Integrand size = 18, antiderivative size = 228 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{3/2}} \, dx=-\frac {2 b \sqrt [4]{c} \arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {\sqrt {2} b \sqrt [4]{c} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}+\frac {2 b \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d} \left (1+\sqrt {c} x\right )}\right )}{d^{3/2}} \] Output:
-2*b*c^(1/4)*arctan(c^(1/4)*(d*x)^(1/2)/d^(1/2))/d^(3/2)+2^(1/2)*b*c^(1/4) *arctan(-1+2^(1/2)*c^(1/4)*(d*x)^(1/2)/d^(1/2))/d^(3/2)+2^(1/2)*b*c^(1/4)* arctan(1+2^(1/2)*c^(1/4)*(d*x)^(1/2)/d^(1/2))/d^(3/2)-2*(a+b*arctanh(c*x^2 ))/d/(d*x)^(1/2)+2*b*c^(1/4)*arctanh(c^(1/4)*(d*x)^(1/2)/d^(1/2))/d^(3/2)- 2^(1/2)*b*c^(1/4)*arctanh(2^(1/2)*c^(1/4)*(d*x)^(1/2)/d^(1/2)/(1+c^(1/2)*x ))/d^(3/2)
Time = 0.09 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.18 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{3/2}} \, dx=-\frac {x \left (4 a+2 \sqrt {2} b \sqrt [4]{c} \sqrt {x} \arctan \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-2 \sqrt {2} b \sqrt [4]{c} \sqrt {x} \arctan \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+4 b \sqrt [4]{c} \sqrt {x} \arctan \left (\sqrt [4]{c} \sqrt {x}\right )+4 b \text {arctanh}\left (c x^2\right )+2 b \sqrt [4]{c} \sqrt {x} \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-2 b \sqrt [4]{c} \sqrt {x} \log \left (1+\sqrt [4]{c} \sqrt {x}\right )-\sqrt {2} b \sqrt [4]{c} \sqrt {x} \log \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )+\sqrt {2} b \sqrt [4]{c} \sqrt {x} \log \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )\right )}{2 (d x)^{3/2}} \] Input:
Integrate[(a + b*ArcTanh[c*x^2])/(d*x)^(3/2),x]
Output:
-1/2*(x*(4*a + 2*Sqrt[2]*b*c^(1/4)*Sqrt[x]*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqrt [x]] - 2*Sqrt[2]*b*c^(1/4)*Sqrt[x]*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] + 4 *b*c^(1/4)*Sqrt[x]*ArcTan[c^(1/4)*Sqrt[x]] + 4*b*ArcTanh[c*x^2] + 2*b*c^(1 /4)*Sqrt[x]*Log[1 - c^(1/4)*Sqrt[x]] - 2*b*c^(1/4)*Sqrt[x]*Log[1 + c^(1/4) *Sqrt[x]] - Sqrt[2]*b*c^(1/4)*Sqrt[x]*Log[1 - Sqrt[2]*c^(1/4)*Sqrt[x] + Sq rt[c]*x] + Sqrt[2]*b*c^(1/4)*Sqrt[x]*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqr t[c]*x]))/(d*x)^(3/2)
Time = 0.59 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.42, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6464, 851, 27, 829, 826, 827, 218, 221, 1476, 1082, 217, 1479, 25, 27, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6464 |
| \(\displaystyle \frac {4 b c \int \frac {\sqrt {d x}}{1-c^2 x^4}dx}{d^2}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {8 b c \int \frac {d^5 x}{d^4-c^2 d^4 x^4}d\sqrt {d x}}{d^3}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 b c d \int \frac {d x}{d^4-c^2 d^4 x^4}d\sqrt {d x}-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 829 |
