Integrand size = 18, antiderivative size = 234 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx=-\frac {b^3 c^3}{2 \sqrt {x}}+\frac {1}{2} b^3 c^4 \text {arctanh}\left (c \sqrt {x}\right )-\frac {b^2 c^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{2 x}+2 b c^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {b c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x^{3/2}}-\frac {3 b c^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} c^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{2 x^2}+4 b^2 c^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \log \left (2-\frac {2}{1+c \sqrt {x}}\right )-2 b^3 c^4 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c \sqrt {x}}\right ) \]
1/2*b^3*c^4*arctanh(c*x^(1/2))-1/2*b^2*c^2*(a+b*arctanh(c*x^(1/2)))/x+2*b* c^4*(a+b*arctanh(c*x^(1/2)))^2-1/2*b*c*(a+b*arctanh(c*x^(1/2)))^2/x^(3/2)+ 1/2*c^4*(a+b*arctanh(c*x^(1/2)))^3-1/2*(a+b*arctanh(c*x^(1/2)))^3/x^2+4*b^ 2*c^4*(a+b*arctanh(c*x^(1/2)))*ln(2-2/(1+c*x^(1/2)))-2*b^3*c^4*polylog(2,- 1+2/(1+c*x^(1/2)))-1/2*b^3*c^3/x^(1/2)-3/2*b*c^3*(a+b*arctanh(c*x^(1/2)))^ 2/x^(1/2)
Time = 0.51 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx=-\frac {2 a^3+2 a^2 b c \sqrt {x}+2 a b^2 c^2 x+6 a^2 b c^3 x^{3/2}+2 b^3 c^3 x^{3/2}-2 a b^2 c^4 x^2-2 b^2 \left (b c \sqrt {x} \left (-1-3 c^2 x+4 c^3 x^{3/2}\right )+3 a \left (-1+c^4 x^2\right )\right ) \text {arctanh}\left (c \sqrt {x}\right )^2-2 b^3 \left (-1+c^4 x^2\right ) \text {arctanh}\left (c \sqrt {x}\right )^3+2 b \text {arctanh}\left (c \sqrt {x}\right ) \left (3 a^2+b^2 c^2 x \left (1-c^2 x\right )+2 a b c \sqrt {x} \left (1+3 c^2 x\right )-8 b^2 c^4 x^2 \log \left (1-e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )\right )+3 a^2 b c^4 x^2 \log \left (1-c \sqrt {x}\right )-3 a^2 b c^4 x^2 \log \left (1+c \sqrt {x}\right )-16 a b^2 c^4 x^2 \log \left (\frac {c \sqrt {x}}{\sqrt {1-c^2 x}}\right )+8 b^3 c^4 x^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}\left (c \sqrt {x}\right )}\right )}{4 x^2} \]
-1/4*(2*a^3 + 2*a^2*b*c*Sqrt[x] + 2*a*b^2*c^2*x + 6*a^2*b*c^3*x^(3/2) + 2* b^3*c^3*x^(3/2) - 2*a*b^2*c^4*x^2 - 2*b^2*(b*c*Sqrt[x]*(-1 - 3*c^2*x + 4*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 + b^2*c^2*x*(1 - c^2*x) + 2*a*b*c*Sqrt[x]*(1 + 3*c^2*x) - 8*b^2*c^4*x^2*Log[1 - E^(-2*ArcTa nh[c*Sqrt[x]])]) + 3*a^2*b*c^4*x^2*Log[1 - c*Sqrt[x]] - 3*a^2*b*c^4*x^2*Lo g[1 + c*Sqrt[x]] - 16*a*b^2*c^4*x^2*Log[(c*Sqrt[x])/Sqrt[1 - c^2*x]] + 8*b ^3*c^4*x^2*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])])/x^2
Time = 1.91 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.31, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6454, 6452, 6544, 6452, 6544, 6452, 264, 219, 6510, 6550, 6494, 2897}
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.
