Integrand size = 16, antiderivative size = 129 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \, dx=\frac {a b \sqrt {x}}{c^3}+\frac {b^2 x}{6 c^2}+\frac {b^2 \sqrt {x} \text {arctanh}\left (c \sqrt {x}\right )}{c^3}+\frac {b x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{3 c}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 c^4}+\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2+\frac {2 b^2 \log \left (1-c^2 x\right )}{3 c^4} \] Output:
a*b*x^(1/2)/c^3+1/6*b^2*x/c^2+b^2*x^(1/2)*arctanh(c*x^(1/2))/c^3+1/3*b*x^( 3/2)*(a+b*arctanh(c*x^(1/2)))/c-1/2*(a+b*arctanh(c*x^(1/2)))^2/c^4+1/2*x^2 *(a+b*arctanh(c*x^(1/2)))^2+2/3*b^2*ln(-c^2*x+1)/c^4
Time = 0.06 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.24 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \, dx=\frac {6 a b c \sqrt {x}+b^2 c^2 x+2 a b c^3 x^{3/2}+3 a^2 c^4 x^2+2 b c \sqrt {x} \left (3 a c^3 x^{3/2}+b \left (3+c^2 x\right )\right ) \text {arctanh}\left (c \sqrt {x}\right )+3 b^2 \left (-1+c^4 x^2\right ) \text {arctanh}\left (c \sqrt {x}\right )^2+b (3 a+4 b) \log \left (1-c \sqrt {x}\right )-3 a b \log \left (1+c \sqrt {x}\right )+4 b^2 \log \left (1+c \sqrt {x}\right )}{6 c^4} \] Input:
Integrate[x*(a + b*ArcTanh[c*Sqrt[x]])^2,x]
Output:
(6*a*b*c*Sqrt[x] + b^2*c^2*x + 2*a*b*c^3*x^(3/2) + 3*a^2*c^4*x^2 + 2*b*c*S qrt[x]*(3*a*c^3*x^(3/2) + b*(3 + c^2*x))*ArcTanh[c*Sqrt[x]] + 3*b^2*(-1 + c^4*x^2)*ArcTanh[c*Sqrt[x]]^2 + b*(3*a + 4*b)*Log[1 - c*Sqrt[x]] - 3*a*b*L og[1 + c*Sqrt[x]] + 4*b^2*Log[1 + c*Sqrt[x]])/(6*c^4)
Time = 1.06 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.23, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6454, 6452, 6542, 6452, 243, 49, 2009, 6542, 2009, 6510}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 6454 |
| \(\displaystyle 2 \int x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2d\sqrt {x}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {1}{2} b c \int \frac {x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {1}{2} b c \left (\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {1}{2} b c \left (\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{3} b c \int \frac {x^{3/2}}{1-c^2 x}d\sqrt {x}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {1}{2} b c \left (\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \int \frac {x}{1-c^2 x}dx}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {1}{2} b c \left (\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (c^2 x-1\right )}\right )dx}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {1}{2} b c \left (\frac {\int \frac {x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (-\frac {x}{c^2}-\frac {\log \left (1-c^2 x\right )}{c^4}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {1}{2} b c \left (\frac {\frac {\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {\int \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )d\sqrt {x}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (-\frac {x}{c^2}-\frac {\log \left (1-c^2 x\right )}{c^4}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {1}{2} b c \left (\frac {\frac {\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {a \sqrt {x}+b \sqrt {x} \text {arctanh}\left (c \sqrt {x}\right )+\frac {b \log \left (1-c^2 x\right )}{2 c}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (-\frac {x}{c^2}-\frac {\log \left (1-c^2 x\right )}{c^4}\right )}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle 2 \left (\frac {1}{4} x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {1}{2} b c \left (\frac {\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b c^3}-\frac {a \sqrt {x}+b \sqrt {x} \text {arctanh}\left (c \sqrt {x}\right )+\frac {b \log \left (1-c^2 x\right )}{2 c}}{c^2}}{c^2}-\frac {\frac {1}{3} x^{3/2} \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (-\frac {x}{c^2}-\frac {\log \left (1-c^2 x\right )}{c^4}\right )}{c^2}\right )\right )\) |
Input:
Int[x*(a + b*ArcTanh[c*Sqrt[x]])^2,x]
Output:
2*((x^2*(a + b*ArcTanh[c*Sqrt[x]])^2)/4 - (b*c*(-(((x^(3/2)*(a + b*ArcTanh [c*Sqrt[x]]))/3 - (b*c*(-(x/c^2) - Log[1 - c^2*x]/c^4))/6)/c^2) + ((a + b* ArcTanh[c*Sqrt[x]])^2/(2*b*c^3) - (a*Sqrt[x] + b*Sqrt[x]*ArcTanh[c*Sqrt[x] ] + (b*Log[1 - c^2*x])/(2*c))/c^2)/c^2))/2)
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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] - Simp[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]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[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[Simpl ify[(m + 1)/n]]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> 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]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[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]
Leaf count of result is larger than twice the leaf count of optimal. \(248\) vs. \(2(105)=210\).