| \(\displaystyle 8 b c d \left (\frac {\int \frac {d x}{d^2-c d^2 x^2}d\sqrt {d x}}{2 d^2}+\frac {\int \frac {d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 8 b c d \left (\frac {\int \frac {d x}{d^2-c d^2 x^2}d\sqrt {d x}}{2 d^2}+\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle 8 b c d \left (\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {1}{\sqrt {c} x d+d}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 8 b c d \left (\frac {\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}+\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 8 b c d \left (\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 8 b c d \left (\frac {\frac {\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 8 b c d \left (\frac {\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 8 b c d \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 8 b c d \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 8 b c d \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 b c d \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c} \sqrt {d}}}{2 \sqrt {c}}}{2 d^2}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 8 b c d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 d^2}+\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {c} d x+\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\log \left (\sqrt {c} d x-\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 d^2}\right )-\frac {2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d \sqrt {d x}}\) |
Input:
Int[(a + b*ArcTanh[c*x^2])/(d*x)^(3/2),x]
Output:
(-2*(a + b*ArcTanh[c*x^2]))/(d*Sqrt[d*x]) + 8*b*c*d*((-1/2*ArcTan[(c^(1/4) *Sqrt[d*x])/Sqrt[d]]/(c^(3/4)*Sqrt[d]) + ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[ d]]/(2*c^(3/4)*Sqrt[d]))/(2*d^2) + ((-(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[d* x])/Sqrt[d]]/(Sqrt[2]*c^(1/4)*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt [d*x])/Sqrt[d]]/(Sqrt[2]*c^(1/4)*Sqrt[d]))/(2*Sqrt[c]) - (-1/2*Log[d + Sqr t[c]*d*x - Sqrt[2]*c^(1/4)*Sqrt[d]*Sqrt[d*x]]/(Sqrt[2]*c^(1/4)*Sqrt[d]) + Log[d + Sqrt[c]*d*x + Sqrt[2]*c^(1/4)*Sqrt[d]*Sqrt[d*x]]/(2*Sqrt[2]*c^(1/4 )*Sqrt[d]))/(2*Sqrt[c]))/(2*d^2))
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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt [-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[x^m/(r + s*x^ (n/2)), x], x] + Simp[r/(2*a) Int[x^m/(r - s*x^(n/2)), x], x]] /; FreeQ[{ a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] && !GtQ[a/b, 0]
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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))*((d_)*(x_))^(m_), x_Symbol] : > Simp[(d*x)^(m + 1)*((a + b*ArcTanh[c*x^n])/(d*(m + 1))), x] - Simp[b*c*(n /(d^n*(m + 1))) Int[(d*x)^(m + n)/(1 - c^2*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[n] && NeQ[m, -1]
Time = 0.40 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.08
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a}{\sqrt {d x}}+2 b \left (-\frac {\operatorname {arctanh}\left (c \,x^{2}\right )}{\sqrt {d x}}+4 c \,d^{2} \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 d^{2} c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{8 d^{2} c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{d}\) | \(246\) |
| default | \(\frac {-\frac {2 a}{\sqrt {d x}}+2 b \left (-\frac {\operatorname {arctanh}\left (c \,x^{2}\right )}{\sqrt {d x}}+4 c \,d^{2} \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 d^{2} c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{8 d^{2} c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{d}\) | \(246\) |
