Optimal. Leaf size=112 \[ -\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {a b x}{2 c^3}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {b^2 x \tan ^{-1}(c x)}{2 c^3}+\frac {b^2 x^2}{12 c^2}-\frac {b^2 \log \left (c^2 x^2+1\right )}{3 c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4852, 4916, 266, 43, 4846, 260, 4884} \[ \frac {a b x}{2 c^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {b^2 x^2}{12 c^2}-\frac {b^2 \log \left (c^2 x^2+1\right )}{3 c^4}+\frac {b^2 x \tan ^{-1}(c x)}{2 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 260
Rule 266
Rule 4846
Rule 4852
Rule 4884
Rule 4916
Rubi steps
\begin {align*} \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} (b c) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c}+\frac {b \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c}\\ &=-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{6} b^2 \int \frac {x^3}{1+c^2 x^2} \, dx+\frac {b \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac {b \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c^3}\\ &=\frac {a b x}{2 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{12} b^2 \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )+\frac {b^2 \int \tan ^{-1}(c x) \, dx}{2 c^3}\\ &=\frac {a b x}{2 c^3}+\frac {b^2 x \tan ^{-1}(c x)}{2 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{12} b^2 \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {b^2 \int \frac {x}{1+c^2 x^2} \, dx}{2 c^2}\\ &=\frac {a b x}{2 c^3}+\frac {b^2 x^2}{12 c^2}+\frac {b^2 x \tan ^{-1}(c x)}{2 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b^2 \log \left (1+c^2 x^2\right )}{3 c^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 111, normalized size = 0.99 \[ \frac {c x \left (3 a^2 c^3 x^3-2 a b c^2 x^2+6 a b+b^2 c x\right )-2 b \tan ^{-1}(c x) \left (a \left (3-3 c^4 x^4\right )+b c x \left (c^2 x^2-3\right )\right )+3 b^2 \left (c^4 x^4-1\right ) \tan ^{-1}(c x)^2-4 b^2 \log \left (c^2 x^2+1\right )}{12 c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 121, normalized size = 1.08 \[ \frac {3 \, a^{2} c^{4} x^{4} - 2 \, a b c^{3} x^{3} + b^{2} c^{2} x^{2} + 6 \, a b c x + 3 \, {\left (b^{2} c^{4} x^{4} - b^{2}\right )} \arctan \left (c x\right )^{2} - 4 \, b^{2} \log \left (c^{2} x^{2} + 1\right ) + 2 \, {\left (3 \, a b c^{4} x^{4} - b^{2} c^{3} x^{3} + 3 \, b^{2} c x - 3 \, a b\right )} \arctan \left (c x\right )}{12 \, c^{4}} \]
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 = 135, normalized size = 1.21 \[ \frac {a^{2} x^{4}}{4}+\frac {b^{2} x^{4} \arctan \left (c x \right )^{2}}{4}-\frac {b^{2} \arctan \left (c x \right ) x^{3}}{6 c}+\frac {b^{2} x \arctan \left (c x \right )}{2 c^{3}}-\frac {b^{2} \arctan \left (c x \right )^{2}}{4 c^{4}}+\frac {b^{2} x^{2}}{12 c^{2}}-\frac {b^{2} \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}}+\frac {x^{4} a b \arctan \left (c x \right )}{2}-\frac {a b \,x^{3}}{6 c}+\frac {a b x}{2 c^{3}}-\frac {a b \arctan \left (c x \right )}{2 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 136, normalized size = 1.21 \[ \frac {1}{4} \, b^{2} x^{4} \arctan \left (c x\right )^{2} + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{6} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b - \frac {1}{12} \, {\left (2 \, c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )} \arctan \left (c x\right ) - \frac {c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.32, size = 134, normalized size = 1.20 \[ \frac {3\,a^2\,c^4\,x^4-4\,b^2\,\ln \left (c^2\,x^2+1\right )-3\,b^2\,{\mathrm {atan}\left (c\,x\right )}^2+b^2\,c^2\,x^2-6\,a\,b\,\mathrm {atan}\left (c\,x\right )-2\,b^2\,c^3\,x^3\,\mathrm {atan}\left (c\,x\right )+6\,b^2\,c\,x\,\mathrm {atan}\left (c\,x\right )+3\,b^2\,c^4\,x^4\,{\mathrm {atan}\left (c\,x\right )}^2-2\,a\,b\,c^3\,x^3+6\,a\,b\,c\,x+6\,a\,b\,c^4\,x^4\,\mathrm {atan}\left (c\,x\right )}{12\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.44, size = 155, normalized size = 1.38 \[ \begin {cases} \frac {a^{2} x^{4}}{4} + \frac {a b x^{4} \operatorname {atan}{\left (c x \right )}}{2} - \frac {a b x^{3}}{6 c} + \frac {a b x}{2 c^{3}} - \frac {a b \operatorname {atan}{\left (c x \right )}}{2 c^{4}} + \frac {b^{2} x^{4} \operatorname {atan}^{2}{\left (c x \right )}}{4} - \frac {b^{2} x^{3} \operatorname {atan}{\left (c x \right )}}{6 c} + \frac {b^{2} x^{2}}{12 c^{2}} + \frac {b^{2} x \operatorname {atan}{\left (c x \right )}}{2 c^{3}} - \frac {b^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{3 c^{4}} - \frac {b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{4 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________