Integrand size = 23, antiderivative size = 114 \[ \int x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {d^3 \sqrt {d+e x^2}}{7 e^{7/2}}-\frac {d^2 \left (d+e x^2\right )^{3/2}}{7 e^{7/2}}+\frac {3 d \left (d+e x^2\right )^{5/2}}{35 e^{7/2}}-\frac {\left (d+e x^2\right )^{7/2}}{49 e^{7/2}}+\frac {1}{7} x^7 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
-1/7*d^2*(e*x^2+d)^(3/2)/e^(7/2)+3/35*d*(e*x^2+d)^(5/2)/e^(7/2)-1/49*(e*x^ 2+d)^(7/2)/e^(7/2)+1/7*x^7*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))+1/7*d^3*(e*x ^2+d)^(1/2)/e^(7/2)
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.69 \[ \int x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {\sqrt {d+e x^2} \left (16 d^3-8 d^2 e x^2+6 d e^2 x^4-5 e^3 x^6\right )}{245 e^{7/2}}+\frac {1}{7} x^7 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
(Sqrt[d + e*x^2]*(16*d^3 - 8*d^2*e*x^2 + 6*d*e^2*x^4 - 5*e^3*x^6))/(245*e^ (7/2)) + (x^7*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/7
Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6775, 243, 53, 2009}
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.
\(\displaystyle \int x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx\) |
\(\Big \downarrow \) 6775 |
\(\displaystyle \frac {1}{7} x^7 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{7} \sqrt {e} \int \frac {x^7}{\sqrt {e x^2+d}}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{7} x^7 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{14} \sqrt {e} \int \frac {x^6}{\sqrt {e x^2+d}}dx^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{7} x^7 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{14} \sqrt {e} \int \left (-\frac {d^3}{e^3 \sqrt {e x^2+d}}+\frac {3 \sqrt {e x^2+d} d^2}{e^3}-\frac {3 \left (e x^2+d\right )^{3/2} d}{e^3}+\frac {\left (e x^2+d\right )^{5/2}}{e^3}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{7} x^7 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{14} \sqrt {e} \left (-\frac {2 d^3 \sqrt {d+e x^2}}{e^4}+\frac {2 d^2 \left (d+e x^2\right )^{3/2}}{e^4}+\frac {2 \left (d+e x^2\right )^{7/2}}{7 e^4}-\frac {6 d \left (d+e x^2\right )^{5/2}}{5 e^4}\right )\) |
-1/14*(Sqrt[e]*((-2*d^3*Sqrt[d + e*x^2])/e^4 + (2*d^2*(d + e*x^2)^(3/2))/e ^4 - (6*d*(d + e*x^2)^(5/2))/(5*e^4) + (2*(d + e*x^2)^(7/2))/(7*e^4))) + ( x^7*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/7
3.1.9.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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. \(223\) vs. \(2(84)=168\).