\(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx\) |
\(\Big \downarrow \) 6454 |
\(\displaystyle 2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^{5/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle 2 \left (\frac {3}{4} b c \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2 \left (1-c^2 x\right )}d\sqrt {x}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
\(\Big \downarrow \) 6544 |
\(\displaystyle 2 \left (\frac {3}{4} b c \left (c^2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x \left (1-c^2 x\right )}d\sqrt {x}+\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2}d\sqrt {x}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle 2 \left (\frac {3}{4} b c \left (\frac {2}{3} b c \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{3/2} \left (1-c^2 x\right )}d\sqrt {x}+c^2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x \left (1-c^2 x\right )}d\sqrt {x}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
\(\Big \downarrow \) 6544 |
\(\displaystyle 2 \left (\frac {3}{4} b c \left (\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}+\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^{3/2}}d\sqrt {x}\right )+c^2 \left (c^2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}+\int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x}d\sqrt {x}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle 2 \left (\frac {3}{4} b c \left (c^2 \left (c^2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}+2 b c \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{\sqrt {x}}\right )+\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}+\frac {1}{2} b c \int \frac {1}{x \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
\(\Big \downarrow \) 264 |
\(\displaystyle 2 \left (\frac {3}{4} b c \left (c^2 \left (c^2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}+2 b c \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{\sqrt {x}}\right )+\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x}d\sqrt {x}-\frac {1}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (\frac {3}{4} b c \left (c^2 \left (c^2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{1-c^2 x}d\sqrt {x}+2 b c \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{\sqrt {x}}\right )+\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}+\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
\(\Big \downarrow \) 6510 |
\(\displaystyle 2 \left (\frac {3}{4} b c \left (c^2 \left (2 b c \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}+\frac {c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{\sqrt {x}}\right )+\frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}+\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
\(\Big \downarrow \) 6550 |
\(\displaystyle 2 \left (\frac {3}{4} b c \left (c^2 \left (2 b c \left (\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\left (\sqrt {x} c+1\right ) \sqrt {x}}d\sqrt {x}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}\right )+\frac {c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{\sqrt {x}}\right )+\frac {2}{3} b c \left (c^2 \left (\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\left (\sqrt {x} c+1\right ) \sqrt {x}}d\sqrt {x}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}+\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
\(\Big \downarrow \) 6494 |
\(\displaystyle 2 \left (\frac {3}{4} b c \left (c^2 \left (2 b c \left (-b c \int \frac {\log \left (2-\frac {2}{\sqrt {x} c+1}\right )}{1-c^2 x}d\sqrt {x}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}+\log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )\right )+\frac {c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{\sqrt {x}}\right )+\frac {2}{3} b c \left (c^2 \left (-b c \int \frac {\log \left (2-\frac {2}{\sqrt {x} c+1}\right )}{1-c^2 x}d\sqrt {x}+\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}+\log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}+\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
