Optimal. Leaf size=148 \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a^4}+\frac{x^2 \cot ^{-1}(a x)}{4 a^2}+\frac{x}{4 a^3}-\frac{\tan ^{-1}(a x)}{4 a^4}-\frac{3 x \cot ^{-1}(a x)^2}{4 a^3}-\frac{\cot ^{-1}(a x)^3}{4 a^4}-\frac{i \cot ^{-1}(a x)^2}{a^4}+\frac{2 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{x^3 \cot ^{-1}(a x)^2}{4 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.385544, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4853, 4917, 321, 203, 4921, 4855, 2402, 2315, 4847, 4885} \[ -\frac{i \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a^4}+\frac{x^2 \cot ^{-1}(a x)}{4 a^2}+\frac{x}{4 a^3}-\frac{\tan ^{-1}(a x)}{4 a^4}-\frac{3 x \cot ^{-1}(a x)^2}{4 a^3}-\frac{\cot ^{-1}(a x)^3}{4 a^4}-\frac{i \cot ^{-1}(a x)^2}{a^4}+\frac{2 \log \left (\frac{2}{1+i a x}\right ) \cot ^{-1}(a x)}{a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{x^3 \cot ^{-1}(a x)^2}{4 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4853
Rule 4917
Rule 321
Rule 203
Rule 4921
Rule 4855
Rule 2402
Rule 2315
Rule 4847
Rule 4885
Rubi steps
\begin{align*} \int x^3 \cot ^{-1}(a x)^3 \, dx &=\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{1}{4} (3 a) \int \frac{x^4 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{3 \int x^2 \cot ^{-1}(a x)^2 \, dx}{4 a}-\frac{3 \int \frac{x^2 \cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a}\\ &=\frac{x^3 \cot ^{-1}(a x)^2}{4 a}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{1}{2} \int \frac{x^3 \cot ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{3 \int \cot ^{-1}(a x)^2 \, dx}{4 a^3}+\frac{3 \int \frac{\cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a^3}\\ &=-\frac{3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac{x^3 \cot ^{-1}(a x)^2}{4 a}-\frac{\cot ^{-1}(a x)^3}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{\int x \cot ^{-1}(a x) \, dx}{2 a^2}-\frac{\int \frac{x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac{3 \int \frac{x \cot ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}\\ &=\frac{x^2 \cot ^{-1}(a x)}{4 a^2}-\frac{i \cot ^{-1}(a x)^2}{a^4}-\frac{3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac{x^3 \cot ^{-1}(a x)^2}{4 a}-\frac{\cot ^{-1}(a x)^3}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{\int \frac{\cot ^{-1}(a x)}{i-a x} \, dx}{2 a^3}+\frac{3 \int \frac{\cot ^{-1}(a x)}{i-a x} \, dx}{2 a^3}+\frac{\int \frac{x^2}{1+a^2 x^2} \, dx}{4 a}\\ &=\frac{x}{4 a^3}+\frac{x^2 \cot ^{-1}(a x)}{4 a^2}-\frac{i \cot ^{-1}(a x)^2}{a^4}-\frac{3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac{x^3 \cot ^{-1}(a x)^2}{4 a}-\frac{\cot ^{-1}(a x)^3}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3+\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4}-\frac{\int \frac{1}{1+a^2 x^2} \, dx}{4 a^3}+\frac{\int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}+\frac{3 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}\\ &=\frac{x}{4 a^3}+\frac{x^2 \cot ^{-1}(a x)}{4 a^2}-\frac{i \cot ^{-1}(a x)^2}{a^4}-\frac{3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac{x^3 \cot ^{-1}(a x)^2}{4 a}-\frac{\cot ^{-1}(a x)^3}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3-\frac{\tan ^{-1}(a x)}{4 a^4}+\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4}-\frac{i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{2 a^4}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{2 a^4}\\ &=\frac{x}{4 a^3}+\frac{x^2 \cot ^{-1}(a x)}{4 a^2}-\frac{i \cot ^{-1}(a x)^2}{a^4}-\frac{3 x \cot ^{-1}(a x)^2}{4 a^3}+\frac{x^3 \cot ^{-1}(a x)^2}{4 a}-\frac{\cot ^{-1}(a x)^3}{4 a^4}+\frac{1}{4} x^4 \cot ^{-1}(a x)^3-\frac{\tan ^{-1}(a x)}{4 a^4}+\frac{2 \cot ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4}-\frac{i \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^4}\\ \end{align*}
Mathematica [A] time = 0.332578, size = 96, normalized size = 0.65 \[ \frac{-4 i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )+\left (a^4 x^4-1\right ) \cot ^{-1}(a x)^3+\left (a^3 x^3-3 a x-4 i\right ) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left (a^2 x^2+8 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )+1\right )+a x}{4 a^4} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.451, size = 209, normalized size = 1.4 \begin{align*}{\frac{{x}^{4} \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{4}}-{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{4\,{a}^{4}}}+{\frac{{x}^{3} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{4\,a}}-{\frac{3\,x \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{4\,{a}^{3}}}+{\frac{{x}^{2}{\rm arccot} \left (ax\right )}{4\,{a}^{2}}}-{\frac{i \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{{a}^{4}}}+{\frac{{\rm arccot} \left (ax\right )}{4\,{a}^{4}}}+{\frac{x}{4\,{a}^{3}}}-{\frac{{\frac{i}{4}}}{{a}^{4}}}+2\,{\frac{{\rm arccot} \left (ax\right )}{{a}^{4}}\ln \left ( 1+{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{2\,i}{{a}^{4}}{\it polylog} \left ( 2,-{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) }+2\,{\frac{{\rm arccot} \left (ax\right )}{{a}^{4}}\ln \left ( 1-{\frac{ax+i}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{2\,i}{{a}^{4}}{\it polylog} \left ( 2,{(ax+i){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \operatorname{arccot}\left (a x\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{acot}^{3}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arccot}\left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]