3.3.2 \(\int x^3 (a+b \tanh ^{-1}(c \sqrt {x}))^3 \, dx\) [202]

Optimal. Leaf size=374 \[ \frac {47 b^3 \sqrt {x}}{70 c^7}+\frac {23 b^3 x^{3/2}}{420 c^5}+\frac {b^3 x^{5/2}}{140 c^3}-\frac {47 b^3 \tanh ^{-1}\left (c \sqrt {x}\right )}{70 c^8}+\frac {71 b^2 x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{140 c^6}+\frac {9 b^2 x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{70 c^4}+\frac {b^2 x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{28 c^2}+\frac {44 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{35 c^8}+\frac {3 b \sqrt {x} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{4 c^7}+\frac {b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{4 c^5}+\frac {3 b x^{5/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{20 c^3}+\frac {3 b x^{7/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{28 c}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{4 c^8}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {88 b^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{35 c^8}-\frac {44 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{35 c^8} \]

[Out]

23/420*b^3*x^(3/2)/c^5+1/140*b^3*x^(5/2)/c^3-47/70*b^3*arctanh(c*x^(1/2))/c^8+71/140*b^2*x*(a+b*arctanh(c*x^(1
/2)))/c^6+9/70*b^2*x^2*(a+b*arctanh(c*x^(1/2)))/c^4+1/28*b^2*x^3*(a+b*arctanh(c*x^(1/2)))/c^2+44/35*b*(a+b*arc
tanh(c*x^(1/2)))^2/c^8+1/4*b*x^(3/2)*(a+b*arctanh(c*x^(1/2)))^2/c^5+3/20*b*x^(5/2)*(a+b*arctanh(c*x^(1/2)))^2/
c^3+3/28*b*x^(7/2)*(a+b*arctanh(c*x^(1/2)))^2/c-1/4*(a+b*arctanh(c*x^(1/2)))^3/c^8+1/4*x^4*(a+b*arctanh(c*x^(1
/2)))^3-88/35*b^2*(a+b*arctanh(c*x^(1/2)))*ln(2/(1-c*x^(1/2)))/c^8-44/35*b^3*polylog(2,1-2/(1-c*x^(1/2)))/c^8+
47/70*b^3*x^(1/2)/c^7+3/4*b*(a+b*arctanh(c*x^(1/2)))^2*x^(1/2)/c^7

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Rubi [A]
time = 1.09, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 54, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6039, 6037, 6127, 308, 212, 327, 6131, 6055, 2449, 2352, 6021, 6095} \begin {gather*} -\frac {88 b^2 \log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{35 c^8}+\frac {71 b^2 x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{140 c^6}+\frac {9 b^2 x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{70 c^4}+\frac {b^2 x^3 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{28 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{4 c^8}+\frac {44 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{35 c^8}+\frac {3 b \sqrt {x} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{4 c^7}+\frac {b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{4 c^5}+\frac {3 b x^{5/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{20 c^3}+\frac {3 b x^{7/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{28 c}+\frac {1}{4} x^4 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {44 b^3 \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{35 c^8}-\frac {47 b^3 \tanh ^{-1}\left (c \sqrt {x}\right )}{70 c^8}+\frac {47 b^3 \sqrt {x}}{70 c^7}+\frac {23 b^3 x^{3/2}}{420 c^5}+\frac {b^3 x^{5/2}}{140 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(47*b^3*Sqrt[x])/(70*c^7) + (23*b^3*x^(3/2))/(420*c^5) + (b^3*x^(5/2))/(140*c^3) - (47*b^3*ArcTanh[c*Sqrt[x]])
/(70*c^8) + (71*b^2*x*(a + b*ArcTanh[c*Sqrt[x]]))/(140*c^6) + (9*b^2*x^2*(a + b*ArcTanh[c*Sqrt[x]]))/(70*c^4)
+ (b^2*x^3*(a + b*ArcTanh[c*Sqrt[x]]))/(28*c^2) + (44*b*(a + b*ArcTanh[c*Sqrt[x]])^2)/(35*c^8) + (3*b*Sqrt[x]*
(a + b*ArcTanh[c*Sqrt[x]])^2)/(4*c^7) + (b*x^(3/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(4*c^5) + (3*b*x^(5/2)*(a + b
*ArcTanh[c*Sqrt[x]])^2)/(20*c^3) + (3*b*x^(7/2)*(a + b*ArcTanh[c*Sqrt[x]])^2)/(28*c) - (a + b*ArcTanh[c*Sqrt[x
]])^3/(4*c^8) + (x^4*(a + b*ArcTanh[c*Sqrt[x]])^3)/4 - (88*b^2*(a + b*ArcTanh[c*Sqrt[x]])*Log[2/(1 - c*Sqrt[x]
)])/(35*c^8) - (44*b^3*PolyLog[2, 1 - 2/(1 - c*Sqrt[x])])/(35*c^8)