Time = 1.04 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.93
| method | result | size |
| parts | \(\frac {a^{2} x^{2}}{2}+\frac {2 b^{2} \left (\frac {c^{4} x^{2} \operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{4}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{3} x^{\frac {3}{2}}}{6}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c \sqrt {x}}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{4}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{4}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{16}+\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{16}-\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}+\frac {c^{2} x}{12}+\frac {\ln \left (c \sqrt {x}-1\right )}{3}+\frac {\ln \left (1+c \sqrt {x}\right )}{3}\right )}{c^{4}}+\frac {4 a b \left (\frac {c^{4} x^{2} \operatorname {arctanh}\left (c \sqrt {x}\right )}{4}+\frac {c^{3} x^{\frac {3}{2}}}{12}+\frac {c \sqrt {x}}{4}+\frac {\ln \left (c \sqrt {x}-1\right )}{8}-\frac {\ln \left (1+c \sqrt {x}\right )}{8}\right )}{c^{4}}\) | \(249\) |
| derivativedivides | \(\frac {\frac {a^{2} c^{4} x^{2}}{2}+2 b^{2} \left (\frac {c^{4} x^{2} \operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{4}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{3} x^{\frac {3}{2}}}{6}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c \sqrt {x}}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{4}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{4}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{16}+\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{16}-\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}+\frac {c^{2} x}{12}+\frac {\ln \left (c \sqrt {x}-1\right )}{3}+\frac {\ln \left (1+c \sqrt {x}\right )}{3}\right )+4 a b \left (\frac {c^{4} x^{2} \operatorname {arctanh}\left (c \sqrt {x}\right )}{4}+\frac {c^{3} x^{\frac {3}{2}}}{12}+\frac {c \sqrt {x}}{4}+\frac {\ln \left (c \sqrt {x}-1\right )}{8}-\frac {\ln \left (1+c \sqrt {x}\right )}{8}\right )}{c^{4}}\) | \(250\) |
| default | \(\frac {\frac {a^{2} c^{4} x^{2}}{2}+2 b^{2} \left (\frac {c^{4} x^{2} \operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{4}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c^{3} x^{\frac {3}{2}}}{6}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) c \sqrt {x}}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{4}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{4}-\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}+\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{16}+\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{16}-\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}+\frac {c^{2} x}{12}+\frac {\ln \left (c \sqrt {x}-1\right )}{3}+\frac {\ln \left (1+c \sqrt {x}\right )}{3}\right )+4 a b \left (\frac {c^{4} x^{2} \operatorname {arctanh}\left (c \sqrt {x}\right )}{4}+\frac {c^{3} x^{\frac {3}{2}}}{12}+\frac {c \sqrt {x}}{4}+\frac {\ln \left (c \sqrt {x}-1\right )}{8}-\frac {\ln \left (1+c \sqrt {x}\right )}{8}\right )}{c^{4}}\) | \(250\) |