\(\Big \downarrow \) 2897 |
\(\displaystyle 2 \left (\frac {3}{4} b c \left (c^2 \left (2 b c \left (\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}+\log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{\sqrt {x} c+1}-1\right )\right )+\frac {c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{3 b}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{\sqrt {x}}\right )+\frac {2}{3} b c \left (c^2 \left (\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}+\log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{\sqrt {x} c+1}-1\right )\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{2 x}+\frac {1}{2} b c \left (c \text {arctanh}\left (c \sqrt {x}\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{4 x^2}\right )\) |
2*(-1/4*(a + b*ArcTanh[c*Sqrt[x]])^3/x^2 + (3*b*c*(-1/3*(a + b*ArcTanh[c*S qrt[x]])^2/x^(3/2) + c^2*(-((a + b*ArcTanh[c*Sqrt[x]])^2/Sqrt[x]) + (c*(a + b*ArcTanh[c*Sqrt[x]])^3)/(3*b) + 2*b*c*((a + b*ArcTanh[c*Sqrt[x]])^2/(2* b) + (a + b*ArcTanh[c*Sqrt[x]])*Log[2 - 2/(1 + c*Sqrt[x])] - (b*PolyLog[2, -1 + 2/(1 + c*Sqrt[x])])/2)) + (2*b*c*(-1/2*(a + b*ArcTanh[c*Sqrt[x]])/x + (b*c*(-(1/Sqrt[x]) + c*ArcTanh[c*Sqrt[x]]))/2 + c^2*((a + b*ArcTanh[c*Sq rt[x]])^2/(2*b) + (a + b*ArcTanh[c*Sqrt[x]])*Log[2 - 2/(1 + c*Sqrt[x])] - (b*PolyLog[2, -1 + 2/(1 + c*Sqrt[x])])/2)))/3))/4)
3.3.8.3.1 Defintions of rubi rules used
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
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_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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]
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[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) 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] && LtQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.70 (sec) , antiderivative size = 1219, normalized size of antiderivative = 5.21
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1219\) |
default | \(\text {Expression too large to display}\) | \(1219\) |
parts | \(\text {Expression too large to display}\) | \(1275\) |
2*c^4*(-3/4*b^3/c/x^(1/2)*arctanh(c*x^(1/2))^2-2*b^3*dilog((1+c*x^(1/2))/( -c^2*x+1)^(1/2))+1/4*b^3*arctanh(c*x^(1/2))+1/4*b^3*arctanh(c*x^(1/2))^3-b ^3*arctanh(c*x^(1/2))^2-3/8*b^3*arctanh(c*x^(1/2))^2*ln(c*x^(1/2)-1)+3/8*b ^3*arctanh(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))+2*b^3*dilog(1+(1+c*x^(1/2))/(-c^2*x+1)^(1/2) )+3/8*I*b^3*Pi*arctanh(c*x^(1/2))^2-1/4*b^3/c^2/x*arctanh(c*x^(1/2))-1/4*b ^3/c^3/x^(3/2)*arctanh(c*x^(1/2))^2-1/4*b^3/c^4/x^2*arctanh(c*x^(1/2))^3-1 /4*a^3/c^4/x^2+3*a^2*b*(-1/4/c^4/x^2*arctanh(c*x^(1/2))+1/8*ln(1+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))+3*a*b^2*(-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))*ln(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+1/8*(ln(1+c*x^(1/2))-ln(1/2*c*x^(1/2)+1/2))*ln(-1/2*c*x^(1/2)+ 1/2)-1/16*ln(1+c*x^(1/2))^2-1/12/c^2/x+2/3*ln(c*x^(1/2))-1/3*ln(c*x^(1/2)- 1)-1/3*ln(1+c*x^(1/2)))-1/4*b^3/(c*x^(1/2)-(-c^2*x+1)^(1/2)+1)*(-c^2*x+1)^ (1/2)+1/4*b^3/(c*x^(1/2)+(-c^2*x+1)^(1/2)+1)*(-c^2*x+1)^(1/2)-3/16*I*b^3*P i*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1)/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^3*arctan h(c*x^(1/2))^2+3/8*I*b^3*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^3*arctan h(c*x^(1/2))^2-3/16*I*b^3*Pi*csgn(I*(1+c*x^(1/2))^2/(c^2*x-1))^3*arctanh(c *x^(1/2))^2-3/8*I*b^3*Pi*csgn(I/(1-(1+c*x^(1/2))^2/(c^2*x-1)))^2*arctan...
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
integral((b^3*arctanh(c*sqrt(x))^3 + 3*a*b^2*arctanh(c*sqrt(x))^2 + 3*a^2* b*arctanh(c*sqrt(x)) + a^3)/x^3, x)