| parts | \(-\frac {2 a}{\sqrt {d x}\, d}+\frac {2 b \left (-\frac {\operatorname {arctanh}\left (c \,x^{2}\right )}{\sqrt {d x}}+4 c \,d^{2} \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 d^{2} c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{8 d^{2} c \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{d}\) | \(248\) |
Input:
int((a+b*arctanh(c*x^2))/(d*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/d*(-a/(d*x)^(1/2)+b*(-1/(d*x)^(1/2)*arctanh(c*x^2)+4*c*d^2*(1/16/d^2/c/( d^2/c)^(1/4)*2^(1/2)*(ln((d*x-(d^2/c)^(1/4)*(d*x)^(1/2)*2^(1/2)+(d^2/c)^(1 /2))/(d*x+(d^2/c)^(1/4)*(d*x)^(1/2)*2^(1/2)+(d^2/c)^(1/2)))+2*arctan(2^(1/ 2)/(d^2/c)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(d^2/c)^(1/4)*(d*x)^(1/2) -1))-1/8/d^2/c/(d^2/c)^(1/4)*(2*arctan((d*x)^(1/2)/(d^2/c)^(1/4))-ln(((d*x )^(1/2)+(d^2/c)^(1/4))/((d*x)^(1/2)-(d^2/c)^(1/4)))))))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.70 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{3/2}} \, dx=\frac {d^{2} x \left (\frac {b^{4} c}{d^{6}}\right )^{\frac {1}{4}} \log \left (d^{5} \left (\frac {b^{4} c}{d^{6}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c\right ) - i \, d^{2} x \left (\frac {b^{4} c}{d^{6}}\right )^{\frac {1}{4}} \log \left (i \, d^{5} \left (\frac {b^{4} c}{d^{6}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c\right ) + i \, d^{2} x \left (\frac {b^{4} c}{d^{6}}\right )^{\frac {1}{4}} \log \left (-i \, d^{5} \left (\frac {b^{4} c}{d^{6}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c\right ) - d^{2} x \left (\frac {b^{4} c}{d^{6}}\right )^{\frac {1}{4}} \log \left (-d^{5} \left (\frac {b^{4} c}{d^{6}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c\right ) + d^{2} x \left (-\frac {b^{4} c}{d^{6}}\right )^{\frac {1}{4}} \log \left (d^{5} \left (-\frac {b^{4} c}{d^{6}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c\right ) - i \, d^{2} x \left (-\frac {b^{4} c}{d^{6}}\right )^{\frac {1}{4}} \log \left (i \, d^{5} \left (-\frac {b^{4} c}{d^{6}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c\right ) + i \, d^{2} x \left (-\frac {b^{4} c}{d^{6}}\right )^{\frac {1}{4}} \log \left (-i \, d^{5} \left (-\frac {b^{4} c}{d^{6}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c\right ) - d^{2} x \left (-\frac {b^{4} c}{d^{6}}\right )^{\frac {1}{4}} \log \left (-d^{5} \left (-\frac {b^{4} c}{d^{6}}\right )^{\frac {3}{4}} + \sqrt {d x} b^{3} c\right ) - \sqrt {d x} {\left (b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a\right )}}{d^{2} x} \] Input:
integrate((a+b*arctanh(c*x^2))/(d*x)^(3/2),x, algorithm="fricas")
Output:
(d^2*x*(b^4*c/d^6)^(1/4)*log(d^5*(b^4*c/d^6)^(3/4) + sqrt(d*x)*b^3*c) - I* d^2*x*(b^4*c/d^6)^(1/4)*log(I*d^5*(b^4*c/d^6)^(3/4) + sqrt(d*x)*b^3*c) + I *d^2*x*(b^4*c/d^6)^(1/4)*log(-I*d^5*(b^4*c/d^6)^(3/4) + sqrt(d*x)*b^3*c) - d^2*x*(b^4*c/d^6)^(1/4)*log(-d^5*(b^4*c/d^6)^(3/4) + sqrt(d*x)*b^3*c) + d ^2*x*(-b^4*c/d^6)^(1/4)*log(d^5*(-b^4*c/d^6)^(3/4) + sqrt(d*x)*b^3*c) - I* d^2*x*(-b^4*c/d^6)^(1/4)*log(I*d^5*(-b^4*c/d^6)^(3/4) + sqrt(d*x)*b^3*c) + I*d^2*x*(-b^4*c/d^6)^(1/4)*log(-I*d^5*(-b^4*c/d^6)^(3/4) + sqrt(d*x)*b^3* c) - d^2*x*(-b^4*c/d^6)^(1/4)*log(-d^5*(-b^4*c/d^6)^(3/4) + sqrt(d*x)*b^3* c) - sqrt(d*x)*(b*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*a))/(d^2*x)