Input:
int(x*(a+b*arctanh(c*x^(1/2)))^2,x,method=_RETURNVERBOSE)
Output:
1/2*a^2*x^2+2*b^2/c^4*(1/4*c^4*x^2*arctanh(c*x^(1/2))^2+1/6*arctanh(c*x^(1 /2))*c^3*x^(3/2)+1/2*arctanh(c*x^(1/2))*c*x^(1/2)+1/4*arctanh(c*x^(1/2))*l n(c*x^(1/2)-1)-1/4*arctanh(c*x^(1/2))*ln(1+c*x^(1/2))-1/8*ln(c*x^(1/2)-1)* ln(1/2*c*x^(1/2)+1/2)+1/16*ln(c*x^(1/2)-1)^2+1/16*ln(1+c*x^(1/2))^2-1/8*(l n(1+c*x^(1/2))-ln(1/2*c*x^(1/2)+1/2))*ln(-1/2*c*x^(1/2)+1/2)+1/12*c^2*x+1/ 3*ln(c*x^(1/2)-1)+1/3*ln(1+c*x^(1/2)))+4*a*b/c^4*(1/4*c^4*x^2*arctanh(c*x^ (1/2))+1/12*c^3*x^(3/2)+1/4*c*x^(1/2)+1/8*ln(c*x^(1/2)-1)-1/8*ln(1+c*x^(1/ 2)))
Time = 0.11 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.60 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \, dx=\frac {12 \, a^{2} c^{4} x^{2} + 4 \, b^{2} c^{2} x + 3 \, {\left (b^{2} c^{4} x^{2} - b^{2}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )^{2} + 4 \, {\left (3 \, a b c^{4} - 3 \, a b + 4 \, b^{2}\right )} \log \left (c \sqrt {x} + 1\right ) - 4 \, {\left (3 \, a b c^{4} - 3 \, a b - 4 \, b^{2}\right )} \log \left (c \sqrt {x} - 1\right ) + 4 \, {\left (3 \, a b c^{4} x^{2} - 3 \, a b c^{4} + {\left (b^{2} c^{3} x + 3 \, b^{2} c\right )} \sqrt {x}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) + 8 \, {\left (a b c^{3} x + 3 \, a b c\right )} \sqrt {x}}{24 \, c^{4}} \] Input:
integrate(x*(a+b*arctanh(c*x^(1/2)))^2,x, algorithm="fricas")
Output:
1/24*(12*a^2*c^4*x^2 + 4*b^2*c^2*x + 3*(b^2*c^4*x^2 - b^2)*log(-(c^2*x + 2 *c*sqrt(x) + 1)/(c^2*x - 1))^2 + 4*(3*a*b*c^4 - 3*a*b + 4*b^2)*log(c*sqrt( x) + 1) - 4*(3*a*b*c^4 - 3*a*b - 4*b^2)*log(c*sqrt(x) - 1) + 4*(3*a*b*c^4* x^2 - 3*a*b*c^4 + (b^2*c^3*x + 3*b^2*c)*sqrt(x))*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1)) + 8*(a*b*c^3*x + 3*a*b*c)*sqrt(x))/c^4
\[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \, dx=\int x \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{2}\, dx \] Input:
integrate(x*(a+b*atanh(c*x**(1/2)))**2,x)
Output:
Integral(x*(a + b*atanh(c*sqrt(x)))**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (105) = 210\).
Time = 0.03 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.67 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + \frac {1}{6} \, {\left (6 \, x^{2} \operatorname {artanh}\left (c \sqrt {x}\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{\frac {3}{2}} + 3 \, \sqrt {x}\right )}}{c^{4}} - \frac {3 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}} + \frac {3 \, \log \left (c \sqrt {x} - 1\right )}{c^{5}}\right )}\right )} a b + \frac {1}{24} \, {\left (4 \, c {\left (\frac {2 \, {\left (c^{2} x^{\frac {3}{2}} + 3 \, \sqrt {x}\right )}}{c^{4}} - \frac {3 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}} + \frac {3 \, \log \left (c \sqrt {x} - 1\right )}{c^{5}}\right )} \operatorname {artanh}\left (c \sqrt {x}\right ) + \frac {4 \, c^{2} x - 2 \, {\left (3 \, \log \left (c \sqrt {x} - 1\right ) - 8\right )} \log \left (c \sqrt {x} + 1\right ) + 3 \, \log \left (c \sqrt {x} + 1\right )^{2} + 3 \, \log \left (c \sqrt {x} - 1\right )^{2} + 16 \, \log \left (c \sqrt {x} - 1\right )}{c^{4}}\right )} b^{2} \] Input:
