} a b^{2}}{192 \, c^{4}} - \frac {{\left (9 \, {\left (32 \, \log \left (-c \sqrt {x} + 1\right )^{3} - 24 \, \log \left (-c \sqrt {x} + 1\right )^{2} + 12 \, \log \left (-c \sqrt {x} + 1\right ) - 3\right )} {\left (c \sqrt {x} - 1\right )}^{4} + 128 \, {\left (9 \, \log \left (-c \sqrt {x} + 1\right )^{3} - 9 \, \log \left (-c \sqrt {x} + 1\right )^{2} + 6 \, \log \left (-c \sqrt {x} + 1\right ) - 2\right )} {\left (c \sqrt {x} - 1\right )}^{3} + 432 \, {\left (4 \, \log \left (-c \sqrt {x} + 1\right )^{3} - 6 \, \log \left (-c \sqrt {x} + 1\right )^{2} + 6 \, \log \left (-c \sqrt {x} + 1\right ) - 3\right )} {\left (c \sqrt {x} - 1\right )}^{2} + 1152 \, {\left (\log \left (-c \sqrt {x} + 1\right )^{3} - 3 \, \log \left (-c \sqrt {x} + 1\right )^{2} + 6 \, \log \left (-c \sqrt {x} + 1\right ) - 6\right )} {\left (c \sqrt {x} - 1\right )}\right )} b^{3}}{4608 \, c^{4}} + \frac {2 \, {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b^{3}}{c^{4}} - \frac {319 \, b^{3} \log \left (c \sqrt {x} - 1\right )}{384 \, c^{4}} + \frac {{\left (25 \, a b^{2} - 4 \, b^{3}\right )} \log \left (c \sqrt {x} + 1\right )}{16 \, c^{4}} + \frac {288 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{3} + 27 \, {\left (8 \, a b^{2} c^{4} - b^{3} c^{4}\right )} x^{2} + 576 \, {\left (3 \, a b^{2} c^{4} x^{2} + b^{3} c^{3} x^{\frac {3}{2}} + 3 \, b^{3} c \sqrt {x} - 3 \, a b^{2} + 4 \, b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{2} - 72 \, {\left (3 \, b^{3} c^{4} x^{2} - 4 \, b^{3} c^{3} x^{\frac {3}{2}} + 6 \, b^{3} c^{2} x - 12 \, b^{3} c \sqrt {x} + 7 \, b^{3} - 12 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )^{2} - 4 \, {\left (168 \, a b^{2} c^{3} + 37 \, b^{3} c^{3}\right )} x^{\frac {3}{2}} + 6 \, {\left (312 \, a b^{2} c^{2} - 115 \, b^{3} c^{2}\right )} x - 288 \, {\left (3 \, a b^{2} c^{4} x^{2} - 4 \, a b^{2} c^{3} x^{\frac {3}{2}} - 12 \, a b^{2} c \sqrt {x} + 2 \, {\left (3 \, a b^{2} c^{2} - 2 \, b^{3} c^{2}\right )} x\right )} \log \left (c \sqrt {x} + 1\right ) + 12 \, {\left (9 \, b^{3} c^{4} x^{2} + 28 \, b^{3} c^{3} x^{\frac {3}{2}} - 18 \, b^{3} c^{2} x + 300 \, b^{3} c \sqrt {x} - 72 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{2} - 96 \, {\left (b^{3} c^{3} x^{\frac {3}{2}} + 3 \, b^{3} c \sqrt {x} + 4 \, b^{3}\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right ) - 12 \, {\left (600 \, a b^{2} c + 223 \, b^{3} c\right )} \sqrt {x}}{4608 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^3*x^2 - 1/32*a*b^2*c*((3*c^3*x^2 + 10*c*x - 2*(3*c^3*x^2 + 4*c^2*x^(3/2) + 6*c*x + 12*sqrt(x))*log(c*sqr
t(x) + 1))/c^4 - 14*log(c*sqrt(x) + 1)/c^5 - 14*log(c*sqrt(x) - 1)/c^5) - 1/16*(12*x^2*log(c*sqrt(x) + 1) - c*
((3*c^3*x^2 - 4*c^2*x^(3/2) + 6*c*x - 12*sqrt(x))/c^4 + 12*log(c*sqrt(x) + 1)/c^5))*a*b^2*log(-c*sqrt(x) + 1)
