Integrand size = 25, antiderivative size = 297 \[ \int x^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 e^{3/2}}-\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {e}}-\frac {28 d^2 \sqrt {x} \sqrt {d+e x^2}}{135 e^2 \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {2}{9} x^{9/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {28 d^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{9/4} \sqrt {d+e x^2}}-\frac {14 d^{9/4} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{135 e^{9/4} \sqrt {d+e x^2}} \]
2/9*x^(9/2)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))+28/405*d*x^(3/2)*(e*x^2+d)^ (1/2)/e^(3/2)-4/81*x^(7/2)*(e*x^2+d)^(1/2)/e^(1/2)-28/135*d^2*x^(1/2)*(e*x ^2+d)^(1/2)/e^2/(d^(1/2)+x*e^(1/2))+28/135*d^(9/4)*(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)))*EllipticE (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^(9/4)/(e*x^2+d)^(1/2)-14/135*d^(9 /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^(9/4)/ (e*x^2+d)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.42 \[ \int x^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {2 x^{3/2} \left (14 d^2+4 d e x^2-10 e^2 x^4+45 e^{3/2} x^3 \sqrt {d+e x^2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-14 d^2 \sqrt {1+\frac {e x^2}{d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {e x^2}{d}\right )\right )}{405 e^{3/2} \sqrt {d+e x^2}} \]
(2*x^(3/2)*(14*d^2 + 4*d*e*x^2 - 10*e^2*x^4 + 45*e^(3/2)*x^3*Sqrt[d + e*x^ 2]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] - 14*d^2*Sqrt[1 + (e*x^2)/d]*Hyper geometric2F1[1/2, 3/4, 7/4, -((e*x^2)/d)]))/(405*e^(3/2)*Sqrt[d + e*x^2])
Time = 0.43 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6775, 262, 262, 266, 834, 27, 761, 1510}
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.
| \(\displaystyle \int x^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx\) |
| \(\Big \downarrow \) 6775 |
| \(\displaystyle \frac {2}{9} x^{9/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {e} \int \frac {x^{9/2}}{\sqrt {e x^2+d}}dx\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2}{9} x^{9/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \int \frac {x^{5/2}}{\sqrt {e x^2+d}}dx}{9 e}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2}{9} x^{9/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {3 d \int \frac {\sqrt {x}}{\sqrt {e x^2+d}}dx}{5 e}\right )}{9 e}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2}{9} x^{9/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {6 d \int \frac {x}{\sqrt {e x^2+d}}d\sqrt {x}}{5 e}\right )}{9 e}\right )\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {2}{9} x^{9/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {6 d \left (\frac {\sqrt {d} \int \frac {1}{\sqrt {e x^2+d}}d\sqrt {x}}{\sqrt {e}}-\frac {\sqrt {d} \int \frac {\sqrt {d}-\sqrt {e} x}{\sqrt {d} \sqrt {e x^2+d}}d\sqrt {x}}{\sqrt {e}}\right )}{5 e}\right )}{9 e}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} x^{9/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {6 d \left (\frac {\sqrt {d} \int \frac {1}{\sqrt {e x^2+d}}d\sqrt {x}}{\sqrt {e}}-\frac {\int \frac {\sqrt {d}-\sqrt {e} x}{\sqrt {e x^2+d}}d\sqrt {x}}{\sqrt {e}}\right )}{5 e}\right )}{9 e}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2}{9} x^{9/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {6 d \left (\frac {\sqrt [4]{d} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{2 e^{3/4} \sqrt {d+e x^2}}-\frac {\int \frac {\sqrt {d}-\sqrt {e} x}{\sqrt {e x^2+d}}d\sqrt {x}}{\sqrt {e}}\right )}{5 e}\right )}{9 e}\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2}{9} x^{9/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {6 d \left (\frac {\sqrt [4]{d} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{2 e^{3/4} \sqrt {d+e x^2}}-\frac {\frac {\sqrt [4]{d} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{\sqrt [4]{e} \sqrt {d+e x^2}}-\frac {\sqrt {x} \sqrt {d+e x^2}}{\sqrt {d}+\sqrt {e} x}}{\sqrt {e}}\right )}{5 e}\right )}{9 e}\right )\) |
(2*x^(9/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/9 - (2*Sqrt[e]*((2*x^(7/2 )*Sqrt[d + e*x^2])/(9*e) - (7*d*((2*x^(3/2)*Sqrt[d + e*x^2])/(5*e) - (6*d* (-((-((Sqrt[x]*Sqrt[d + e*x^2])/(Sqrt[d] + Sqrt[e]*x)) + (d^(1/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTan [(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(e^(1/4)*Sqrt[d + e*x^2]))/Sqrt[e]) + ( d^(1/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*El lipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(2*e^(3/4)*Sqrt[d + e*x ^2])))/(5*e)))/(9*e)))/9
3.1.23.3.1 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[((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[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/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]
\[\int x^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.31 \[ \int x^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {45 \, e^{2} x^{\frac {9}{2}} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right ) + 84 \, d^{2} {\rm weierstrassZeta}\left (-\frac {4 \, d}{e}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, d}{e}, 0, x\right )\right ) - 4 \, {\left (5 \, e x^{3} - 7 \, d x\right )} \sqrt {e x^{2} + d} \sqrt {e} \sqrt {x}}{405 \, e^{2}} \]
1/405*(45*e^2*x^(9/2)*log((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d) + 84*d^2*weierstrassZeta(-4*d/e, 0, weierstrassPInverse(-4*d/e, 0, x)) - 4* (5*e*x^3 - 7*d*x)*sqrt(e*x^2 + d)*sqrt(e)*sqrt(x))/e^2
Timed out. \[ \int x^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\text {Timed out} \]
\[ \int x^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { x^{\frac {7}{2}} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \]
1/9*x^(9/2)*log(sqrt(e)*x + sqrt(e*x^2 + d)) - 1/9*x^(9/2)*log(-sqrt(e)*x + sqrt(e*x^2 + d)) - 2*d*sqrt(e)*integrate(-1/9*x*e^(1/2*log(e*x^2 + d) + 7/2*log(x))/(e^2*x^4 + d*e*x^2 - (e*x^2 + d)^2), x)
\[ \int x^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { x^{\frac {7}{2}} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \]
Timed out. \[ \int x^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int x^{7/2}\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]