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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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

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Mathematica [A]
time = 0.83, size = 418, normalized size = 1.12 \begin {gather*} \frac {-564 a b^2+630 a^2 b c \sqrt {x}+564 b^3 c \sqrt {x}+426 a b^2 c^2 x+210 a^2 b c^3 x^{3/2}+46 b^3 c^3 x^{3/2}+108 a b^2 c^4 x^2+126 a^2 b c^5 x^{5/2}+6 b^3 c^5 x^{5/2}+30 a b^2 c^6 x^3+90 a^2 b c^7 x^{7/2}+210 a^3 c^8 x^4+6 b^2 \left (b \left (-176+105 c \sqrt {x}+35 c^3 x^{3/2}+21 c^5 x^{5/2}+15 c^7 x^{7/2}\right )+105 a \left (-1+c^8 x^4\right )\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^2+210 b^3 \left (-1+c^8 x^4\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^3+6 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (105 a^2 c^8 x^4+b^2 \left (-94+71 c^2 x+18 c^4 x^2+5 c^6 x^3\right )+2 a b c \sqrt {x} \left (105+35 c^2 x+21 c^4 x^2+15 c^6 x^3\right )-352 b^2 \log \left (1+e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+315 a^2 b \log \left (1-c \sqrt {x}\right )-315 a^2 b \log \left (1+c \sqrt {x}\right )+1056 a b^2 \log \left (1-c^2 x\right )+1056 b^3 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )}{840 c^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-564*a*b^2 + 630*a^2*b*c*Sqrt[x] + 564*b^3*c*Sqrt[x] + 426*a*b^2*c^2*x + 210*a^2*b*c^3*x^(3/2) + 46*b^3*c^3*x
^(3/2) + 108*a*b^2*c^4*x^2 + 126*a^2*b*c^5*x^(5/2) + 6*b^3*c^5*x^(5/2) + 30*a*b^2*c^6*x^3 + 90*a^2*b*c^7*x^(7/
2) + 210*a^3*c^8*x^4 + 6*b^2*(b*(-176 + 105*c*Sqrt[x] + 35*c^3*x^(3/2) + 21*c^5*x^(5/2) + 15*c^7*x^(7/2)) + 10
5*a*(-1 + c^8*x^4))*ArcTanh[c*Sqrt[x]]^2 + 210*b^3*(-1 + c^8*x^4)*ArcTanh[c*Sqrt[x]]^3 + 6*b*ArcTanh[c*Sqrt[x]
]*(105*a^2*c^8*x^4 + b^2*(-94 + 71*c^2*x + 18*c^4*x^2 + 5*c^6*x^3) + 2*a*b*c*Sqrt[x]*(105 + 35*c^2*x + 21*c^4*
x^2 + 15*c^6*x^3) - 352*b^2*Log[1 + E^(-2*ArcTanh[c*Sqrt[x]])]) + 315*a^2*b*Log[1 - c*Sqrt[x]] - 315*a^2*b*Log
[1 + c*Sqrt[x]] + 1056*a*b^2*Log[1 - c^2*x] + 1056*b^3*PolyLog[2, -E^(-2*ArcTanh[c*Sqrt[x]])])/(840*c^8)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 19.47, size = 1431, normalized size = 3.83