integrate(x*(a+b*arctanh(c*x^(1/2)))^2,x, algorithm="maxima")
Output:
1/2*b^2*x^2*arctanh(c*sqrt(x))^2 + 1/2*a^2*x^2 + 1/6*(6*x^2*arctanh(c*sqrt (x)) + c*(2*(c^2*x^(3/2) + 3*sqrt(x))/c^4 - 3*log(c*sqrt(x) + 1)/c^5 + 3*l og(c*sqrt(x) - 1)/c^5))*a*b + 1/24*(4*c*(2*(c^2*x^(3/2) + 3*sqrt(x))/c^4 - 3*log(c*sqrt(x) + 1)/c^5 + 3*log(c*sqrt(x) - 1)/c^5)*arctanh(c*sqrt(x)) + (4*c^2*x - 2*(3*log(c*sqrt(x) - 1) - 8)*log(c*sqrt(x) + 1) + 3*log(c*sqrt (x) + 1)^2 + 3*log(c*sqrt(x) - 1)^2 + 16*log(c*sqrt(x) - 1))/c^4)*b^2
\[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(a+b*arctanh(c*x^(1/2)))^2,x, algorithm="giac")
Output:
integrate((b*arctanh(c*sqrt(x)) + a)^2*x, x)
Time = 3.98 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.11 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \, dx=\frac {a^2\,x^2}{2}-\frac {b^2\,{\mathrm {atanh}\left (c\,\sqrt {x}\right )}^2}{2\,c^4}+\frac {2\,b^2\,\ln \left (c^2\,x-1\right )}{3\,c^4}+\frac {b^2\,x^2\,{\mathrm {atanh}\left (c\,\sqrt {x}\right )}^2}{2}+\frac {b^2\,x}{6\,c^2}+\frac {b^2\,x^{3/2}\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{3\,c}+\frac {b^2\,\sqrt {x}\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{c^3}+\frac {a\,b\,x^{3/2}}{3\,c}+\frac {a\,b\,\sqrt {x}}{c^3}-\frac {a\,b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{c^4}+a\,b\,x^2\,\mathrm {atanh}\left (c\,\sqrt {x}\right ) \] Input:
int(x*(a + b*atanh(c*x^(1/2)))^2,x)
Output:
(a^2*x^2)/2 - (b^2*atanh(c*x^(1/2))^2)/(2*c^4) + (2*b^2*log(c^2*x - 1))/(3 *c^4) + (b^2*x^2*atanh(c*x^(1/2))^2)/2 + (b^2*x)/(6*c^2) + (b^2*x^(3/2)*at anh(c*x^(1/2)))/(3*c) + (b^2*x^(1/2)*atanh(c*x^(1/2)))/c^3 + (a*b*x^(3/2)) /(3*c) + (a*b*x^(1/2))/c^3 - (a*b*atanh(c*x^(1/2)))/c^4 + a*b*x^2*atanh(c* x^(1/2))
Time = 0.17 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.14 \[ \int x \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2 \, dx=\frac {3 \mathit {atanh} \left (\sqrt {x}\, c \right )^{2} b^{2} c^{4} x^{2}-3 \mathit {atanh} \left (\sqrt {x}\, c \right )^{2} b^{2}+2 \sqrt {x}\, \mathit {atanh} \left (\sqrt {x}\, c \right ) b^{2} c^{3} x +6 \sqrt {x}\, \mathit {atanh} \left (\sqrt {x}\, c \right ) b^{2} c +6 \mathit {atanh} \left (\sqrt {x}\, c \right ) a b \,c^{4} x^{2}-6 \mathit {atanh} \left (\sqrt {x}\, c \right ) a b +8 \mathit {atanh} \left (\sqrt {x}\, c \right ) b^{2}+2 \sqrt {x}\, a b \,c^{3} x +6 \sqrt {x}\, a b c +8 \,\mathrm {log}\left (\sqrt {x}\, c -1\right ) b^{2}+3 a^{2} c^{4} x^{2}+b^{2} c^{2} x}{6 c^{4}} \] Input:
int(x*(a+b*atanh(c*x^(1/2)))^2,x)
Output:
(3*atanh(sqrt(x)*c)**2*b**2*c**4*x**2 - 3*atanh(sqrt(x)*c)**2*b**2 + 2*sqr t(x)*atanh(sqrt(x)*c)*b**2*c**3*x + 6*sqrt(x)*atanh(sqrt(x)*c)*b**2*c + 6* atanh(sqrt(x)*c)*a*b*c**4*x**2 - 6*atanh(sqrt(x)*c)*a*b + 8*atanh(sqrt(x)* c)*b**2 + 2*sqrt(x)*a*b*c**3*x + 6*sqrt(x)*a*b*c + 8*log(sqrt(x)*c - 1)*b* *2 + 3*a**2*c**4*x**2 + b**2*c**2*x)/(6*c**4)