Optimal. Leaf size=37 \[ -\frac {a+b \tan ^{-1}(c x)}{2 x^2}-\frac {1}{2} b c^2 \tan ^{-1}(c x)-\frac {b c}{2 x} \]
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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 = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4852, 325, 203} \[ -\frac {a+b \tan ^{-1}(c x)}{2 x^2}-\frac {1}{2} b c^2 \tan ^{-1}(c x)-\frac {b c}{2 x} \]
Antiderivative was successfully verified.
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Rule 203
Rule 325
Rule 4852
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx &=-\frac {a+b \tan ^{-1}(c x)}{2 x^2}+\frac {1}{2} (b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c}{2 x}-\frac {a+b \tan ^{-1}(c x)}{2 x^2}-\frac {1}{2} \left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {b c}{2 x}-\frac {1}{2} b c^2 \tan ^{-1}(c x)-\frac {a+b \tan ^{-1}(c x)}{2 x^2}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 46, normalized size = 1.24 \[ -\frac {a}{2 x^2}-\frac {b c \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-c^2 x^2\right )}{2 x}-\frac {b \tan ^{-1}(c x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 26, normalized size = 0.70 \[ -\frac {b c x + {\left (b c^{2} x^{2} + b\right )} \arctan \left (c x\right ) + a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [A] time = 0.01, size = 35, normalized size = 0.95 \[ -\frac {a}{2 x^{2}}-\frac {b \arctan \left (c x \right )}{2 x^{2}}-\frac {b c}{2 x}-\frac {b \,c^{2} \arctan \left (c x \right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 31, normalized size = 0.84 \[ -\frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b - \frac {a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 42, normalized size = 1.14 \[ -\frac {\frac {a}{2}+\frac {b\,\mathrm {atan}\left (c\,x\right )}{2}+\frac {b\,c\,x}{2}}{x^2}-\frac {b\,c\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {c^2}}\right )\,\sqrt {c^2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 37, normalized size = 1.00 \[ - \frac {a}{2 x^{2}} - \frac {b c^{2} \operatorname {atan}{\left (c x \right )}}{2} - \frac {b c}{2 x} - \frac {b \operatorname {atan}{\left (c x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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