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1431\)
default	\(\text {Expression too large to display}\)	\(1431\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/c^8*(-3/32*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)))+3/16*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))^2-11/30*b^3+3/8*a*b^2*c^8*x^4*arctanh(c*x
^(1/2))^2+3/28*a*b^2*arctanh(c*x^(1/2))*c^7*x^(7/2)+3/20*a*b^2*arctanh(c*x^(1/2))*c^5*x^(5/2)+9/140*a*b^2*c^4*
x^2+3/56*a^2*b*c^7*x^(7/2)+3/40*a^2*b*c^5*x^(5/2)+1/8*a^2*b*c^3*x^(3/2)+3/8*a^2*b*c*x^(1/2)-3/16*I*b^3*arctanh
(c*x^(1/2))^2*Pi+3/8*a*b^2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)-3/8*a*b^2*arctanh(c*x^(1/2))*ln(1+c*x^(1/2))-3/1
6*a*b^2*ln(c*x^(1/2)-1)*ln(1/2*c*x^(1/2)+1/2)-3/16*a*b^2*ln(-1/2*c*x^(1/2)+1/2)*ln(1+c*x^(1/2))+3/16*a*b^2*ln(
-1/2*c*x^(1/2)+1/2)*ln(1/2*c*x^(1/2)+1/2)+1/56*a*b^2*c^6*x^3+9/140*b^3*arctanh(c*x^(1/2))*c^4*x^2+71/280*b^3*a
rctanh(c*x^(1/2))*c^2*x+1/8*b^3*c^8*x^4*arctanh(c*x^(1/2))^3+3/56*b^3*arctanh(c*x^(1/2))^2*c^7*x^(7/2)+3/40*b^
3*arctanh(c*x^(1/2))^2*c^5*x^(5/2)+1/8*b^3*arctanh(c*x^(1/2))^2*c^3*x^(3/2)+3/8*b^3*arctanh(c*x^(1/2))^2*c*x^(
1/2)+1/56*b^3*arctanh(c*x^(1/2))*c^6*x^3-3/32*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+3/32*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))+3/32*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+1/280*b^3*c^
5*x^(5/2)+23/840*b^3*c^3*x^(3/2)+47/140*b^3*c*x^(1/2)+1/8*c^8*x^4*a^3+3/32*a*b^2*ln(c*x^(1/2)-1)^2+3/32*a*b^2*
ln(1+c*x^(1/2))^2+22/35*a*b^2*ln(c*x^(1/2)-1)+22/35*a*b^2*ln(1+c*x^(1/2))+3/16*a^2*b*ln(c*x^(1/2)-1)-3/16*a^2*
b*ln(1+c*x^(1/2))-44/35*b^3*arctanh(c*x^(1/2))*ln(1+I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-44/35*b^3*arctanh(c*x^(1
/2))*ln(1-I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))+3/16*b^3*arctanh(c*x^(1/2))^2*ln(c*x^(1/2)-1)-3/16*b^3*arctanh(c*x
^(1/2))^2*ln(1+c*x^(1/2))+3/8*b^3*arctanh(c*x^(1/2))^2*ln((1+c*x^(1/2))/(-c^2*x+1)^(1/2))+71/280*a*b^2*x*c^2+3
/32*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+3/32*I*b^
3*arctanh(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+(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+(1+c*x^(1/2))^2/(-c^2*x+1)))^2+3/8*a^
2*b*c^8*x^4*arctanh(c*x^(1/2))+1/4*a*b^2*arctanh(c*x^(1/2))*c^3*x^(3/2)+3/4*a*b^2*arctanh(c*x^(1/2))*c*x^(1/2)
+22/35*b^3*arctanh(c*x^(1/2))^2-1/8*b^3*arctanh(c*x^(1/2))^3-44/35*b^3*dilog(1-I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2