Time = 0.02 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.96
method | result | size |
default | \(\frac {x^{7} \operatorname {arctanh}\left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{7}+\frac {e^{\frac {3}{2}} \left (\frac {x^{8} \sqrt {e \,x^{2}+d}}{9 e}-\frac {8 d \left (\frac {x^{6} \sqrt {e \,x^{2}+d}}{7 e}-\frac {6 d \left (\frac {x^{4} \sqrt {e \,x^{2}+d}}{5 e}-\frac {4 d \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{5 e}\right )}{7 e}\right )}{9 e}\right )}{7 d}-\frac {\sqrt {e}\, \left (\frac {x^{6} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{9 e}-\frac {2 d \left (\frac {x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{7 e}-\frac {4 d \left (\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}\right )}{7 e}\right )}{3 e}\right )}{7 d}\) | \(224\) |
parts | \(\frac {x^{7} \operatorname {arctanh}\left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{7}+\frac {e^{\frac {3}{2}} \left (\frac {x^{8} \sqrt {e \,x^{2}+d}}{9 e}-\frac {8 d \left (\frac {x^{6} \sqrt {e \,x^{2}+d}}{7 e}-\frac {6 d \left (\frac {x^{4} \sqrt {e \,x^{2}+d}}{5 e}-\frac {4 d \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{5 e}\right )}{7 e}\right )}{9 e}\right )}{7 d}-\frac {\sqrt {e}\, \left (\frac {x^{6} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{9 e}-\frac {2 d \left (\frac {x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{7 e}-\frac {4 d \left (\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}\right )}{7 e}\right )}{3 e}\right )}{7 d}\) | \(224\) |
1/7*x^7*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))+1/7*e^(3/2)/d*(1/9*x^8/e*(e*x^2 +d)^(1/2)-8/9*d/e*(1/7*x^6/e*(e*x^2+d)^(1/2)-6/7*d/e*(1/5*x^4/e*(e*x^2+d)^ (1/2)-4/5*d/e*(1/3*x^2/e*(e*x^2+d)^(1/2)-2/3*d/e^2*(e*x^2+d)^(1/2)))))-1/7 *e^(1/2)/d*(1/9*x^6*(e*x^2+d)^(3/2)/e-2/3*d/e*(1/7*x^4*(e*x^2+d)^(3/2)/e-4 /7*d/e*(1/5*x^2*(e*x^2+d)^(3/2)/e-2/15*d/e^2*(e*x^2+d)^(3/2))))
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77 \[ \int x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {35 \, e^{4} x^{7} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right ) - 2 \, {\left (5 \, e^{3} x^{6} - 6 \, d e^{2} x^{4} + 8 \, d^{2} e x^{2} - 16 \, d^{3}\right )} \sqrt {e x^{2} + d} \sqrt {e}}{490 \, e^{4}} \]
1/490*(35*e^4*x^7*log((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d) - 2*( 5*e^3*x^6 - 6*d*e^2*x^4 + 8*d^2*e*x^2 - 16*d^3)*sqrt(e*x^2 + d)*sqrt(e))/e ^4
Time = 1.39 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.02 \[ \int x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\begin {cases} \frac {16 d^{3} \sqrt {d + e x^{2}}}{245 e^{\frac {7}{2}}} - \frac {8 d^{2} x^{2} \sqrt {d + e x^{2}}}{245 e^{\frac {5}{2}}} + \frac {6 d x^{4} \sqrt {d + e x^{2}}}{245 e^{\frac {3}{2}}} + \frac {x^{7} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{7} - \frac {x^{6} \sqrt {d + e x^{2}}}{49 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((16*d**3*sqrt(d + e*x**2)/(245*e**(7/2)) - 8*d**2*x**2*sqrt(d + e*x**2)/(245*e**(5/2)) + 6*d*x**4*sqrt(d + e*x**2)/(245*e**(3/2)) + x**7*a tanh(sqrt(e)*x/sqrt(d + e*x**2))/7 - x**6*sqrt(d + e*x**2)/(49*sqrt(e)), N e(e, 0)), (0, True))
Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.36 \[ \int x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {1}{7} \, x^{7} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ) - \frac {35 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} - 135 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d + 189 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} - 105 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3}}{2205 \, d e^{\frac {7}{2}}} + \frac {35 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x^{2} + d} d^{4}}{2205 \, d e^{\frac {7}{2}}} \]
1/7*x^7*arctanh(sqrt(e)*x/sqrt(e*x^2 + d)) - 1/2205*(35*(e*x^2 + d)^(9/2) - 135*(e*x^2 + d)^(7/2)*d + 189*(e*x^2 + d)^(5/2)*d^2 - 105*(e*x^2 + d)^(3 /2)*d^3)/(d*e^(7/2)) + 1/2205*(35*(e*x^2 + d)^(9/2) - 180*(e*x^2 + d)^(7/2 )*d + 378*(e*x^2 + d)^(5/2)*d^2 - 420*(e*x^2 + d)^(3/2)*d^3 + 315*sqrt(e*x ^2 + d)*d^4)/(d*e^(7/2))
\[ \int x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { x^{6} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \]
Timed out. \[ \int x^6 \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int x^6\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]