Optimal. Leaf size=48 \[ -\frac {a+b \tan ^{-1}(c x)}{4 x^4}+\frac {1}{4} b c^4 \tan ^{-1}(c x)+\frac {b c^3}{4 x}-\frac {b c}{12 x^3} \]
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Rubi [A] time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4852, 325, 203} \[ -\frac {a+b \tan ^{-1}(c x)}{4 x^4}+\frac {b c^3}{4 x}+\frac {1}{4} b c^4 \tan ^{-1}(c x)-\frac {b c}{12 x^3} \]
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^5} \, dx &=-\frac {a+b \tan ^{-1}(c x)}{4 x^4}+\frac {1}{4} (b c) \int \frac {1}{x^4 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c}{12 x^3}-\frac {a+b \tan ^{-1}(c x)}{4 x^4}-\frac {1}{4} \left (b c^3\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c}{12 x^3}+\frac {b c^3}{4 x}-\frac {a+b \tan ^{-1}(c x)}{4 x^4}+\frac {1}{4} \left (b c^5\right ) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {b c}{12 x^3}+\frac {b c^3}{4 x}+\frac {1}{4} b c^4 \tan ^{-1}(c x)-\frac {a+b \tan ^{-1}(c x)}{4 x^4}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 46, normalized size = 0.96 \[ -\frac {a}{4 x^4}-\frac {b c \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-c^2 x^2\right )}{12 x^3}-\frac {b \tan ^{-1}(c x)}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 41, normalized size = 0.85 \[ \frac {3 \, b c^{3} x^{3} - b c x + 3 \, {\left (b c^{4} x^{4} - b\right )} \arctan \left (c x\right ) - 3 \, a}{12 \, x^{4}} \]
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 = 44, normalized size = 0.92 \[ -\frac {a}{4 x^{4}}-\frac {b \arctan \left (c x \right )}{4 x^{4}}-\frac {b c}{12 x^{3}}+\frac {b \,c^{3}}{4 x}+\frac {b \,c^{4} \arctan \left (c x \right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 46, normalized size = 0.96 \[ \frac {1}{12} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b - \frac {a}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 42, normalized size = 0.88 \[ \frac {b\,c^4\,\mathrm {atan}\left (c\,x\right )}{4}-\frac {-b\,c^3\,x^3+\frac {b\,c\,x}{3}+a}{4\,x^4}-\frac {b\,\mathrm {atan}\left (c\,x\right )}{4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.83, size = 46, normalized size = 0.96 \[ - \frac {a}{4 x^{4}} + \frac {b c^{4} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b c^{3}}{4 x} - \frac {b c}{12 x^{3}} - \frac {b \operatorname {atan}{\left (c x \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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