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )^{3}}{x^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (193) = 386\).
Time = 1.33 (sec) , antiderivative size = 703, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx=-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} - 2 \, {\left (\log \left (c \sqrt {x}\right ) \log \left (-c \sqrt {x} + 1\right ) + {\rm Li}_2\left (-c \sqrt {x} + 1\right )\right )} b^{3} c^{4} + 2 \, {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x}\right ) + {\rm Li}_2\left (c \sqrt {x} + 1\right )\right )} b^{3} c^{4} - \frac {1}{8} \, {\left ({\left (6 \, c^{3} \log \left (c \sqrt {x} - 1\right ) - 3 \, c^{3} \log \left (x\right ) + \frac {6 \, c^{2} x + 3 \, c \sqrt {x} + 2}{x^{\frac {3}{2}}}\right )} c - \frac {6 \, \log \left (-c \sqrt {x} + 1\right )}{x^{2}}\right )} a^{2} b + \frac {1}{4} \, {\left (3 \, a^{2} b c^{4} - 8 \, a b^{2} c^{4} + b^{3} c^{4}\right )} \log \left (c \sqrt {x} + 1\right ) - \frac {1}{4} \, {\left (8 \, a b^{2} c^{4} + b^{3} c^{4}\right )} \log \left (c \sqrt {x} - 1\right ) - \frac {1}{8} \, {\left (3 \, a^{2} b c^{4} - 16 \, a b^{2} c^{4}\right )} \log \left (x\right ) - \frac {a^{3}}{2 \, x^{2}} - \frac {4 \, a^{2} b c \sqrt {x} - {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{3} + {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (-c \sqrt {x} + 1\right )^{3} + 2 \, {\left (3 \, b^{3} c^{3} x^{\frac {3}{2}} + b^{3} c \sqrt {x} + 3 \, a b^{2} - {\left (3 \, a b^{2} c^{4} - 4 \, b^{3} c^{4}\right )} x^{2}\right )} \log \left (c \sqrt {x} + 1\right )^{2} + {\left (6 \, b^{3} c^{3} x^{\frac {3}{2}} + 2 \, b^{3} c \sqrt {x} + 6 \, a b^{2} - 2 \, {\left (3 \, a b^{2} c^{4} + 4 \, b^{3} c^{4}\right )} x^{2} - 3 \, {\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 (3 \, a^{2} b c^{3} + 2 \, b^{3} c^{3}\right )} x^{\frac {3}{2}} - 2 \, {\left (3 \, a^{2} b c^{2} - 4 \, a b^{2} c^{2}\right )} x + 4 \, {\left (6 \, a b^{2} c^{3} x^{\frac {3}{2}} + b^{3} c^{2} x + 2 \, a b^{2} c \sqrt {x} + 3 \, a^{2} b\right )} \log \left (c \sqrt {x} + 1\right ) - {\left (24 \, a b^{2} c^{3} x^{\frac {3}{2}} + 4 \, b^{3} c^{2} x + 8 \, a b^{2} c \sqrt {x} - 3 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{2} + 4 \, {\left (3 \, b^{3} c^{3} x^{\frac {3}{2}} + b^{3} c \sqrt {x} + 3 \, a b^{2} - {\left (3 \, a b^{2} c^{4} - 4 \, b^{3} c^{4}\right )} x^{2}\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )}{16 \, x^{2}} \]
-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 - 2*(log(c*sqrt(x))*log(-c*sqrt(x) + 1) + dilog(-c*sqrt(x) + 1))*b^3*c^4 + 2*(log(c*sqrt(x) + 1)*log(-c*sqrt(x)) + dilog(c*sqrt(x) + 1) )*b^3*c^4 - 1/8*((6*c^3*log(c*sqrt(x) - 1) - 3*c^3*log(x) + (6*c^2*x + 3*c *sqrt(x) + 2)/x^(3/2))*c - 6*log(-c*sqrt(x) + 1)/x^2)*a^2*b + 1/4*(3*a^2*b *c^4 - 8*a*b^2*c^4 + b^3*c^4)*log(c*sqrt(x) + 1) - 1/4*(8*a*b^2*c^4 + b^3* c^4)*log(c*sqrt(x) - 1) - 1/8*(3*a^2*b*c^4 - 16*a*b^2*c^4)*log(x) - 1/2*a^ 3/x^2 - 1/16*(4*a^2*b*c*sqrt(x) - (b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1)^3 + (b^3*c^4*x^2 - b^3)*log(-c*sqrt(x) + 1)^3 + 2*(3*b^3*c^3*x^(3/2) + b^3* c*sqrt(x) + 3*a*b^2 - (3*a*b^2*c^4 - 4*b^3*c^4)*x^2)*log(c*sqrt(x) + 1)^2 + (6*b^3*c^3*x^(3/2) + 2*b^3*c*sqrt(x) + 6*a*b^2 - 2*(3*a*b^2*c^4 + 4*b^3* c^4)*x^2 - 3*(b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1))*log(-c*sqrt(x) + 1)^2 + 4*(3*a^2*b*c^3 + 2*b^3*c^3)*x^(3/2) - 2*(3*a^2*b*c^2 - 4*a*b^2*c^2)*x + 4*(6*a*b^2*c^3*x^(3/2) + b^3*c^2*x + 2*a*b^2*c*sqrt(x) + 3*a^2*b)*log(c*s qrt(x) + 1) - (24*a*b^2*c^3*x^(3/2) + 4*b^3*c^2*x + 8*a*b^2*c*sqrt(x) - 3* (b^3*c^4*x^2 - b^3)*log(c*sqrt(x) + 1)^2 + 4*(3*b^3*c^3*x^(3/2) + b^3*c*sq rt(x) + 3*a*b^2 - (3*a*b^2*c^4 - 4*b^3*c^4)*x^2)*log(c*sqrt(x) + 1))*log(- c*sqrt(x) + 1))/x^2
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3}{x^3} \,d x \]