Optimal. Leaf size=291 \[ -\frac {245}{768 a \left (1-a^2 x^2\right )}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}-\frac {1}{216 a \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}+\frac {245 \tanh ^{-1}(a x)^2}{768 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.33, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5964, 5956, 5994, 261, 5960} \[ -\frac {245}{768 a \left (1-a^2 x^2\right )}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}-\frac {1}{216 a \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}-\frac {15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}+\frac {245 \tanh ^{-1}(a x)^2}{768 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 5956
Rule 5960
Rule 5964
Rule 5994
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^4} \, dx &=-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {1}{6} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^4} \, dx+\frac {5}{6} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx\\ &=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5}{36} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\frac {5}{16} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\frac {5}{8} \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}+\frac {5}{48} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\frac {15}{64} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{16} (15 a) \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {65 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {65 \tanh ^{-1}(a x)^2}{768 a}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}+\frac {15}{16} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{96} (5 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{128} (15 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}-\frac {65}{768 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {245 \tanh ^{-1}(a x)^2}{768 a}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}-\frac {1}{32} (15 a) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {1}{216 a \left (1-a^2 x^2\right )^3}-\frac {65}{2304 a \left (1-a^2 x^2\right )^2}-\frac {245}{768 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)}{36 \left (1-a^2 x^2\right )^3}+\frac {65 x \tanh ^{-1}(a x)}{576 \left (1-a^2 x^2\right )^2}+\frac {245 x \tanh ^{-1}(a x)}{384 \left (1-a^2 x^2\right )}+\frac {245 \tanh ^{-1}(a x)^2}{768 a}-\frac {\tanh ^{-1}(a x)^2}{12 a \left (1-a^2 x^2\right )^3}-\frac {5 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )^2}-\frac {15 \tanh ^{-1}(a x)^2}{32 a \left (1-a^2 x^2\right )}+\frac {x \tanh ^{-1}(a x)^3}{6 \left (1-a^2 x^2\right )^3}+\frac {5 x \tanh ^{-1}(a x)^3}{24 \left (1-a^2 x^2\right )^2}+\frac {5 x \tanh ^{-1}(a x)^3}{16 \left (1-a^2 x^2\right )}+\frac {5 \tanh ^{-1}(a x)^4}{64 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 143, normalized size = 0.49 \[ \frac {2205 a^4 x^4-4605 a^2 x^2+540 \left (a^2 x^2-1\right )^3 \tanh ^{-1}(a x)^4-144 a x \left (15 a^4 x^4-40 a^2 x^2+33\right ) \tanh ^{-1}(a x)^3-6 a x \left (735 a^4 x^4-1600 a^2 x^2+897\right ) \tanh ^{-1}(a x)+9 \left (245 a^6 x^6-375 a^4 x^4-105 a^2 x^2+299\right ) \tanh ^{-1}(a x)^2+2432}{6912 a \left (a^2 x^2-1\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 216, normalized size = 0.74 \[ \frac {8820 \, a^{4} x^{4} + 135 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} - 18420 \, a^{2} x^{2} - 72 \, {\left (15 \, a^{5} x^{5} - 40 \, a^{3} x^{3} + 33 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} + 9 \, {\left (245 \, a^{6} x^{6} - 375 \, a^{4} x^{4} - 105 \, a^{2} x^{2} + 299\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 12 \, {\left (735 \, a^{5} x^{5} - 1600 \, a^{3} x^{3} + 897 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 9728}{27648 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.97, size = 3550, normalized size = 12.20 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.37, size = 871, normalized size = 2.99 \[ -\frac {1}{96} \, {\left (\frac {2 \, {\left (15 \, a^{4} x^{5} - 40 \, a^{2} x^{3} + 33 \, x\right )}}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1} - \frac {15 \, \log \left (a x + 1\right )}{a} + \frac {15 \, \log \left (a x - 1\right )}{a}\right )} \operatorname {artanh}\left (a x\right )^{3} + \frac {{\left (180 \, a^{4} x^{4} - 420 \, a^{2} x^{2} - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} + 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 45 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 