Integrand size = 18, antiderivative size = 301 \[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {2 b \sqrt {d} \arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}-\frac {\sqrt {2} b \sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {\sqrt {2} b \sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {2 b \sqrt {d} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{3 c^{3/4}}+\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}}-\frac {b \sqrt {d} \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{3 \sqrt {2} c^{3/4}} \]
2/3*(d*x)^(3/2)*(a+b*arctanh(c*x^2))/d+2/3*b*arctan(c^(1/4)*(d*x)^(1/2)/d^ (1/2))*d^(1/2)/c^(3/4)-2/3*b*arctanh(c^(1/4)*(d*x)^(1/2)/d^(1/2))*d^(1/2)/ c^(3/4)+1/6*b*ln(d^(1/2)+x*c^(1/2)*d^(1/2)-c^(1/4)*2^(1/2)*(d*x)^(1/2))*d^ (1/2)/c^(3/4)*2^(1/2)-1/6*b*ln(d^(1/2)+x*c^(1/2)*d^(1/2)+c^(1/4)*2^(1/2)*( d*x)^(1/2))*d^(1/2)/c^(3/4)*2^(1/2)+1/3*b*arctan(-1+c^(1/4)*2^(1/2)*(d*x)^ (1/2)/d^(1/2))*2^(1/2)*d^(1/2)/c^(3/4)+1/3*b*arctan(1+c^(1/4)*2^(1/2)*(d*x )^(1/2)/d^(1/2))*2^(1/2)*d^(1/2)/c^(3/4)
Time = 0.08 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.75 \[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {\sqrt {d x} \left (4 a c^{3/4} x^{3/2}-2 \sqrt {2} b \arctan \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+2 \sqrt {2} b \arctan \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+4 b \arctan \left (\sqrt [4]{c} \sqrt {x}\right )+4 b c^{3/4} x^{3/2} \text {arctanh}\left (c x^2\right )+2 b \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-2 b \log \left (1+\sqrt [4]{c} \sqrt {x}\right )+\sqrt {2} b \log \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )-\sqrt {2} b \log \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )\right )}{6 c^{3/4} \sqrt {x}} \]
(Sqrt[d*x]*(4*a*c^(3/4)*x^(3/2) - 2*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*c^(1/4)*S qrt[x]] + 2*Sqrt[2]*b*ArcTan[1 + Sqrt[2]*c^(1/4)*Sqrt[x]] + 4*b*ArcTan[c^( 1/4)*Sqrt[x]] + 4*b*c^(3/4)*x^(3/2)*ArcTanh[c*x^2] + 2*b*Log[1 - c^(1/4)*S qrt[x]] - 2*b*Log[1 + c^(1/4)*Sqrt[x]] + Sqrt[2]*b*Log[1 - Sqrt[2]*c^(1/4) *Sqrt[x] + Sqrt[c]*x] - Sqrt[2]*b*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c ]*x]))/(6*c^(3/4)*Sqrt[x])
Time = 0.61 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.09, 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, 830, 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.
| \(\displaystyle \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6464 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {4 b c \int \frac {(d x)^{5/2}}{1-c^2 x^4}dx}{3 d^2}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8 b c \int \frac {d^7 x^3}{d^4-c^2 d^4 x^4}d\sqrt {d x}}{3 d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \int \frac {d^3 x^3}{d^4-c^2 d^4 x^4}d\sqrt {d x}\) |
| \(\Big \downarrow \) 830 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\int \frac {d x}{d^2-c d^2 x^2}d\sqrt {d x}}{2 c}-\frac {\int \frac {d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 c}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\int \frac {d x}{d^2-c d^2 x^2}d\sqrt {d x}}{2 c}-\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {1}{\sqrt {c} x d+d}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}-\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 \sqrt {c}}}{2 c}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c} \sqrt {d}}}{2 \sqrt {c}}}{2 c}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 (d x)^{3/2} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{3 d}-\frac {8}{3} b c d \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 c^{3/4} \sqrt {d}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {c} d x+\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\log \left (\sqrt {c} d x-\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}}{2 c}\right )\) |
(2*(d*x)^(3/2)*(a + b*ArcTanh[c*x^2]))/(3*d) - (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])/S qrt[d]]/(2*c^(3/4)*Sqrt[d]))/(2*c) - ((-(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)*Sq rt[d*x])/Sqrt[d]]/(Sqrt[2]*c^(1/4)*Sqrt[d]))/(2*Sqrt[c]) - (-1/2*Log[d + S qrt[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*c)))/3
3.1.84.3.1 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[(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[s/(2*b) Int[x^(m - n/2)/( r + s*x^(n/2)), x], x] - Simp[s/(2*b) Int[x^(m - n/2)/(r - s*x^(n/2)), x] , x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && L tQ[m, n] && !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 = 0.63 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.80
