Integrand size = 23, antiderivative size = 127 \[ \int x^5 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=-\frac {5 d^2 x \sqrt {d+e x^2}}{96 e^{5/2}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 e^{3/2}}-\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {e}}+\frac {5 d^3 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{96 e^3}+\frac {1}{6} x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
5/96*d^3*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^3+1/6*x^6*arctanh(x*e^(1/2)/ (e*x^2+d)^(1/2))-5/96*d^2*x*(e*x^2+d)^(1/2)/e^(5/2)+5/144*d*x^3*(e*x^2+d)^ (1/2)/e^(3/2)-1/36*x^5*(e*x^2+d)^(1/2)/e^(1/2)
Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.78 \[ \int x^5 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {\sqrt {e} x \sqrt {d+e x^2} \left (-15 d^2+10 d e x^2-8 e^2 x^4\right )+48 e^3 x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+15 d^3 \log \left (\sqrt {e} x+\sqrt {d+e x^2}\right )}{288 e^3} \]
(Sqrt[e]*x*Sqrt[d + e*x^2]*(-15*d^2 + 10*d*e*x^2 - 8*e^2*x^4) + 48*e^3*x^6 *ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + 15*d^3*Log[Sqrt[e]*x + Sqrt[d + e* x^2]])/(288*e^3)
Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6775, 262, 262, 262, 224, 219}
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.
\(\displaystyle \int x^5 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx\) |
\(\Big \downarrow \) 6775 |
\(\displaystyle \frac {1}{6} x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \int \frac {x^6}{\sqrt {e x^2+d}}dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{6} x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \left (\frac {x^5 \sqrt {d+e x^2}}{6 e}-\frac {5 d \int \frac {x^4}{\sqrt {e x^2+d}}dx}{6 e}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{6} x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \left (\frac {x^5 \sqrt {d+e x^2}}{6 e}-\frac {5 d \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \int \frac {x^2}{\sqrt {e x^2+d}}dx}{4 e}\right )}{6 e}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{6} x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \left (\frac {x^5 \sqrt {d+e x^2}}{6 e}-\frac {5 d \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \int \frac {1}{\sqrt {e x^2+d}}dx}{2 e}\right )}{4 e}\right )}{6 e}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{6} x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \left (\frac {x^5 \sqrt {d+e x^2}}{6 e}-\frac {5 d \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 e}\right )}{4 e}\right )}{6 e}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{6} x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {e} \left (\frac {x^5 \sqrt {d+e x^2}}{6 e}-\frac {5 d \left (\frac {x^3 \sqrt {d+e x^2}}{4 e}-\frac {3 d \left (\frac {x \sqrt {d+e x^2}}{2 e}-\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 e^{3/2}}\right )}{4 e}\right )}{6 e}\right )\) |
(x^6*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/6 - (Sqrt[e]*((x^5*Sqrt[d + e*x ^2])/(6*e) - (5*d*((x^3*Sqrt[d + e*x^2])/(4*e) - (3*d*((x*Sqrt[d + e*x^2]) /(2*e) - (d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*e^(3/2))))/(4*e)))/(6 *e)))/6
3.1.1.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(252\) vs. \(2(97)=194\).
Time = 0.02 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.99
method | result | size |
default | \(\frac {x^{6} \operatorname {arctanh}\left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{6}+\frac {e^{\frac {3}{2}} \left (\frac {x^{7} \sqrt {e \,x^{2}+d}}{8 e}-\frac {7 d \left (\frac {x^{5} \sqrt {e \,x^{2}+d}}{6 e}-\frac {5 d \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )}{6 e}\right )}{8 e}\right )}{6 d}-\frac {\sqrt {e}\, \left (\frac {x^{5} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{8 e}-\frac {5 d \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 e}\right )}{8 e}\right )}{6 d}\) | \(253\) |
parts | \(\frac {x^{6} \operatorname {arctanh}\left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{6}+\frac {e^{\frac {3}{2}} \left (\frac {x^{7} \sqrt {e \,x^{2}+d}}{8 e}-\frac {7 d \left (\frac {x^{5} \sqrt {e \,x^{2}+d}}{6 e}-\frac {5 d \left (\frac {x^{3} \sqrt {e \,x^{2}+d}}{4 e}-\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2 e}-\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 e^{\frac {3}{2}}}\right )}{4 e}\right )}{6 e}\right )}{8 e}\right )}{6 d}-\frac {\sqrt {e}\, \left (\frac {x^{5} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{8 e}-\frac {5 d \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 e}\right )}{8 e}\right )}{6 d}\) | \(253\) |
1/6*x^6*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))+1/6*e^(3/2)/d*(1/8*x^7/e*(e*x^2 +d)^(1/2)-7/8*d/e*(1/6*x^5/e*(e*x^2+d)^(1/2)-5/6*d/e*(1/4*x^3/e*(e*x^2+d)^ (1/2)-3/4*d/e*(1/2*x/e*(e*x^2+d)^(1/2)-1/2*d/e^(3/2)*ln(x*e^(1/2)+(e*x^2+d )^(1/2))))))-1/6*e^(1/2)/d*(1/8*x^5*(e*x^2+d)^(3/2)/e-5/8*d/e*(1/6*x^3*(e* x^2+d)^(3/2)/e-1/2*d/e*(1/4*x*(e*x^2+d)^(3/2)/e-1/4*d/e*(1/2*x*(e*x^2+d)^( 1/2)+1/2*d/e^(1/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))))))
Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.68 \[ \int x^5 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=-\frac {2 \, {\left (8 \, e^{2} x^{5} - 10 \, d e x^{3} + 15 \, d^{2} x\right )} \sqrt {e x^{2} + d} \sqrt {e} - 3 \, {\left (16 \, e^{3} x^{6} + 5 \, d^{3}\right )} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right )}{576 \, e^{3}} \]
-1/576*(2*(8*e^2*x^5 - 10*d*e*x^3 + 15*d^2*x)*sqrt(e*x^2 + d)*sqrt(e) - 3* (16*e^3*x^6 + 5*d^3)*log((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d))/e ^3
Time = 0.91 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95 \[ \int x^5 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\begin {cases} \frac {5 d^{3} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{96 e^{3}} - \frac {5 d^{2} x \sqrt {d + e x^{2}}}{96 e^{\frac {5}{2}}} + \frac {5 d x^{3} \sqrt {d + e x^{2}}}{144 e^{\frac {3}{2}}} + \frac {x^{6} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{6} - \frac {x^{5} \sqrt {d + e x^{2}}}{36 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((5*d**3*atanh(sqrt(e)*x/sqrt(d + e*x**2))/(96*e**3) - 5*d**2*x*s qrt(d + e*x**2)/(96*e**(5/2)) + 5*d*x**3*sqrt(d + e*x**2)/(144*e**(3/2)) + x**6*atanh(sqrt(e)*x/sqrt(d + e*x**2))/6 - x**5*sqrt(d + e*x**2)/(36*sqrt (e)), Ne(e, 0)), (0, True))
\[ \int x^5 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { x^{5} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \]
1/12*x^6*log(sqrt(e)*x + sqrt(e*x^2 + d)) - 1/12*x^6*log(-sqrt(e)*x + sqrt (e*x^2 + d)) - 1/2*d*sqrt(e)*integrate(-1/3*sqrt(e*x^2 + d)*x^6/(e^2*x^4 + d*e*x^2 - (e*x^2 + d)^2), x)
\[ \int x^5 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { x^{5} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \]
Timed out. \[ \int x^5 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int x^5\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]