Integrand size = 25, antiderivative size = 142 \[ \int \sqrt {x} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=-\frac {4 \sqrt {x} \sqrt {d+e x^2}}{9 \sqrt {e}}+\frac {2}{3} x^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {2 d^{3/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{9 e^{3/4} \sqrt {d+e x^2}} \]
2/3*x^(3/2)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))-4/9*x^(1/2)*(e*x^2+d)^(1/2) /e^(1/2)+2/9*d^(3/4)*(cos(2*arctan(e^(1/4)*x^(1/2)/d^(1/4)))^2)^(1/2)/cos( 2*arctan(e^(1/4)*x^(1/2)/d^(1/4)))*EllipticF(sin(2*arctan(e^(1/4)*x^(1/2)/ d^(1/4))),1/2*2^(1/2))*(d^(1/2)+x*e^(1/2))*((e*x^2+d)/(d^(1/2)+x*e^(1/2))^ 2)^(1/2)/e^(3/4)/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.95 \[ \int \sqrt {x} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {2}{9} \sqrt {x} \left (-\frac {2 \sqrt {d+e x^2}}{\sqrt {e}}+3 x \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\right )+\frac {4 \sqrt {d} \sqrt {\frac {i \sqrt {d}}{\sqrt {e}}} \sqrt {1+\frac {d}{e x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {d}}{\sqrt {e}}}}{\sqrt {x}}\right ),-1\right )}{9 \sqrt {d+e x^2}} \]
(2*Sqrt[x]*((-2*Sqrt[d + e*x^2])/Sqrt[e] + 3*x*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]))/9 + (4*Sqrt[d]*Sqrt[(I*Sqrt[d])/Sqrt[e]]*Sqrt[1 + d/(e*x^2)]*x *EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1])/(9*Sqrt[d + e*x^2])
Time = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6775, 262, 266, 761}
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.
\(\displaystyle \int \sqrt {x} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx\) |
\(\Big \downarrow \) 6775 |
\(\displaystyle \frac {2}{3} x^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{3} \sqrt {e} \int \frac {x^{3/2}}{\sqrt {e x^2+d}}dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {2}{3} x^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{3} \sqrt {e} \left (\frac {2 \sqrt {x} \sqrt {d+e x^2}}{3 e}-\frac {d \int \frac {1}{\sqrt {x} \sqrt {e x^2+d}}dx}{3 e}\right )\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2}{3} x^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{3} \sqrt {e} \left (\frac {2 \sqrt {x} \sqrt {d+e x^2}}{3 e}-\frac {2 d \int \frac {1}{\sqrt {e x^2+d}}d\sqrt {x}}{3 e}\right )\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {2}{3} x^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{3} \sqrt {e} \left (\frac {2 \sqrt {x} \sqrt {d+e x^2}}{3 e}-\frac {d^{3/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{3 e^{5/4} \sqrt {d+e x^2}}\right )\) |
(2*x^(3/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/3 - (2*Sqrt[e]*((2*Sqrt[x ]*Sqrt[d + e*x^2])/(3*e) - (d^(3/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2) /(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1 /2])/(3*e^(5/4)*Sqrt[d + e*x^2])))/3
3.1.18.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[ArcTanh[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_ Symbol] :> Simp[(d*x)^(m + 1)*(ArcTanh[(c*x)/Sqrt[a + b*x^2]]/(d*(m + 1))), x] - Simp[c/(d*(m + 1)) Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; Fre eQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]
\[\int \sqrt {x}\, \operatorname {arctanh}\left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.49 \[ \int \sqrt {x} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {3 \, e x^{\frac {3}{2}} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right ) + 4 \, d {\rm weierstrassPInverse}\left (-\frac {4 \, d}{e}, 0, x\right ) - 4 \, \sqrt {e x^{2} + d} \sqrt {e} \sqrt {x}}{9 \, e} \]
1/9*(3*e*x^(3/2)*log((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d) + 4*d* weierstrassPInverse(-4*d/e, 0, x) - 4*sqrt(e*x^2 + d)*sqrt(e)*sqrt(x))/e
\[ \int \sqrt {x} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int \sqrt {x} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}\, dx \]
\[ \int \sqrt {x} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { \sqrt {x} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \]
-2*d*sqrt(e)*integrate(-1/3*x*e^(1/2*log(e*x^2 + d) + 1/2*log(x))/(e^2*x^4 + d*e*x^2 - (e*x^2 + d)^2), x) + 1/3*x^(3/2)*log(sqrt(e)*x + sqrt(e*x^2 + d)) - 1/3*x^(3/2)*log(-sqrt(e)*x + sqrt(e*x^2 + d))
\[ \int \sqrt {x} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { \sqrt {x} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \]
Timed out. \[ \int \sqrt {x} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int \sqrt {x}\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]