Optimal. Leaf size=105 \[ \frac {3}{2} i a^2 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+3 a^2 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} i a^2 \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4947, 5039,
5045, 4989, 2497, 5005} \begin {gather*} \frac {3}{2} i a^2 \text {Li}_2\left (\frac {2}{1-i a x}-1\right )-\frac {1}{2} a^2 \cot ^{-1}(a x)^3+\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+3 a^2 \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)-\frac {\cot ^{-1}(a x)^3}{2 x^2}+\frac {3 a \cot ^{-1}(a x)^2}{2 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2497
Rule 4947
Rule 4989
Rule 5005
Rule 5039
Rule 5045
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)^3}{x^3} \, dx &=-\frac {\cot ^{-1}(a x)^3}{2 x^2}-\frac {1}{2} (3 a) \int \frac {\cot ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(a x)^3}{2 x^2}-\frac {1}{2} (3 a) \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx+\frac {1}{2} \left (3 a^3\right ) \int \frac {\cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+\left (3 i a^2\right ) \int \frac {\cot ^{-1}(a x)}{x (i+a x)} \, dx\\ &=\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+3 a^2 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\left (3 a^3\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac {3}{2} i a^2 \cot ^{-1}(a x)^2+\frac {3 a \cot ^{-1}(a x)^2}{2 x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^3-\frac {\cot ^{-1}(a x)^3}{2 x^2}+3 a^2 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {3}{2} i a^2 \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 90, normalized size = 0.86 \begin {gather*} -\frac {\cot ^{-1}(a x) \left (3 i a x (i+a x) \cot ^{-1}(a x)+\left (1+a^2 x^2\right ) \cot ^{-1}(a x)^2-6 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right )}{2 x^2}-\frac {3}{2} i a^2 \text {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.82, size = 5367, normalized size = 51.11
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(5367\) |
default | \(\text {Expression too large to display}\) | \(5367\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acot}^{3}{\left (a x \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 29, normalized size = 0.28 \begin {gather*} -\frac {1}{2} \, a \arctan \left (\frac {1}{a x}\right )^{3} - \frac {\arctan \left (\frac {1}{a x}\right )^{3}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acot}\left (a\,x\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________