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d x \right )^{\frac {3}{2}} \operatorname {arctanh}\left (c \,x^{2}\right )}{3}-\frac {4 c \,d^{2} \left (-\frac {2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{8 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3}\right )}{d}\) | \(240\) |
| default | \(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d x \right )^{\frac {3}{2}} \operatorname {arctanh}\left (c \,x^{2}\right )}{3}-\frac {4 c \,d^{2} \left (-\frac {2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{8 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3}\right )}{d}\) | \(240\) |
| parts | \(\frac {2 a \left (d x \right )^{\frac {3}{2}}}{3 d}+\frac {2 b \left (\frac {\left (d x \right )^{\frac {3}{2}} \operatorname {arctanh}\left (c \,x^{2}\right )}{3}-\frac {4 c \,d^{2} \left (-\frac {2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{8 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2} \left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )}{3}\right )}{d}\) | \(242\) |
2/d*(1/3*(d*x)^(3/2)*a+b*(1/3*(d*x)^(3/2)*arctanh(c*x^2)-4/3*c*d^2*(-1/8/c ^2/(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))))-1/16/c^2/(d^2/c)^(1/4)*2^(1/2)*(l n((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)))))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.28 \[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {1}{3} \, {\left (b x \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a x\right )} \sqrt {d x} - \frac {1}{3} \, \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d + \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) + \frac {1}{3} i \, \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d + i \, \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) - \frac {1}{3} i \, \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d - i \, \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) + \frac {1}{3} \, \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d - \left (\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) + \frac {1}{3} \, \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d + \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) - \frac {1}{3} i \, \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d + i \, \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) + \frac {1}{3} i \, \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d - i \, \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) - \frac {1}{3} \, \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {1}{4}} \log \left (\sqrt {d x} b^{3} d - \left (-\frac {b^{4} d^{2}}{c^{3}}\right )^{\frac {3}{4}} c^{2}\right ) \]
1/3*(b*x*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*a*x)*sqrt(d*x) - 1/3*(b^4*d^2/c ^3)^(1/4)*log(sqrt(d*x)*b^3*d + (b^4*d^2/c^3)^(3/4)*c^2) + 1/3*I*(b^4*d^2/ c^3)^(1/4)*log(sqrt(d*x)*b^3*d + I*(b^4*d^2/c^3)^(3/4)*c^2) - 1/3*I*(b^4*d ^2/c^3)^(1/4)*log(sqrt(d*x)*b^3*d - I*(b^4*d^2/c^3)^(3/4)*c^2) + 1/3*(b^4* d^2/c^3)^(1/4)*log(sqrt(d*x)*b^3*d - (b^4*d^2/c^3)^(3/4)*c^2) + 1/3*(-b^4* d^2/c^3)^(1/4)*log(sqrt(d*x)*b^3*d + (-b^4*d^2/c^3)^(3/4)*c^2) - 1/3*I*(-b ^4*d^2/c^3)^(1/4)*log(sqrt(d*x)*b^3*d + I*(-b^4*d^2/c^3)^(3/4)*c^2) + 1/3* I*(-b^4*d^2/c^3)^(1/4)*log(sqrt(d*x)*b^3*d - I*(-b^4*d^2/c^3)^(3/4)*c^2) - 1/3*(-b^4*d^2/c^3)^(1/4)*log(sqrt(d*x)*b^3*d - (-b^4*d^2/c^3)^(3/4)*c^2)
\[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int \sqrt {d x} \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )\, dx \]
Time = 0.27 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00 \[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {4 \, \left (d x\right )^{\frac {3}{2}} a + {\left (4 \, \left (d x\right )^{\frac {3}{2}} \operatorname {artanh}\left (c x^{2}\right ) + \frac {{\left (\frac {d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}} + \frac {\sqrt {2} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {3}{4}} \sqrt {d}}\right )}}{c} + \frac {2 \, d^{4} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d} \sqrt {c}}\right )}}{c}\right )} c}{d^{2}}\right )} b}{6 \, d} \]
1/6*(4*(d*x)^(3/2)*a + (4*(d*x)^(3/2)*arctanh(c*x^2) + (d^4*(2*sqrt(2)*arc tan(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)*s qrt(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)*s qrt(d) + d)/(c^(3/4)*sqrt(d)))/c + 2*d^4*(2*arctan(sqrt(d*x)*sqrt(c)/sqrt( sqrt(c)*d))/(sqrt(sqrt(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)*c/d^2)*b)/d
\[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int { \sqrt {d x} {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )} \,d x } \]
Timed out. \[ \int \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int \sqrt {d\,x}\,\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right ) \,d x \]