Integrand size = 18, antiderivative size = 285 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{\sqrt {d x}} \, dx=-\frac {2 b \arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}-\frac {\sqrt {2} b \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}+\frac {\sqrt {2} b \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}+\frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-\frac {2 b \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt [4]{c} \sqrt {d}}-\frac {b \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {b \log \left (\sqrt {d}+\sqrt {c} \sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}} \]
-2*b*arctan(c^(1/4)*(d*x)^(1/2)/d^(1/2))/c^(1/4)/d^(1/2)-2*b*arctanh(c^(1/ 4)*(d*x)^(1/2)/d^(1/2))/c^(1/4)/d^(1/2)-1/2*b*ln(d^(1/2)+x*c^(1/2)*d^(1/2) -c^(1/4)*2^(1/2)*(d*x)^(1/2))/c^(1/4)*2^(1/2)/d^(1/2)+1/2*b*ln(d^(1/2)+x*c ^(1/2)*d^(1/2)+c^(1/4)*2^(1/2)*(d*x)^(1/2))/c^(1/4)*2^(1/2)/d^(1/2)+b*arct an(-1+c^(1/4)*2^(1/2)*(d*x)^(1/2)/d^(1/2))*2^(1/2)/c^(1/4)/d^(1/2)+b*arcta n(1+c^(1/4)*2^(1/2)*(d*x)^(1/2)/d^(1/2))*2^(1/2)/c^(1/4)/d^(1/2)+2*(a+b*ar ctanh(c*x^2))*(d*x)^(1/2)/d
Time = 0.05 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{\sqrt {d x}} \, dx=\frac {\sqrt {x} \left (4 a \sqrt [4]{c} \sqrt {x}-2 \sqrt {2} b \arctan \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )+2 \sqrt {2} b \arctan \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}\right )-4 b \arctan \left (\sqrt [4]{c} \sqrt {x}\right )+4 b \sqrt [4]{c} \sqrt {x} \text {arctanh}\left (c x^2\right )+2 b \log \left (1-\sqrt [4]{c} \sqrt {x}\right )-2 b \log \left (1+\sqrt [4]{c} \sqrt {x}\right )-\sqrt {2} b \log \left (1-\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )+\sqrt {2} b \log \left (1+\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )\right )}{2 \sqrt [4]{c} \sqrt {d x}} \]
(Sqrt[x]*(4*a*c^(1/4)*Sqrt[x] - 2*Sqrt[2]*b*ArcTan[1 - Sqrt[2]*c^(1/4)*Sqr t[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^(1/4)*Sqrt[x]*ArcTanh[c*x^2] + 2*b*Log[1 - c^(1/4)*Sqr t[x]] - 2*b*Log[1 + c^(1/4)*Sqrt[x]] - Sqrt[2]*b*Log[1 - Sqrt[2]*c^(1/4)*S qrt[x] + Sqrt[c]*x] + Sqrt[2]*b*Log[1 + Sqrt[2]*c^(1/4)*Sqrt[x] + Sqrt[c]* x]))/(2*c^(1/4)*Sqrt[d*x])
Time = 0.58 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.12, 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, 755, 756, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^2\right )}{\sqrt {d x}} \, dx\) |
| \(\Big \downarrow \) 6464 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-\frac {4 b c \int \frac {(d x)^{3/2}}{1-c^2 x^4}dx}{d^2}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-\frac {8 b c \int \frac {d^6 x^2}{d^4-c^2 d^4 x^4}d\sqrt {d x}}{d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \int \frac {d^2 x^2}{d^4-c^2 d^4 x^4}d\sqrt {d x}\) |
| \(\Big \downarrow \) 830 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\int \frac {1}{d^2-c d^2 x^2}d\sqrt {d x}}{2 c}-\frac {\int \frac {1}{c x^2 d^2+d^2}d\sqrt {d x}}{2 c}\right )\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\int \frac {1}{d^2-c d^2 x^2}d\sqrt {d x}}{2 c}-\frac {\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}}{2 c}\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 d}+\frac {\int \frac {1}{\sqrt {c} x d+d}d\sqrt {d x}}{2 d}}{2 c}-\frac {\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}}{2 c}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\frac {\int \frac {1}{d-\sqrt {c} d x}d\sqrt {d x}}{2 d}+\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}}{2 c}-\frac {\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}}{2 c}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}}{2 c}-\frac {\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\int \frac {\sqrt {c} x d+d}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}}{2 c}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}}{2 c}-\frac {\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c}}}{2 d}}{2 c}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}}{2 c}-\frac {\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}}{2 c}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}}{2 c}-\frac {\frac {\int \frac {d-\sqrt {c} d x}{c x^2 d^2+d^2}d\sqrt {d x}}{2 d}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}}{2 c}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}}{2 c}-\frac {\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}}{2 