Optimal. Leaf size=37 \[ \frac {1}{2} x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {b \tan ^{-1}(c x)}{2 c^2}-\frac {b x}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4852, 321, 203} \[ \frac {1}{2} x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {b \tan ^{-1}(c x)}{2 c^2}-\frac {b x}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 321
Rule 4852
Rubi steps
\begin {align*} \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {1}{2} x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{2} (b c) \int \frac {x^2}{1+c^2 x^2} \, dx\\ &=-\frac {b x}{2 c}+\frac {1}{2} x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{2 c}\\ &=-\frac {b x}{2 c}+\frac {b \tan ^{-1}(c x)}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tan ^{-1}(c x)\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 42, normalized size = 1.14 \[ \frac {a x^2}{2}+\frac {b \tan ^{-1}(c x)}{2 c^2}+\frac {1}{2} b x^2 \tan ^{-1}(c x)-\frac {b x}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 34, normalized size = 0.92 \[ \frac {a c^{2} x^{2} - b c x + {\left (b c^{2} x^{2} + b\right )} \arctan \left (c x\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 35, normalized size = 0.95 \[ \frac {a \,x^{2}}{2}+\frac {b \,x^{2} \arctan \left (c x \right )}{2}-\frac {b x}{2 c}+\frac {b \arctan \left (c x \right )}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 37, normalized size = 1.00 \[ \frac {1}{2} \, a x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, size = 34, normalized size = 0.92 \[ \frac {a\,x^2}{2}+\frac {b\,\mathrm {atan}\left (c\,x\right )}{2\,c^2}+\frac {b\,x^2\,\mathrm {atan}\left (c\,x\right )}{2}-\frac {b\,x}{2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.42, size = 42, normalized size = 1.14 \[ \begin {cases} \frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {atan}{\left (c x \right )}}{2} - \frac {b x}{2 c} + \frac {b \operatorname {atan}{\left (c x \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\\frac {a x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________