+ 1/16*(12*x^2*log(c*sqrt(x) + 1) - c*((3*c^3*x^2 - 4*c^2*x^(3/2) + 6*c*x - 12*sqrt(x))/c^4 + 12*log(c*sqrt(x)
+ 1)/c^5))*a^2*b - 1/16*(12*x^2*log(-c*sqrt(x) + 1) - c*((3*c^3*x^2 + 4*c^2*x^(3/2) + 6*c*x + 12*sqrt(x))/c^4
+ 12*log(c*sqrt(x) - 1)/c^5))*a^2*b + 1/192*(9*(8*log(-c*sqrt(x) + 1)^2 - 4*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(
x) - 1)^4 + 32*(9*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^3 + 216*(2*log(-c*sqrt(x)
+ 1)^2 - 2*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^2 + 288*(log(-c*sqrt(x) + 1)^2 - 2*log(-c*sqrt(x) + 1) +
2)*(c*sqrt(x) - 1))*a*b^2/c^4 - 1/4608*(9*(32*log(-c*sqrt(x) + 1)^3 - 24*log(-c*sqrt(x) + 1)^2 + 12*log(-c*sqr
t(x) + 1) - 3)*(c*sqrt(x) - 1)^4 + 128*(9*log(-c*sqrt(x) + 1)^3 - 9*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) +
1) - 2)*(c*sqrt(x) - 1)^3 + 432*(4*log(-c*sqrt(x) + 1)^3 - 6*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) -
3)*(c*sqrt(x) - 1)^2 + 1152*(log(-c*sqrt(x) + 1)^3 - 3*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 6)*(c*s
qrt(x) - 1))*b^3/c^4 + 2*(log(c*sqrt(x) + 1)*log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*sqrt(x) + 1/2))*b^3/c^4 -
319/384*b^3*log(c*sqrt(x) - 1)/c^4 + 1/16*(25*a*b^2 - 4*b^3)*log(c*sqrt(x) + 1)/c^4 + 1/4608*(288*(b^3*c^4*x^
2 - b^3)*log(c*sqrt(x) + 1)^3 + 27*(8*a*b^2*c^4 - b^3*c^4)*x^2 + 576*(3*a*b^2*c^4*x^2 + b^3*c^3*x^(3/2) + 3*b^
3*c*sqrt(x) - 3*a*b^2 + 4*b^3)*log(c*sqrt(x) + 1)^2 - 72*(3*b^3*c^4*x^2 - 4*b^3*c^3*x^(3/2) + 6*b^3*c^2*x - 12
*b^3*c*sqrt(x) + 7*b^3 - 12*(b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1)^2 - 4*(168*a*b^2*c^3 +
37*b^3*c^3)*x^(3/2) + 6*(312*a*b^2*c^2 - 115*b^3*c^2)*x - 288*(3*a*b^2*c^4*x^2 - 4*a*b^2*c^3*x^(3/2) - 12*a*b
^2*c*sqrt(x) + 2*(3*a*b^2*c^2 - 2*b^3*c^2)*x)*log(c*sqrt(x) + 1) + 12*(9*b^3*c^4*x^2 + 28*b^3*c^3*x^(3/2) - 18
*b^3*c^2*x + 300*b^3*c*sqrt(x) - 72*(b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1)^2 - 96*(b^3*c^3*x^(3/2) + 3*b^3*c*s
qrt(x) + 4*b^3)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1) - 12*(600*a*b^2*c + 223*b^3*c)*sqrt(x))/c^4

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Fricas [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*(a+b*arctanh(c*x^(1/2)))^3,x, algorithm="fricas")

[Out]

integral(b^3*x*arctanh(c*sqrt(x))^3 + 3*a*b^2*x*arctanh(c*sqrt(x))^2 + 3*a^2*b*x*arctanh(c*sqrt(x)) + a^3*x, x
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*atanh(c*x**(1/2)))**3,x)

[Out]

Integral(x*(a + b*atanh(c*sqrt(x)))**3, x)

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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*(a+b*arctanh(c*x^(1/2)))^3,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*atanh(c*x^(1/2)))^3,x)

[Out]

int(x*(a + b*atanh(c*x^(1/2)))^3, x)

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