))-44/35*b^3*dilog(1+I*(1+c*x^(1/2))/(-c^2*x+1)^(1/2))-47/140*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. 1972 vs. \(2 (297) = 594\).
time = 0.85, size = 1972, normalized size = 5.27 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^3*x^4 - 1/26880*a*b^2*c*((315*c^7*x^4 + 500*c^5*x^3 + 1002*c^3*x^2 + 3684*c*x - 12*(105*c^7*x^4 + 120*c^
6*x^(7/2) + 140*c^5*x^3 + 168*c^4*x^(5/2) + 210*c^3*x^2 + 280*c^2*x^(3/2) + 420*c*x + 840*sqrt(x))*log(c*sqrt(
x) + 1))/c^8 - 6396*log(c*sqrt(x) + 1)/c^9 - 6396*log(c*sqrt(x) - 1)/c^9) - 1/2240*(840*x^4*log(c*sqrt(x) + 1)
- c*((105*c^7*x^4 - 120*c^6*x^(7/2) + 140*c^5*x^3 - 168*c^4*x^(5/2) + 210*c^3*x^2 - 280*c^2*x^(3/2) + 420*c*x
- 840*sqrt(x))/c^8 + 840*log(c*sqrt(x) + 1)/c^9))*a*b^2*log(-c*sqrt(x) + 1) + 1/2240*(840*x^4*log(c*sqrt(x) +
1) - c*((105*c^7*x^4 - 120*c^6*x^(7/2) + 140*c^5*x^3 - 168*c^4*x^(5/2) + 210*c^3*x^2 - 280*c^2*x^(3/2) + 420*
c*x - 840*sqrt(x))/c^8 + 840*log(c*sqrt(x) + 1)/c^9))*a^2*b - 1/2240*(840*x^4*log(-c*sqrt(x) + 1) - c*((105*c^
7*x^4 + 120*c^6*x^(7/2) + 140*c^5*x^3 + 168*c^4*x^(5/2) + 210*c^3*x^2 + 280*c^2*x^(3/2) + 420*c*x + 840*sqrt(x
))/c^8 + 840*log(c*sqrt(x) - 1)/c^9))*a^2*b + 1/1881600*(11025*(32*log(-c*sqrt(x) + 1)^2 - 8*log(-c*sqrt(x) +
1) + 1)*(c*sqrt(x) - 1)^8 + 57600*(49*log(-c*sqrt(x) + 1)^2 - 14*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^7 +
548800*(18*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^6 + 790272*(25*log(-c*sqrt(x) +
1)^2 - 10*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^5 + 3087000*(8*log(-c*sqrt(x) + 1)^2 - 4*log(-c*sqrt(x) + 1
) + 1)*(c*sqrt(x) - 1)^4 + 2195200*(9*log(-c*sqrt(x) + 1)^2 - 6*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1)^3 + 4
939200*(2*log(-c*sqrt(x) + 1)^2 - 2*log(-c*sqrt(x) + 1) + 1)*(c*sqrt(x) - 1)^2 + 2822400*(log(-c*sqrt(x) + 1)^
2 - 2*log(-c*sqrt(x) + 1) + 2)*(c*sqrt(x) - 1))*a*b^2/c^8 - 1/3161088000*(385875*(256*log(-c*sqrt(x) + 1)^3 -
96*log(-c*sqrt(x) + 1)^2 + 24*log(-c*sqrt(x) + 1) - 3)*(c*sqrt(x) - 1)^8 + 2304000*(343*log(-c*sqrt(x) + 1)^3
- 147*log(-c*sqrt(x) + 1)^2 + 42*log(-c*sqrt(x) + 1) - 6)*(c*sqrt(x) - 1)^7 + 76832000*(36*log(-c*sqrt(x) + 1)
^3 - 18*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 1)*(c*sqrt(x) - 1)^6 + 44255232*(125*log(-c*sqrt(x) +
1)^3 - 75*log(-c*sqrt(x) + 1)^2 + 30*log(-c*sqrt(x) + 1) - 6)*(c*sqrt(x) - 1)^5 + 216090000*(32*log(-c*sqrt(x)