c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}}{2 c}-\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}\right )}{\sqrt [4]{c} \left (x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}}{2 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}}{2 c}-\frac {\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt {c} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{x d+\frac {d}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {d x} \sqrt {d}}{\sqrt [4]{c}}}d\sqrt {d x}}{2 \sqrt {c} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}}{2 c}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \sqrt {d x} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d}-8 b c d \left (\frac {\frac {\arctan \left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{c} d^{3/2}}}{2 c}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {c} d x+\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}-\frac {\log \left (\sqrt {c} d x-\sqrt {2} \sqrt [4]{c} \sqrt {d} \sqrt {d x}+d\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}}{2 d}}{2 c}\right )\) |
(2*Sqrt[d*x]*(a + b*ArcTanh[c*x^2]))/d - 8*b*c*d*((ArcTan[(c^(1/4)*Sqrt[d* x])/Sqrt[d]]/(2*c^(1/4)*d^(3/2)) + ArcTanh[(c^(1/4)*Sqrt[d*x])/Sqrt[d]]/(2 *c^(1/4)*d^(3/2)))/(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)*Sqrt[d*x])/S qrt[d]]/(Sqrt[2]*c^(1/4)*Sqrt[d]))/(2*d) + (-1/2*Log[d + Sqrt[c]*d*x - Sqr t[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 *d))/(2*c))
3.1.85.3.1 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) 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 = 0.61 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {d x}\, a +2 b \left (\sqrt {d x}\, \operatorname {arctanh}\left (c \,x^{2}\right )-4 c \,d^{2} \left (-\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c \,d^{2}}+\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{8 c \,d^{2}}\right )\right )}{d}\) | \(242\) |
| default | \(\frac {2 \sqrt {d x}\, a +2 b \left (\sqrt {d x}\, \operatorname {arctanh}\left (c \,x^{2}\right )-4 c \,d^{2} \left (-\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c \,d^{2}}+\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{8 c \,d^{2}}\right )\right )}{d}\) | \(242\) |
| parts | \(\frac {2 a \sqrt {d x}}{d}+\frac {2 b \left (\sqrt {d x}\, \operatorname {arctanh}\left (c \,x^{2}\right )-4 c \,d^{2} \left (-\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}{d x -\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {d^{2}}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c \,d^{2}}+\frac {\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{c}\right )^{\frac {1}{4}}}\right )\right )}{8 c \,d^{2}}\right )\right )}{d}\) | \(245\) |
2/d*((d*x)^(1/2)*a+b*((d*x)^(1/2)*arctanh(c*x^2)-4*c*d^2*(-1/16/c*(d^2/c)^ (1/4)/d^2*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/c*(d^2/c)^(1/4)/d^2*(ln(((d*x)^(1/2)+(d^2/c)^(1/4))/((d*x)^(1/2)-(d^ 2/c)^(1/4)))+2*arctan((d*x)^(1/2)/(d^2/c)^(1/4))))))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.24 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{\sqrt {d x}} \, dx=-\frac {\left (\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d \log \left (\sqrt {d x} b + \left (\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d\right ) + i \, \left (\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d \log \left (\sqrt {d x} b + i \, \left (\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d\right ) - i \, \left (\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d \log \left (\sqrt {d x} b - i \, \left (\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d\right ) - \left (\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d \log \left (\sqrt {d x} b - \left (\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d\right ) - \left (-\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d \log \left (\sqrt {d x} b + \left (-\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d\right ) - i \, \left (-\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d \log \left (\sqrt {d x} b + i \, \left (-\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d\right ) + i \, \left (-\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d \log \left (\sqrt {d x} b - i \, \left (-\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d\right ) + \left (-\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d \log \left (\sqrt {d x} b - \left (-\frac {b^{4}}{c d^{2}}\right )^{\frac {1}{4}} d\right ) - \sqrt {d x} {\left (b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a\right )}}{d} \]