272\right )} a \operatorname {artanh}\left (a x\right )^{2}}{384 \, {\left (a^{8} x^{6} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} - a^{2}\right )}} + \frac {1}{27648} \, {\left (\frac {{\left (8820 \, a^{4} x^{4} - 135 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{4} + 540 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} \log \left (a x - 1\right ) - 135 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{4} - 18420 \, a^{2} x^{2} - 45 \, {\left (49 \, a^{6} x^{6} - 147 \, a^{4} x^{4} + 147 \, a^{2} x^{2} + 18 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 49\right )} \log \left (a x + 1\right )^{2} - 2205 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 90 \, {\left (6 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 49 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 9728\right )} a^{2}}{a^{10} x^{6} - 3 \, a^{8} x^{4} + 3 \, a^{6} x^{2} - a^{4}} - \frac {12 \, {\left (1470 \, a^{5} x^{5} - 3200 \, a^{3} x^{3} - 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{3} + 270 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right ) + 90 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{3} + 1794 \, a x - 15 \, {\left (49 \, a^{6} x^{6} - 147 \, a^{4} x^{4} + 147 \, a^{2} x^{2} + 18 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} - 49\right )} \log \left (a x + 1\right ) + 735 \, {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} a \operatorname {artanh}\left (a x\right )}{a^{9} x^{6} - 3 \, a^{7} x^{4} + 3 \, a^{5} x^{2} - a^{3}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.97, size = 1041, normalized size = 3.58 \[ \frac {\frac {1216}{3\,a}-\frac {1535\,a\,x^2}{2}+\frac {735\,a^3\,x^4}{2}}{1152\,a^6\,x^6-3456\,a^4\,x^4+3456\,a^2\,x^2-1152}-{\ln \left (1-a\,x\right )}^3\,\left (\frac {5\,\ln \left (a\,x+1\right )}{256\,a}-\frac {\frac {5\,a^4\,x^5}{16}-\frac {5\,a^2\,x^3}{6}+\frac {11\,x}{16}}{8\,a^6\,x^6-24\,a^4\,x^4+24\,a^2\,x^2-8}\right )+\frac {5\,{\ln \left (a\,x+1\right )}^4}{1024\,a}+\frac {5\,{\ln \left (1-a\,x\right )}^4}{1024\,a}+{\ln \left (1-a\,x\right )}^2\,\left (\frac {15\,{\ln \left (a\,x+1\right )}^2}{512\,a}+\frac {245}{3072\,a}+\frac {\frac {37\,x}{2}-35\,a\,x^2+\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}+15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}-\frac {\frac {37\,x}{2}+35\,a\,x^2-\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}-15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}-\frac {\ln \left (a\,x+1\right )\,\left (30\,a^4\,x^5-80\,a^2\,x^3+66\,x\right )}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}\right )+{\ln \left (a\,x+1\right )}^2\,\left (\frac {\frac {17}{96\,a^2}-\frac {35\,x^2}{128}+\frac {15\,a^2\,x^4}{128}}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6}+\frac {245}{3072\,a}\right )+\ln \left (1-a\,x\right )\,\left (\frac {36\,x+22\,a\,x^2-\frac {23}{2\,a}-67\,a^2\,x^3-\frac {21\,a^3\,x^4}{2}+31\,a^4\,x^5}{768\,a^6\,x^6-2304\,a^4\,x^4+2304\,a^2\,x^2-768}-\frac {5\,{\ln \left (a\,x+1\right )}^3}{256\,a}-\ln \left (a\,x+1\right )\,\left (\frac {\frac {37\,x}{2}-35\,a\,x^2+\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}+15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{128\,a^6\,x^6-384\,a^4\,x^4+384\,a^2\,x^2-128}-\frac {\frac {37\,x}{2}+35\,a\,x^2-\frac {68}{3\,a}-\frac {82\,a^2\,x^3}{3}-15\,a^3\,x^4+\frac {23\,a^4\,x^5}{2}}{128\,a^6\,x^6-384\,a^4\,x^4+384\,a^2\,x^2-128}+\frac {245\,\left (a^6\,x^6-3\,a^4\,x^4+3\,a^2\,x^2-1\right )}{12\,a\,\left (128\,a^6\,x^6-384\,a^4\,x^4+384\,a^2\,x^2-128\right )}\right )+\frac {\frac {227\,x}{2}+173\,a\,x^2-\frac {593}{6\,a}-\frac {599\,a^2\,x^3}{3}-\frac {159\,a^3\,x^4}{2}+\frac {183\,a^4\,x^5}{2}}{768\,a^6\,x^6-2304\,a^4\,x^4+2304\,a^2\,x^2-768}+\frac {\frac {299\,x}{2}-195\,a\,x^2+\frac {331}{3\,a}-\frac {800\,a^2\,x^3}{3}+90\,a^3\,x^4+\frac {245\,a^4\,x^5}{2}}{768\,a^6\,x^6-2304\,a^4\,x^4+2304\,a^2\,x^2-768}+\frac {{\ln \left (a\,x+1\right )}^2\,\left (30\,a^4\,x^5-80\,a^2\,x^3+66\,x\right )}{256\,a^6\,x^6-768\,a^4\,x^4+768\,a^2\,x^2-256}\right )-\frac {\ln \left (a\,x+1\right )\,\left (\frac {299\,x}{768\,a}-\frac {25\,a\,x^3}{36}+\frac {245\,a^3\,x^5}{768}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6}-\frac {{\ln \left (a\,x+1\right )}^3\,\left (\frac {11\,x}{128\,a}-\frac {5\,a\,x^3}{48}+\frac {5\,a^3\,x^5}{128}\right )}{3\,a\,x^2-\frac {1}{a}-3\,a^3\,x^4+a^5\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________