Optimal. Leaf size=80 \[ -\frac {\left (1-3 a^2\right ) \log \left ((a+b x)^2+1\right )}{6 b^3}+\frac {a \left (3-a^2\right ) \tan ^{-1}(a+b x)}{3 b^3}+\frac {(a+b x)^2}{6 b^3}-\frac {a x}{b^2}+\frac {1}{3} x^3 \cot ^{-1}(a+b x) \]
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Rubi [A] time = 0.08, antiderivative size = 80, 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 = {5048, 4863, 702, 635, 203, 260} \[ -\frac {\left (1-3 a^2\right ) \log \left ((a+b x)^2+1\right )}{6 b^3}+\frac {a \left (3-a^2\right ) \tan ^{-1}(a+b x)}{3 b^3}-\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \cot ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 702
Rule 4863
Rule 5048
Rubi steps
\begin {align*} \int x^2 \cot ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \cot ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{3} x^3 \cot ^{-1}(a+b x)+\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3}{1+x^2} \, dx,x,a+b x\right )\\ &=\frac {1}{3} x^3 \cot ^{-1}(a+b x)+\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {3 a}{b^3}+\frac {x}{b^3}+\frac {a \left (3-a^2\right )-\left (1-3 a^2\right ) x}{b^3 \left (1+x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=-\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \cot ^{-1}(a+b x)+\frac {\operatorname {Subst}\left (\int \frac {a \left (3-a^2\right )-\left (1-3 a^2\right ) x}{1+x^2} \, dx,x,a+b x\right )}{3 b^3}\\ &=-\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \cot ^{-1}(a+b x)-\frac {\left (1-3 a^2\right ) \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{3 b^3}+\frac {\left (a \left (3-a^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,a+b x\right )}{3 b^3}\\ &=-\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \cot ^{-1}(a+b x)+\frac {a \left (3-a^2\right ) \tan ^{-1}(a+b x)}{3 b^3}-\frac {\left (1-3 a^2\right ) \log \left (1+(a+b x)^2\right )}{6 b^3}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 114, normalized size = 1.42 \[ \frac {\frac {1}{3} b \left (\frac {a+b x}{b}-\frac {a}{b}\right )^3 \cot ^{-1}(a+b x)+\frac {1}{3} b \left (\frac {(a+b x)^2}{2 b^3}-\frac {(1-i a)^3 \log (a+b x+i)}{2 b^3}-\frac {(1+i a)^3 \log (-a-b x+i)}{2 b^3}-\frac {3 a x}{b^2}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.25, size = 73, normalized size = 0.91 \[ \frac {2 \, b^{3} x^{3} \operatorname {arccot}\left (b x + a\right ) + b^{2} x^{2} - 4 \, a b x - 2 \, {\left (a^{3} - 3 \, a\right )} \arctan \left (b x + a\right ) + {\left (3 \, a^{2} - 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.85, size = 423, normalized size = 5.29 \[ -\frac {12 \, a^{2} \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 6 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{5} + \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{6} + 12 \, a^{2} \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} - 12 \, a^{2} \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 12 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} - 12 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 3 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{5} - 4 \, \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + 6 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) + 12 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 3 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} - \arctan \left (\frac {1}{b x + a}\right ) - \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )}{24 \, b^{3} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 94, normalized size = 1.18 \[ \frac {x^{3} \mathrm {arccot}\left (b x +a \right )}{3}+\frac {x^{2}}{6 b}-\frac {2 a x}{3 b^{2}}-\frac {5 a^{2}}{6 b^{3}}+\frac {\ln \left (1+\left (b x +a \right )^{2}\right ) a^{2}}{2 b^{3}}-\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{6 b^{3}}-\frac {\arctan \left (b x +a \right ) a^{3}}{3 b^{3}}+\frac {\arctan \left (b x +a \right ) a}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 85, normalized size = 1.06 \[ \frac {1}{3} \, x^{3} \operatorname {arccot}\left (b x + a\right ) + \frac {1}{6} \, b {\left (\frac {b x^{2} - 4 \, a x}{b^{3}} - \frac {2 \, {\left (a^{3} - 3 \, a\right )} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{b^{4}} + \frac {{\left (3 \, a^{2} - 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 101, normalized size = 1.26 \[ \frac {x^3\,\mathrm {acot}\left (a+b\,x\right )}{3}-\frac {\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{6\,b^3}+\frac {x^2}{6\,b}+\frac {a^2\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2\,b^3}-\frac {a^3\,\mathrm {atan}\left (a+b\,x\right )}{3\,b^3}+\frac {a\,\mathrm {atan}\left (a+b\,x\right )}{b^3}-\frac {2\,a\,x}{3\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.00, size = 117, normalized size = 1.46 \[ \begin {cases} \frac {a^{3} \operatorname {acot}{\left (a + b x \right )}}{3 b^{3}} + \frac {a^{2} \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b^{3}} - \frac {2 a x}{3 b^{2}} - \frac {a \operatorname {acot}{\left (a + b x \right )}}{b^{3}} + \frac {x^{3} \operatorname {acot}{\left (a + b x \right )}}{3} + \frac {x^{2}}{6 b} - \frac {\log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{6 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {acot}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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