Optimal. Leaf size=86 \[ -\frac {3 \tan ^{-1}(a x)}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {x^3}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 x}{32 a^3 c^3 \left (a^2 x^2+1\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4944, 288, 205} \[ \frac {x^3}{16 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 x}{32 a^3 c^3 \left (a^2 x^2+1\right )}+\frac {x^4 \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 \tan ^{-1}(a x)}{32 a^4 c^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 288
Rule 4944
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {x^4 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{4} a \int \frac {x^4}{\left (c+a^2 c x^2\right )^3} \, dx\\ &=\frac {x^3}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x^4 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \int \frac {x^2}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a c}\\ &=\frac {x^3}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x}{32 a^3 c^3 \left (1+a^2 x^2\right )}+\frac {x^4 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \int \frac {1}{c+a^2 c x^2} \, dx}{32 a^3 c^2}\\ &=\frac {x^3}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 58, normalized size = 0.67 \[ \frac {a x \left (5 a^2 x^2+3\right )+\left (5 a^4 x^4-6 a^2 x^2-3\right ) \tan ^{-1}(a x)}{32 a^4 c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 69, normalized size = 0.80 \[ \frac {5 \, a^{3} x^{3} + 3 \, a x + {\left (5 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 3\right )} \arctan \left (a x\right )}{32 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]
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 = 102, normalized size = 1.19 \[ \frac {\arctan \left (a x \right )}{4 a^{4} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )}{2 a^{4} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {5 x^{3}}{32 a \,c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 x}{32 a^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {5 \arctan \left (a x \right )}{32 a^{4} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 108, normalized size = 1.26 \[ \frac {1}{32} \, a {\left (\frac {5 \, a^{2} x^{3} + 3 \, x}{a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}} + \frac {5 \, \arctan \left (a x\right )}{a^{5} c^{3}}\right )} - \frac {{\left (2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{4 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 62, normalized size = 0.72 \[ \frac {3\,a\,x-3\,\mathrm {atan}\left (a\,x\right )+5\,a^3\,x^3-6\,a^2\,x^2\,\mathrm {atan}\left (a\,x\right )+5\,a^4\,x^4\,\mathrm {atan}\left (a\,x\right )}{32\,a^4\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.69, size = 243, normalized size = 2.83 \[ \begin {cases} \frac {5 a^{4} x^{4} \operatorname {atan}{\left (a x \right )}}{32 a^{8} c^{3} x^{4} + 64 a^{6} c^{3} x^{2} + 32 a^{4} c^{3}} + \frac {5 a^{3} x^{3}}{32 a^{8} c^{3} x^{4} + 64 a^{6} c^{3} x^{2} + 32 a^{4} c^{3}} - \frac {6 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{32 a^{8} c^{3} x^{4} + 64 a^{6} c^{3} x^{2} + 32 a^{4} c^{3}} + \frac {3 a x}{32 a^{8} c^{3} x^{4} + 64 a^{6} c^{3} x^{2} + 32 a^{4} c^{3}} - \frac {3 \operatorname {atan}{\left (a x \right )}}{32 a^{8} c^{3} x^{4} + 64 a^{6} c^{3} x^{2} + 32 a^{4} c^{3}} & \text {for}\: c \neq 0 \\\tilde {\infty } \left (\frac {x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {x^{3}}{12 a} + \frac {x}{4 a^{3}} - \frac {\operatorname {atan}{\left (a x \right )}}{4 a^{4}}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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