\[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{3/2}} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c x^{2} \right )}}{\left (d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*atanh(c*x**2))/(d*x)**(3/2),x)
Output:
Integral((a + b*atanh(c*x**2))/(d*x)**(3/2), x)
Time = 0.11 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.30 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{3/2}} \, dx=-\frac {b {\left (\frac {4 \, \operatorname {artanh}\left (c x^{2}\right )}{\sqrt {d x}} - \frac {{\left (d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}} + \frac {\sqrt {2} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}}\right )} - 2 \, d^{2} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}}\right )}\right )} c}{d^{2}}\right )} + \frac {4 \, a}{\sqrt {d x}}}{2 \, d} \] Input:
integrate((a+b*arctanh(c*x^2))/(d*x)^(3/2),x, algorithm="maxima")
Output:
-1/2*(b*(4*arctanh(c*x^2)/sqrt(d*x) - (d^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*( sqrt(2)*c^(1/4)*sqrt(d) + 2*sqrt(d*x)*sqrt(c))/sqrt(sqrt(c)*d))/(sqrt(sqrt (c)*d)*sqrt(c)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*sqrt(d) - 2*sqrt(d*x)*sqrt(c))/sqrt(sqrt(c)*d))/(sqrt(sqrt(c)*d)*sqrt(c)) - sqrt(2) *log(sqrt(c)*d*x + sqrt(2)*sqrt(d*x)*c^(1/4)*sqrt(d) + d)/(c^(3/4)*sqrt(d) ) + sqrt(2)*log(sqrt(c)*d*x - sqrt(2)*sqrt(d*x)*c^(1/4)*sqrt(d) + d)/(c^(3 /4)*sqrt(d))) - 2*d^2*(2*arctan(sqrt(d*x)*sqrt(c)/sqrt(sqrt(c)*d))/(sqrt(s qrt(c)*d)*sqrt(c)) + log((sqrt(d*x)*sqrt(c) - sqrt(sqrt(c)*d))/(sqrt(d*x)* sqrt(c) + sqrt(sqrt(c)*d)))/(sqrt(sqrt(c)*d)*sqrt(c))))*c/d^2) + 4*a/sqrt( d*x))/d
Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (160) = 320\).
Time = 0.29 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.21 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{3/2}} \, dx=-\frac {\frac {2 \, b \log \left (-\frac {c d^{2} x^{2} + d^{2}}{c d^{2} x^{2} - d^{2}}\right )}{\sqrt {d x}} + \frac {4 \, a}{\sqrt {d x}} - \frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} - \frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} - \frac {2 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} - \frac {2 \, \sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{c^{2} d^{2}} + \frac {\sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x + \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{c^{2} d^{2}} - \frac {\sqrt {2} \left (c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x - \sqrt {2} \sqrt {d x} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {d^{2}}{c}}\right )}{c^{2} d^{2}} + \frac {\sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x + \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{c^{2} d^{2}} - \frac {\sqrt {2} \left (-c^{3} d^{2}\right )^{\frac {3}{4}} b \log \left (d x - \sqrt {2} \sqrt {d x} \left (-\frac {d^{2}}{c}\right )^{\frac {1}{4}} + \sqrt {-\frac {d^{2}}{c}}\right )}{c^{2} d^{2}}}{2 \, d} \] Input:
