Optimal. Leaf size=186 \[ -\frac {23 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{30 a^6}-\frac {19 \tanh ^{-1}(a x)}{60 a^6}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}-\frac {23 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{15 a^6}+\frac {19 x}{60 a^5}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^3}{60 a^3}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3+\frac {x^5 \coth ^{-1}(a x)^2}{10 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.72, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {5917, 5981, 302, 206, 321, 5985, 5919, 2402, 2315, 5911, 5949} \[ -\frac {23 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{30 a^6}+\frac {x^3}{60 a^3}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {19 x}{60 a^5}-\frac {19 \tanh ^{-1}(a x)}{60 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}-\frac {23 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{15 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3+\frac {x^5 \coth ^{-1}(a x)^2}{10 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 5911
Rule 5917
Rule 5919
Rule 5949
Rule 5981
Rule 5985
Rubi steps
\begin {align*} \int x^5 \coth ^{-1}(a x)^3 \, dx &=\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{2} a \int \frac {x^6 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac {1}{6} x^6 \coth ^{-1}(a x)^3+\frac {\int x^4 \coth ^{-1}(a x)^2 \, dx}{2 a}-\frac {\int \frac {x^4 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a}\\ &=\frac {x^5 \coth ^{-1}(a x)^2}{10 a}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {1}{5} \int \frac {x^5 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {\int x^2 \coth ^{-1}(a x)^2 \, dx}{2 a^3}-\frac {\int \frac {x^2 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a^3}\\ &=\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3+\frac {\int \coth ^{-1}(a x)^2 \, dx}{2 a^5}-\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a^5}+\frac {\int x^3 \coth ^{-1}(a x) \, dx}{5 a^2}-\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^2}-\frac {\int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^2}\\ &=\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3+\frac {\int x \coth ^{-1}(a x) \, dx}{5 a^4}-\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^4}+\frac {\int x \coth ^{-1}(a x) \, dx}{3 a^4}-\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^4}-\frac {\int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{a^4}-\frac {\int \frac {x^4}{1-a^2 x^2} \, dx}{20 a}\\ &=\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {\int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{5 a^5}-\frac {\int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{3 a^5}-\frac {\int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{a^5}-\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{10 a^3}-\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{6 a^3}-\frac {\int \left (-\frac {1}{a^4}-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx}{20 a}\\ &=\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {23 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^6}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{20 a^5}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{10 a^5}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{6 a^5}+\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^5}+\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{3 a^5}+\frac {\int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^5}\\ &=\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {19 \tanh ^{-1}(a x)}{60 a^6}-\frac {23 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^6}-\frac {\operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{5 a^6}-\frac {\operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{3 a^6}-\frac {\operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{a^6}\\ &=\frac {19 x}{60 a^5}+\frac {x^3}{60 a^3}+\frac {4 x^2 \coth ^{-1}(a x)}{15 a^4}+\frac {x^4 \coth ^{-1}(a x)}{20 a^2}+\frac {23 \coth ^{-1}(a x)^2}{30 a^6}+\frac {x \coth ^{-1}(a x)^2}{2 a^5}+\frac {x^3 \coth ^{-1}(a x)^2}{6 a^3}+\frac {x^5 \coth ^{-1}(a x)^2}{10 a}-\frac {\coth ^{-1}(a x)^3}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^3-\frac {19 \tanh ^{-1}(a x)}{60 a^6}-\frac {23 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^6}-\frac {23 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{30 a^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.54, size = 117, normalized size = 0.63 \[ \frac {10 \left (a^6 x^6-1\right ) \coth ^{-1}(a x)^3+a x \left (a^2 x^2+19\right )+2 \left (3 a^5 x^5+5 a^3 x^3+15 a x-23\right ) \coth ^{-1}(a x)^2+\coth ^{-1}(a x) \left (3 a^4 x^4+16 a^2 x^2-92 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )-19\right )+46 \text {Li}_2\left (e^{-2 \coth ^{-1}(a x)}\right )}{60 a^6} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{5} \operatorname {arcoth}\left (a x\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \operatorname {arcoth}\left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 3.61, size = 1141, normalized size = 6.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 289, normalized size = 1.55 \[ \frac {1}{6} \, x^{6} \operatorname {arcoth}\left (a x\right )^{3} + \frac {1}{60} \, a {\left (\frac {2 \, {\left (3 \, a^{4} x^{5} + 5 \, a^{2} x^{3} + 15 \, x\right )}}{a^{6}} - \frac {15 \, \log \left (a x + 1\right )}{a^{7}} + \frac {15 \, \log \left (a x - 1\right )}{a^{7}}\right )} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{240} \, a {\left (\frac {\frac {4 \, a^{3} x^{3} + {\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right )^{2} - 5 \, \log \left (a x + 1\right )^{3} + 5 \, \log \left (a x - 1\right )^{3} + 76 \, a x - {\left (15 \, \log \left (a x - 1\right )^{2} - 92 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 46 \, \log \left (a x - 1\right )^{2} + 38 \, \log \left (a x - 1\right )}{a} - \frac {184 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {38 \, \log \left (a x + 1\right )}{a}}{a^{6}} + \frac {2 \, {\left (6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} - 2 \, {\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right ) + 15 \, \log \left (a x + 1\right )^{2} + 15 \, \log \left (a x - 1\right )^{2} + 92 \, \log \left (a x - 1\right )\right )} \operatorname {arcoth}\left (a x\right )}{a^{7}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________