Optimal. Leaf size=23 \[ \frac {1}{3} x^3 \tan ^{-1}(\cot (a+b x))+\frac {b x^4}{12} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2168, 30} \[ \frac {1}{3} x^3 \tan ^{-1}(\cot (a+b x))+\frac {b x^4}{12} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2168
Rubi steps
\begin {align*} \int x^2 \tan ^{-1}(\cot (a+b x)) \, dx &=\frac {1}{3} x^3 \tan ^{-1}(\cot (a+b x))+\frac {1}{3} b \int x^3 \, dx\\ &=\frac {b x^4}{12}+\frac {1}{3} x^3 \tan ^{-1}(\cot (a+b x))\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 20, normalized size = 0.87 \[ \frac {1}{12} x^3 \left (4 \tan ^{-1}(\cot (a+b x))+b x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 17, normalized size = 0.74 \[ -\frac {1}{4} \, b x^{4} + \frac {1}{6} \, {\left (\pi - 2 \, a\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 19, normalized size = 0.83 \[ -\frac {1}{4} \, b x^{4} + \frac {1}{6} \, \pi x^{3} - \frac {1}{3} \, a x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.43, size = 65, normalized size = 2.83 \[ \frac {\pi \,x^{3}}{6}-\frac {x^{3} \mathrm {arccot}\left (\cot \left (b x +a \right )\right )}{3}-\frac {-\frac {\left (b x +a \right )^{4}}{4}+a \left (b x +a \right )^{3}-\frac {3 \left (b x +a \right )^{2} a^{2}}{2}+\left (b x +a \right ) a^{3}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 17, normalized size = 0.74 \[ -\frac {1}{4} \, b x^{4} + \frac {1}{6} \, {\left (\pi - 2 \, a\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.54, size = 25, normalized size = 1.09 \[ \frac {\Pi \,x^3}{6}+\frac {b\,x^4}{12}-\frac {x^3\,\mathrm {acot}\left (\mathrm {cot}\left (a+b\,x\right )\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.39, size = 26, normalized size = 1.13 \[ \frac {b x^{4}}{12} - \frac {x^{3} \operatorname {acot}{\left (\cot {\left (a + b x \right )} \right )}}{3} + \frac {\pi x^{3}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________