-((b^4/(c*d^2))^(1/4)*d*log(sqrt(d*x)*b + (b^4/(c*d^2))^(1/4)*d) + I*(b^4/ (c*d^2))^(1/4)*d*log(sqrt(d*x)*b + I*(b^4/(c*d^2))^(1/4)*d) - I*(b^4/(c*d^ 2))^(1/4)*d*log(sqrt(d*x)*b - I*(b^4/(c*d^2))^(1/4)*d) - (b^4/(c*d^2))^(1/ 4)*d*log(sqrt(d*x)*b - (b^4/(c*d^2))^(1/4)*d) - (-b^4/(c*d^2))^(1/4)*d*log (sqrt(d*x)*b + (-b^4/(c*d^2))^(1/4)*d) - I*(-b^4/(c*d^2))^(1/4)*d*log(sqrt (d*x)*b + I*(-b^4/(c*d^2))^(1/4)*d) + I*(-b^4/(c*d^2))^(1/4)*d*log(sqrt(d* x)*b - I*(-b^4/(c*d^2))^(1/4)*d) + (-b^4/(c*d^2))^(1/4)*d*log(sqrt(d*x)*b - (-b^4/(c*d^2))^(1/4)*d) - sqrt(d*x)*(b*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2 *a))/d
\[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{\sqrt {d x}} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c x^{2} \right )}}{\sqrt {d x}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{\sqrt {d x}} \, dx=\frac {{\left (4 \, \sqrt {d x} \operatorname {artanh}\left (c x^{2}\right ) + \frac {c {\left (\frac {\frac {2 \, \sqrt {2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} + 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} \sqrt {d} - 2 \, \sqrt {d x} \sqrt {c}\right )}}{2 \, \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} + \frac {\sqrt {2} d^{\frac {5}{2}} \log \left (\sqrt {c} d x + \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {1}{4}}} - \frac {\sqrt {2} d^{\frac {5}{2}} \log \left (\sqrt {c} d x - \sqrt {2} \sqrt {d x} c^{\frac {1}{4}} \sqrt {d} + d\right )}{c^{\frac {1}{4}}}}{c} - \frac {2 \, {\left (\frac {2 \, d^{3} \arctan \left (\frac {\sqrt {d x} \sqrt {c}}{\sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}} - \frac {d^{3} \log \left (\frac {\sqrt {d x} \sqrt {c} - \sqrt {\sqrt {c} d}}{\sqrt {d x} \sqrt {c} + \sqrt {\sqrt {c} d}}\right )}{\sqrt {\sqrt {c} d}}\right )}}{c}\right )}}{d^{2}}\right )} b + 4 \, \sqrt {d x} a}{2 \, d} \]
1/2*((4*sqrt(d*x)*arctanh(c*x^2) + c*((2*sqrt(2)*d^3*arctan(1/2*sqrt(2)*(s qrt(2)*c^(1/4)*sqrt(d) + 2*sqrt(d*x)*sqrt(c))/sqrt(sqrt(c)*d))/sqrt(sqrt(c )*d) + 2*sqrt(2)*d^3*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(2)*d^(5/2)*log(sqrt (c)*d*x + sqrt(2)*sqrt(d*x)*c^(1/4)*sqrt(d) + d)/c^(1/4) - sqrt(2)*d^(5/2) *log(sqrt(c)*d*x - sqrt(2)*sqrt(d*x)*c^(1/4)*sqrt(d) + d)/c^(1/4))/c - 2*( 2*d^3*arctan(sqrt(d*x)*sqrt(c)/sqrt(sqrt(c)*d))/sqrt(sqrt(c)*d) - d^3*log( (sqrt(d*x)*sqrt(c) - sqrt(sqrt(c)*d))/(sqrt(d*x)*sqrt(c) + sqrt(sqrt(c)*d) ))/sqrt(sqrt(c)*d))/c)/d^2)*b + 4*sqrt(d*x)*a)/d
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (198) = 396\).
Time = 0.29 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.73 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{\sqrt {d x}} \, dx=\frac {{\left (c d^{2} {\left (\frac {2 \, \sqrt {2} \left (c^{3} d^{2}\right )^{\frac {1}{4}} \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 {1}{4}} \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 {1}{4}} \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 {1}{4}} \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 {1}{4}} \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 {1}{4}} \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 {1}{4}} \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 {1}{4}} \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}}\right )} + 2 \, \sqrt {d x} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )\right )} b + 4 \, \sqrt {d x} a}{2 \, d} \]
1/2*((c*d^2*(2*sqrt(2)*(c^3*d^2)^(1/4)*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)^(1/4) *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)^(1/4)*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)^(1 /4)*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)^(1/4)*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)^(1/4)*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)^ (1/4)*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)^(1/4)*log(d*x - sqrt(2)*sqrt(d*x)*(-d^2/c)^(1/4) + s qrt(-d^2/c))/(c^2*d^2)) + 2*sqrt(d*x)*log(-(c*x^2 + 1)/(c*x^2 - 1)))*b + 4 *sqrt(d*x)*a)/d
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{\sqrt {d x}} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x^2\right )}{\sqrt {d\,x}} \,d x \]