Integrand size = 21, antiderivative size = 191 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx=-\frac {2}{27 a \left (1-a^2 x^2\right )^{3/2}}-\frac {40}{9 a \sqrt {1-a^2 x^2}}+\frac {2 x \text {arctanh}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}}+\frac {40 x \text {arctanh}(a x)}{9 \sqrt {1-a^2 x^2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \text {arctanh}(a x)^3}{3 \sqrt {1-a^2 x^2}} \]
-2/27/a/(-a^2*x^2+1)^(3/2)+2/9*x*arctanh(a*x)/(-a^2*x^2+1)^(3/2)-1/3*arcta nh(a*x)^2/a/(-a^2*x^2+1)^(3/2)+1/3*x*arctanh(a*x)^3/(-a^2*x^2+1)^(3/2)-40/ 9/a/(-a^2*x^2+1)^(1/2)+40/9*x*arctanh(a*x)/(-a^2*x^2+1)^(1/2)-2*arctanh(a* x)^2/a/(-a^2*x^2+1)^(1/2)+2/3*x*arctanh(a*x)^3/(-a^2*x^2+1)^(1/2)
Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.46 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {-122+120 a^2 x^2-6 a x \left (-21+20 a^2 x^2\right ) \text {arctanh}(a x)+9 \left (-7+6 a^2 x^2\right ) \text {arctanh}(a x)^2-9 a x \left (-3+2 a^2 x^2\right ) \text {arctanh}(a x)^3}{27 a \left (1-a^2 x^2\right )^{3/2}} \]
(-122 + 120*a^2*x^2 - 6*a*x*(-21 + 20*a^2*x^2)*ArcTanh[a*x] + 9*(-7 + 6*a^ 2*x^2)*ArcTanh[a*x]^2 - 9*a*x*(-3 + 2*a^2*x^2)*ArcTanh[a*x]^3)/(27*a*(1 - a^2*x^2)^(3/2))
Time = 0.86 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.26, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6526, 6522, 6520, 6524, 6520}
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.
\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6526 |
\(\displaystyle \frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {2}{3} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6522 |
\(\displaystyle \frac {2}{3} \left (\frac {2}{3} \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {2}{3} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6520 |
\(\displaystyle \frac {2}{3} \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 6524 |
\(\displaystyle \frac {2}{3} \left (6 \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}\right )+\frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 6520 |
\(\displaystyle \frac {x \text {arctanh}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\text {arctanh}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {2}{3} \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\) |
-1/3*ArcTanh[a*x]^2/(a*(1 - a^2*x^2)^(3/2)) + (x*ArcTanh[a*x]^3)/(3*(1 - a ^2*x^2)^(3/2)) + (2*(-1/9*1/(a*(1 - a^2*x^2)^(3/2)) + (x*ArcTanh[a*x])/(3* (1 - a^2*x^2)^(3/2)) + (2*(-(1/(a*Sqrt[1 - a^2*x^2])) + (x*ArcTanh[a*x])/S qrt[1 - a^2*x^2]))/3))/3 + (2*((-3*ArcTanh[a*x]^2)/(a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] + 6*(-(1/(a*Sqrt[1 - a^2*x^2])) + (x *ArcTanh[a*x])/Sqrt[1 - a^2*x^2])))/3
3.5.76.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb ol] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTanh[c*x])/(d* Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/( 2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x]) /; Fre eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) ), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 *p*(p - 1) Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^(p - 1)/(4 *c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p /(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1 )*(a + b*ArcTanh[c*x])^p, x], x] + Simp[b^2*p*((p - 1)/(4*(q + 1)^2)) Int [(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
Time = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.55
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (18 \operatorname {arctanh}\left (a x \right )^{3} a^{3} x^{3}+120 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )-54 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )^{2}-27 \operatorname {arctanh}\left (a x \right )^{3} a x -120 a^{2} x^{2}-126 a x \,\operatorname {arctanh}\left (a x \right )+63 \operatorname {arctanh}\left (a x \right )^{2}+122\right )}{27 a \left (a^{2} x^{2}-1\right )^{2}}\) | \(105\) |
-1/27/a*(-a^2*x^2+1)^(1/2)*(18*arctanh(a*x)^3*a^3*x^3+120*a^3*x^3*arctanh( a*x)-54*a^2*x^2*arctanh(a*x)^2-27*arctanh(a*x)^3*a*x-120*a^2*x^2-126*a*x*a rctanh(a*x)+63*arctanh(a*x)^2+122)/(a^2*x^2-1)^2
Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.70 \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\frac {{\left (960 \, a^{2} x^{2} - 9 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 18 \, {\left (6 \, a^{2} x^{2} - 7\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 24 \, {\left (20 \, a^{3} x^{3} - 21 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 976\right )} \sqrt {-a^{2} x^{2} + 1}}{216 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \]
1/216*(960*a^2*x^2 - 9*(2*a^3*x^3 - 3*a*x)*log(-(a*x + 1)/(a*x - 1))^3 + 1 8*(6*a^2*x^2 - 7)*log(-(a*x + 1)/(a*x - 1))^2 - 24*(20*a^3*x^3 - 21*a*x)*l og(-(a*x + 1)/(a*x - 1)) - 976)*sqrt(-a^2*x^2 + 1)/(a^5*x^4 - 2*a^3*x^2 + a)
\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \]