Integrand size = 16, antiderivative size = 188 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^3} \, dx=-\frac {3 b c}{2 \sqrt {x}}+\frac {1}{4} \sqrt {3} b c^{4/3} \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )-\frac {1}{4} \sqrt {3} b c^{4/3} \arctan \left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )+\frac {1}{2} b c^{4/3} \text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}-\frac {1}{8} b c^{4/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{8} b c^{4/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right ) \]
1/2*(-a-b*arctanh(c*x^(3/2)))/x^2+1/2*b*c^(4/3)*arctanh(c^(1/3)*x^(1/2))-1 /8*b*c^(4/3)*ln(1+c^(2/3)*x-c^(1/3)*x^(1/2))+1/8*b*c^(4/3)*ln(1+c^(2/3)*x+ c^(1/3)*x^(1/2))+1/4*b*c^(4/3)*arctan(1/3*(1-2*c^(1/3)*x^(1/2))*3^(1/2))*3 ^(1/2)-1/4*b*c^(4/3)*arctan(1/3*(1+2*c^(1/3)*x^(1/2))*3^(1/2))*3^(1/2)-3/2 *b*c/x^(1/2)
Time = 0.06 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.17 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^3} \, dx=-\frac {a}{2 x^2}-\frac {3 b c}{2 \sqrt {x}}-\frac {1}{4} \sqrt {3} b c^{4/3} \arctan \left (\frac {-1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )-\frac {1}{4} \sqrt {3} b c^{4/3} \arctan \left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )-\frac {b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}-\frac {1}{4} b c^{4/3} \log \left (1-\sqrt [3]{c} \sqrt {x}\right )+\frac {1}{4} b c^{4/3} \log \left (1+\sqrt [3]{c} \sqrt {x}\right )-\frac {1}{8} b c^{4/3} \log \left (1-\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right )+\frac {1}{8} b c^{4/3} \log \left (1+\sqrt [3]{c} \sqrt {x}+c^{2/3} x\right ) \]
-1/2*a/x^2 - (3*b*c)/(2*Sqrt[x]) - (Sqrt[3]*b*c^(4/3)*ArcTan[(-1 + 2*c^(1/ 3)*Sqrt[x])/Sqrt[3]])/4 - (Sqrt[3]*b*c^(4/3)*ArcTan[(1 + 2*c^(1/3)*Sqrt[x] )/Sqrt[3]])/4 - (b*ArcTanh[c*x^(3/2)])/(2*x^2) - (b*c^(4/3)*Log[1 - c^(1/3 )*Sqrt[x]])/4 + (b*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x]])/4 - (b*c^(4/3)*Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x])/8 + (b*c^(4/3)*Log[1 + c^(1/3)*Sqrt[x] + c ^(2/3)*x])/8
Time = 0.41 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6452, 847, 851, 825, 27, 219, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {3}{4} b c \int \frac {1}{x^{3/2} \left (1-c^2 x^3\right )}dx-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \int \frac {x^{3/2}}{1-c^2 x^3}dx-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {3}{4} b c \left (2 c^2 \int \frac {x^2}{1-c^2 x^3}d\sqrt {x}-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {3}{4} b c \left (2 c^2 \left (\frac {\int \frac {1}{1-c^{2/3} x}d\sqrt {x}}{3 c^{4/3}}+\frac {\int -\frac {\sqrt [3]{c} \sqrt {x}+1}{2 \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}d\sqrt {x}}{3 c^{4/3}}+\frac {\int -\frac {1-\sqrt [3]{c} \sqrt {x}}{2 \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}d\sqrt {x}}{3 c^{4/3}}\right )-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} b c \left (2 c^2 \left (\frac {\int \frac {1}{1-c^{2/3} x}d\sqrt {x}}{3 c^{4/3}}-\frac {\int \frac {\sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}-\frac {\int \frac {1-\sqrt [3]{c} \sqrt {x}}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}\right )-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{4} b c \left (2 c^2 \left (-\frac {\int \frac {\sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}-\frac {\int \frac {1-\sqrt [3]{c} \sqrt {x}}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3}{4} b c \left (2 c^2 \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}+\frac {\int -\frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{4} b c \left (2 c^2 \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} b c \left (2 c^2 \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3}{4} b c \left (2 c^2 \left (-\frac {\frac {3 \int \frac {1}{-x-3}d\left (1-2 \sqrt [3]{c} \sqrt {x}\right )}{\sqrt [3]{c}}-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}-\frac {-\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {3 \int \frac {1}{-x-3}d\left (2 \sqrt [3]{c} \sqrt {x}+1\right )}{\sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{4} b c \left (2 c^2 \left (-\frac {-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} \sqrt {x}}{c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {1}{2} \int \frac {2 \sqrt [3]{c} \sqrt {x}+1}{c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1}d\sqrt {x}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3}{4} b c \left (2 c^2 \left (-\frac {\frac {\log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {\log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{3 c^{5/3}}\right )-\frac {2}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{2 x^2}\) |
-1/2*(a + b*ArcTanh[c*x^(3/2)])/x^2 + (3*b*c*(-2/Sqrt[x] + 2*c^2*(ArcTanh[ c^(1/3)*Sqrt[x]]/(3*c^(5/3)) - (-((Sqrt[3]*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/ Sqrt[3]])/c^(1/3)) + Log[1 - c^(1/3)*Sqrt[x] + c^(2/3)*x]/(2*c^(1/3)))/(6* c^(4/3)) - ((Sqrt[3]*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/c^(1/3) - Lo g[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x]/(2*c^(1/3)))/(6*c^(4/3)))))/4
3.3.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k *m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m)) Int[1/ (r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 ] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, 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]
Time = 0.04 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.96
| method | result | size |
| derivativedivides | \(-\frac {a}{2 x^{2}}-\frac {b \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{2 x^{2}}-\frac {3 b c}{2 \sqrt {x}}-\frac {b c \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(180\) |
| default | \(-\frac {a}{2 x^{2}}-\frac {b \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{2 x^{2}}-\frac {3 b c}{2 \sqrt {x}}-\frac {b c \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(180\) |
| parts | \(-\frac {a}{2 x^{2}}-\frac {b \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{2 x^{2}}-\frac {3 b c}{2 \sqrt {x}}-\frac {b c \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(180\) |
-1/2*a/x^2-1/2*b/x^2*arctanh(c*x^(3/2))-3/2*b*c/x^(1/2)-1/4*b*c/(1/c)^(1/3 )*ln(x^(1/2)-(1/c)^(1/3))+1/8*b*c/(1/c)^(1/3)*ln(x+(1/c)^(1/3)*x^(1/2)+(1/ c)^(2/3))-1/4*b*c*3^(1/2)/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x^ (1/2)+1))+1/4*b*c/(1/c)^(1/3)*ln(x^(1/2)+(1/c)^(1/3))-1/8*b*c/(1/c)^(1/3)* ln(x-(1/c)^(1/3)*x^(1/2)+(1/c)^(2/3))-1/4*b*c*3^(1/2)/(1/c)^(1/3)*arctan(1 /3*3^(1/2)*(2/(1/c)^(1/3)*x^(1/2)-1))
Time = 0.27 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.14 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^3} \, dx=-\frac {2 \, \sqrt {3} b \left (-c\right )^{\frac {1}{3}} c x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-c\right )^{\frac {1}{3}} \sqrt {x} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, \sqrt {3} b c^{\frac {4}{3}} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} c^{\frac {1}{3}} \sqrt {x} - \frac {1}{3} \, \sqrt {3}\right ) + b \left (-c\right )^{\frac {1}{3}} c x^{2} \log \left (c x + \left (-c\right )^{\frac {2}{3}} \sqrt {x} - \left (-c\right )^{\frac {1}{3}}\right ) + b c^{\frac {4}{3}} x^{2} \log \left (c x - c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {1}{3}} c x^{2} \log \left (c \sqrt {x} - \left (-c\right )^{\frac {2}{3}}\right ) - 2 \, b c^{\frac {4}{3}} x^{2} \log \left (c \sqrt {x} + c^{\frac {2}{3}}\right ) + 12 \, b c x^{\frac {3}{2}} + 2 \, b \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right ) + 4 \, a}{8 \, x^{2}} \]
-1/8*(2*sqrt(3)*b*(-c)^(1/3)*c*x^2*arctan(2/3*sqrt(3)*(-c)^(1/3)*sqrt(x) - 1/3*sqrt(3)) + 2*sqrt(3)*b*c^(4/3)*x^2*arctan(2/3*sqrt(3)*c^(1/3)*sqrt(x) - 1/3*sqrt(3)) + b*(-c)^(1/3)*c*x^2*log(c*x + (-c)^(2/3)*sqrt(x) - (-c)^( 1/3)) + b*c^(4/3)*x^2*log(c*x - c^(2/3)*sqrt(x) + c^(1/3)) - 2*b*(-c)^(1/3 )*c*x^2*log(c*sqrt(x) - (-c)^(2/3)) - 2*b*c^(4/3)*x^2*log(c*sqrt(x) + c^(2 /3)) + 12*b*c*x^(3/2) + 2*b*log(-(c^2*x^3 + 2*c*x^(3/2) + 1)/(c^2*x^3 - 1) ) + 4*a)/x^2
