Optimal. Leaf size=191 \[ -\frac {40}{9 a \sqrt {1-a^2 x^2}}-\frac {2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^3}{3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {40 x \tanh ^{-1}(a x)}{9 \sqrt {1-a^2 x^2}}+\frac {2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5964, 5962, 5958, 5960} \[ -\frac {40}{9 a \sqrt {1-a^2 x^2}}-\frac {2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^3}{3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {40 x \tanh ^{-1}(a x)}{9 \sqrt {1-a^2 x^2}}+\frac {2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5958
Rule 5960
Rule 5962
Rule 5964
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=-\frac {\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {2}{3} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}}-\frac {\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^3}{3 \sqrt {1-a^2 x^2}}+\frac {4}{9} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+4 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, 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 \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}}+\frac {40 x \tanh ^{-1}(a x)}{9 \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)^3}{3 \sqrt {1-a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 87, normalized size = 0.46 \[ \frac {120 a^2 x^2-9 a x \left (2 a^2 x^2-3\right ) \tanh ^{-1}(a x)^3+9 \left (6 a^2 x^2-7\right ) \tanh ^{-1}(a x)^2-6 a x \left (20 a^2 x^2-21\right ) \tanh ^{-1}(a x)-122}{27 a \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 134, normalized size = 0.70 \[ \frac {{\left (960 \, a^{2} x^{2} - 9 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 18 \, {\left (6 \, a^{2} x^{2} - 7\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 24 \, {\left (20 \, a^{3} x^{3} - 21 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 976\right )} \sqrt {-a^{2} x^{2} + 1}}{216 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.44, size = 105, normalized size = 0.55 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (18 \arctanh \left (a x \right )^{3} x^{3} a^{3}+120 a^{3} x^{3} \arctanh \left (a x \right )-54 a^{2} x^{2} \arctanh \left (a x \right )^{2}-27 \arctanh \left (a x \right )^{3} a x -120 a^{2} x^{2}-126 a x \arctanh \left (a x \right )+63 \arctanh \left (a x \right )^{2}+122\right )}{27 a \left (a^{2} x^{2}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________