Optimal. Leaf size=105 \[ \frac {3}{16 a c^3 \left (a^2 x^2+1\right )}+\frac {1}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac {x \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \tan ^{-1}(a x)^2}{16 a c^3} \]
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Rubi [A] time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4896, 4892, 261} \[ \frac {3}{16 a c^3 \left (a^2 x^2+1\right )}+\frac {1}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac {x \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \tan ^{-1}(a x)^2}{16 a c^3} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4892
Rule 4896
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {1}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac {1}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^2}{16 a c^3}-\frac {(3 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}\\ &=\frac {1}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3}{16 a c^3 \left (1+a^2 x^2\right )}+\frac {x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \tan ^{-1}(a x)^2}{16 a c^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 68, normalized size = 0.65 \[ \frac {3 a^2 x^2+2 a x \left (3 a^2 x^2+5\right ) \tan ^{-1}(a x)+3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+4}{16 a c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 85, normalized size = 0.81 \[ \frac {3 \, a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 2 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right ) + 4}{16 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
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 = 0.04, size = 96, normalized size = 0.91 \[ \frac {1}{16 a \,c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3}{16 a \,c^{3} \left (a^{2} x^{2}+1\right )}+\frac {x \arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{2}}{16 a \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 129, normalized size = 1.23 \[ \frac {1}{8} \, {\left (\frac {3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{2} + c^{3}} + \frac {3 \, \arctan \left (a x\right )}{a c^{3}}\right )} \arctan \left (a x\right ) + \frac {{\left (3 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a}{16 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 85, normalized size = 0.81 \[ \frac {3\,a^4\,x^4\,{\mathrm {atan}\left (a\,x\right )}^2+6\,a^3\,x^3\,\mathrm {atan}\left (a\,x\right )+6\,a^2\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2+3\,a^2\,x^2+10\,a\,x\,\mathrm {atan}\left (a\,x\right )+3\,{\mathrm {atan}\left (a\,x\right )}^2+4}{16\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
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