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^3} \, dx=-\frac {1}{8} \, {\left ({\left (2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) + 2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) - c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right ) + c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right ) - 2 \, c^{\frac {1}{3}} \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right ) + 2 \, c^{\frac {1}{3}} \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right ) + \frac {12}{\sqrt {x}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )}{x^{2}}\right )} b - \frac {a}{2 \, x^{2}} \]
-1/8*((2*sqrt(3)*c^(1/3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*sqrt(x) + c^(1/3))/ c^(1/3)) + 2*sqrt(3)*c^(1/3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*sqrt(x) - c^(1/ 3))/c^(1/3)) - c^(1/3)*log(c^(2/3)*x + c^(1/3)*sqrt(x) + 1) + c^(1/3)*log( c^(2/3)*x - c^(1/3)*sqrt(x) + 1) - 2*c^(1/3)*log((c^(1/3)*sqrt(x) + 1)/c^( 1/3)) + 2*c^(1/3)*log((c^(1/3)*sqrt(x) - 1)/c^(1/3)) + 12/sqrt(x))*c + 4*a rctanh(c*x^(3/2))/x^2)*b - 1/2*a/x^2
Time = 0.38 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.06 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^3} \, dx=-\frac {\sqrt {3} b c^{3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {\sqrt {3} b c^{3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {b c^{3} \log \left (x + \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b c^{3} \log \left (x - \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {b c^{3} \log \left (\sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b c^{3} \log \left ({\left | \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b \log \left (-\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1}\right )}{4 \, x^{2}} - \frac {3 \, b c x^{\frac {3}{2}} + a}{2 \, x^{2}} \]
-1/4*sqrt(3)*b*c^3*arctan(1/3*sqrt(3)*(2*sqrt(x) + 1/abs(c)^(1/3))*abs(c)^ (1/3))/abs(c)^(5/3) - 1/4*sqrt(3)*b*c^3*arctan(1/3*sqrt(3)*(2*sqrt(x) - 1/ abs(c)^(1/3))*abs(c)^(1/3))/abs(c)^(5/3) + 1/8*b*c^3*log(x + sqrt(x)/abs(c )^(1/3) + 1/abs(c)^(2/3))/abs(c)^(5/3) - 1/8*b*c^3*log(x - sqrt(x)/abs(c)^ (1/3) + 1/abs(c)^(2/3))/abs(c)^(5/3) + 1/4*b*c^3*log(sqrt(x) + 1/abs(c)^(1 /3))/abs(c)^(5/3) - 1/4*b*c^3*log(abs(sqrt(x) - 1/abs(c)^(1/3)))/abs(c)^(5 /3) - 1/4*b*log(-(c*x^(3/2) + 1)/(c*x^(3/2) - 1))/x^2 - 1/2*(3*b*c*x^(3/2) + a)/x^2
Time = 10.36 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{x^3} \, dx=\frac {b\,c^{4/3}\,\ln \left (\frac {c^{1/3}\,\sqrt {x}+1}{c^{1/3}\,\sqrt {x}-1}\right )}{4}-\frac {a}{2\,x^2}+\frac {\ln \left (1-c\,x^{3/2}\right )\,\left (\frac {b\,x}{2}-\frac {b\,c^2\,x^4}{2}\right )}{2\,x^3-2\,c^2\,x^6}-\frac {3\,b\,c}{2\,\sqrt {x}}-\frac {b\,\ln \left (c\,x^{3/2}+1\right )}{4\,x^2}+\frac {b\,c^{4/3}\,\ln \left (\frac {\sqrt {3}+c^{2/3}\,x\,1{}\mathrm {i}+c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}\,c^{2/3}\,x+1{}\mathrm {i}}{2\,c^{2/3}\,x+1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}{4}+\frac {b\,c^{4/3}\,\ln \left (\frac {\sqrt {3}\,c^{2/3}\,x+c^{2/3}\,x\,1{}\mathrm {i}-c^{1/3}\,\sqrt {x}\,4{}\mathrm {i}-\sqrt {3}+1{}\mathrm {i}}{2\,c^{2/3}\,x+1-\sqrt {3}\,1{}\mathrm {i}}\right )\,\sqrt {\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{4} \]
(b*c^(4/3)*log((c^(1/3)*x^(1/2) + 1)/(c^(1/3)*x^(1/2) - 1)))/4 - a/(2*x^2) + (log(1 - c*x^(3/2))*((b*x)/2 - (b*c^2*x^4)/2))/(2*x^3 - 2*c^2*x^6) - (3 *b*c)/(2*x^(1/2)) - (b*log(c*x^(3/2) + 1))/(4*x^2) + (b*c^(4/3)*log((3^(1/ 2) + c^(2/3)*x*1i + c^(1/3)*x^(1/2)*4i - 3^(1/2)*c^(2/3)*x + 1i)/(3^(1/2)* 1i + 2*c^(2/3)*x + 1))*((3^(1/2)*1i)/2 - 1/2)^(1/2))/4 + (b*c^(4/3)*log((c ^(2/3)*x*1i - 3^(1/2) - c^(1/3)*x^(1/2)*4i + 3^(1/2)*c^(2/3)*x + 1i)/(2*c^ (2/3)*x - 3^(1/2)*1i + 1))*((3^(1/2)*1i)/2 + 1/2)^(1/2)*1i)/4