integrate((a+b*arctanh(c*x^2))/(d*x)^(3/2),x, algorithm="giac")
Output:
-1/2*(2*b*log(-(c*d^2*x^2 + d^2)/(c*d^2*x^2 - d^2))/sqrt(d*x) + 4*a/sqrt(d *x) - 2*sqrt(2)*(c^3*d^2)^(3/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(d^2/c)^(1/4 ) + 2*sqrt(d*x))/(d^2/c)^(1/4))/(c^2*d^2) - 2*sqrt(2)*(c^3*d^2)^(3/4)*b*ar ctan(-1/2*sqrt(2)*(sqrt(2)*(d^2/c)^(1/4) - 2*sqrt(d*x))/(d^2/c)^(1/4))/(c^ 2*d^2) - 2*sqrt(2)*(-c^3*d^2)^(3/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(-d^2/c) ^(1/4) + 2*sqrt(d*x))/(-d^2/c)^(1/4))/(c^2*d^2) - 2*sqrt(2)*(-c^3*d^2)^(3/ 4)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*(-d^2/c)^(1/4) - 2*sqrt(d*x))/(-d^2/c)^( 1/4))/(c^2*d^2) + sqrt(2)*(c^3*d^2)^(3/4)*b*log(d*x + sqrt(2)*sqrt(d*x)*(d ^2/c)^(1/4) + sqrt(d^2/c))/(c^2*d^2) - sqrt(2)*(c^3*d^2)^(3/4)*b*log(d*x - sqrt(2)*sqrt(d*x)*(d^2/c)^(1/4) + sqrt(d^2/c))/(c^2*d^2) + sqrt(2)*(-c^3* d^2)^(3/4)*b*log(d*x + sqrt(2)*sqrt(d*x)*(-d^2/c)^(1/4) + sqrt(-d^2/c))/(c ^2*d^2) - sqrt(2)*(-c^3*d^2)^(3/4)*b*log(d*x - sqrt(2)*sqrt(d*x)*(-d^2/c)^ (1/4) + sqrt(-d^2/c))/(c^2*d^2))/d
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x^2\right )}{{\left (d\,x\right )}^{3/2}} \,d x \] Input:
int((a + b*atanh(c*x^2))/(d*x)^(3/2),x)
Output:
int((a + b*atanh(c*x^2))/(d*x)^(3/2), x)
Time = 0.17 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{(d x)^{3/2}} \, dx=\frac {\sqrt {d}\, \left (-2 \sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} \sqrt {2}}\right ) b +2 \sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} \sqrt {2}}\right ) b -4 \sqrt {x}\, c^{\frac {1}{4}} \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}}}\right ) b -2 \sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}\, \mathit {atanh} \left (c \,x^{2}\right ) b -4 \mathit {atanh} \left (c \,x^{2}\right ) b -\sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b -\sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b +2 \sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x +1\right ) b -\sqrt {x}\, c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {c}\, x +1\right ) b +2 \sqrt {x}\, c^{\frac {1}{4}} \mathrm {log}\left (c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b -2 \sqrt {x}\, c^{\frac {1}{4}} \mathrm {log}\left (-c^{\frac {1}{4}}+\sqrt {x}\, \sqrt {c}\right ) b -4 a \right )}{2 \sqrt {x}\, d^{2}} \] Input:
int((a+b*atanh(c*x^2))/(d*x)^(3/2),x)
Output:
(sqrt(d)*( - 2*sqrt(x)*c**(1/4)*sqrt(2)*atan((c**(1/4)*sqrt(2) - 2*sqrt(x) *sqrt(c))/(c**(1/4)*sqrt(2)))*b + 2*sqrt(x)*c**(1/4)*sqrt(2)*atan((c**(1/4 )*sqrt(2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)*sqrt(2)))*b - 4*sqrt(x)*c**(1/4)* atan((sqrt(x)*sqrt(c))/c**(1/4))*b - 2*sqrt(x)*c**(1/4)*sqrt(2)*atanh(c*x* *2)*b - 4*atanh(c*x**2)*b - sqrt(x)*c**(1/4)*sqrt(2)*log(c**(1/4) + sqrt(x )*sqrt(c))*b - sqrt(x)*c**(1/4)*sqrt(2)*log( - c**(1/4) + sqrt(x)*sqrt(c)) *b + 2*sqrt(x)*c**(1/4)*sqrt(2)*log( - sqrt(x)*c**(1/4)*sqrt(2) + sqrt(c)* x + 1)*b - sqrt(x)*c**(1/4)*sqrt(2)*log(sqrt(c)*x + 1)*b + 2*sqrt(x)*c**(1 /4)*log(c**(1/4) + sqrt(x)*sqrt(c))*b - 2*sqrt(x)*c**(1/4)*log( - c**(1/4) + sqrt(x)*sqrt(c))*b - 4*a))/(2*sqrt(x)*d**2)