+ 1)^3 - 24*log(-c*sqrt(x) + 1)^2 + 12*log(-c*sqrt(x) + 1) - 3)*(c*sqrt(x) - 1)^4 + 614656000*(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 + 691488000*(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 + 790272000*(log(-c*sqrt(x)
+ 1)^3 - 3*log(-c*sqrt(x) + 1)^2 + 6*log(-c*sqrt(x) + 1) - 6)*(c*sqrt(x) - 1))*b^3/c^8 + 44/35*(log(c*sqrt(x)
+ 1)*log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*sqrt(x) + 1/2))*b^3/c^8 - 1881559/3763200*b^3*log(c*sqrt(x) - 1)
/c^8 + 1/2240*(2283*a*b^2 - 752*b^3)*log(c*sqrt(x) + 1)/c^8 + 1/3161088000*(1157625*(16*a*b^2*c^8 - b^3*c^8)*x
^4 - 27000*(1680*a*b^2*c^7 + 169*b^3*c^7)*x^(7/2) + 3500*(24528*a*b^2*c^6 - 3565*b^3*c^6)*x^3 + 98784000*(b^3*
c^8*x^4 - b^3)*log(c*sqrt(x) + 1)^3 - 168*(895440*a*b^2*c^5 + 44269*b^3*c^5)*x^(5/2) + 210*(1248240*a*b^2*c^4
- 334699*b^3*c^4)*x^2 + 5644800*(105*a*b^2*c^8*x^4 + 15*b^3*c^7*x^(7/2) + 21*b^3*c^5*x^(5/2) + 35*b^3*c^3*x^(3
/2) + 105*b^3*c*sqrt(x) - 105*a*b^2 + 176*b^3)*log(c*sqrt(x) + 1)^2 - 352800*(105*b^3*c^8*x^4 - 120*b^3*c^7*x^
(7/2) + 140*b^3*c^6*x^3 - 168*b^3*c^5*x^(5/2) + 210*b^3*c^4*x^2 - 280*b^3*c^3*x^(3/2) + 420*b^3*c^2*x - 840*b^
3*c*sqrt(x) + 533*b^3 - 840*(b^3*c^8*x^4 - b^3)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1)^2 - 280*(1718640*a*b^2
*c^3 + 2899*b^3*c^3)*x^(3/2) + 420*(2424240*a*b^2*c^2 - 1227199*b^3*c^2)*x - 1411200*(105*a*b^2*c^8*x^4 - 120*
a*b^2*c^7*x^(7/2) - 168*a*b^2*c^5*x^(5/2) - 280*a*b^2*c^3*x^(3/2) - 840*a*b^2*c*sqrt(x) + 20*(7*a*b^2*c^6 - 2*
b^3*c^6)*x^3 + 6*(35*a*b^2*c^4 - 24*b^3*c^4)*x^2 + 4*(105*a*b^2*c^2 - 142*b^3*c^2)*x)*log(c*sqrt(x) + 1) + 840
*(11025*b^3*c^8*x^4 + 27000*b^3*c^7*x^(7/2) - 16100*b^3*c^6*x^3 + 89544*b^3*c^5*x^(5/2) - 85890*b^3*c^4*x^2 +
286440*b^3*c^3*x^(3/2) - 348180*b^3*c^2*x + 1917720*b^3*c*sqrt(x) - 352800*(b^3*c^8*x^4 - b^3)*log(c*sqrt(x) +
1)^2 - 13440*(15*b^3*c^7*x^(7/2) + 21*b^3*c^5*x^(5/2) + 35*b^3*c^3*x^(3/2) + 105*b^3*c*sqrt(x) + 176*b^3)*log
(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1) - 840*(3835440*a*b^2*c + 618199*b^3*c)*sqrt(x))/c^8

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \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**3*(a+b*atanh(c*x**(1/2)))**3,x)

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\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^3*(a + b*atanh(c*x^(1/2)))^3,x)

[Out]

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

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