Optimal. Leaf size=116 \[ \frac {1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{6} a^5 c^3 x^2+3 a^4 c^3 x \tan ^{-1}(a x)+\frac {8}{3} a^3 c^3 \log (x)-\frac {3 a^2 c^3 \tan ^{-1}(a x)}{x}-\frac {8}{3} a^3 c^3 \log \left (a^2 x^2+1\right )-\frac {c^3 \tan ^{-1}(a x)}{3 x^3}-\frac {a c^3}{6 x^2} \]
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Rubi [A] time = 0.16, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4948, 4846, 260, 4852, 266, 44, 36, 29, 31, 43} \[ -\frac {1}{6} a^5 c^3 x^2-\frac {8}{3} a^3 c^3 \log \left (a^2 x^2+1\right )+\frac {1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac {8}{3} a^3 c^3 \log (x)+3 a^4 c^3 x \tan ^{-1}(a x)-\frac {3 a^2 c^3 \tan ^{-1}(a x)}{x}-\frac {a c^3}{6 x^2}-\frac {c^3 \tan ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 43
Rule 44
Rule 260
Rule 266
Rule 4846
Rule 4852
Rule 4948
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)}{x^4} \, dx &=\int \left (3 a^4 c^3 \tan ^{-1}(a x)+\frac {c^3 \tan ^{-1}(a x)}{x^4}+\frac {3 a^2 c^3 \tan ^{-1}(a x)}{x^2}+a^6 c^3 x^2 \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int \frac {\tan ^{-1}(a x)}{x^4} \, dx+\left (3 a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{x^2} \, dx+\left (3 a^4 c^3\right ) \int \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx\\ &=-\frac {c^3 \tan ^{-1}(a x)}{3 x^3}-\frac {3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac {1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac {1}{3} \left (a c^3\right ) \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (3 a^3 c^3\right ) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx-\left (3 a^5 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^7 c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx\\ &=-\frac {c^3 \tan ^{-1}(a x)}{3 x^3}-\frac {3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac {1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)-\frac {3}{2} a^3 c^3 \log \left (1+a^2 x^2\right )+\frac {1}{6} \left (a c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{2} \left (3 a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )-\frac {1}{6} \left (a^7 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {c^3 \tan ^{-1}(a x)}{3 x^3}-\frac {3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac {1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)-\frac {3}{2} a^3 c^3 \log \left (1+a^2 x^2\right )+\frac {1}{6} \left (a c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^2}-\frac {a^2}{x}+\frac {a^4}{1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{2} \left (3 a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (3 a^5 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{6} \left (a^7 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {a c^3}{6 x^2}-\frac {1}{6} a^5 c^3 x^2-\frac {c^3 \tan ^{-1}(a x)}{3 x^3}-\frac {3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+\frac {1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac {8}{3} a^3 c^3 \log (x)-\frac {8}{3} a^3 c^3 \log \left (1+a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.72 \[ \frac {c^3 \left (2 \left (a^6 x^6+9 a^4 x^4-9 a^2 x^2-1\right ) \tan ^{-1}(a x)-a x \left (a^4 x^4-16 a^2 x^2 \log (x)+16 a^2 x^2 \log \left (a^2 x^2+1\right )+1\right )\right )}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 100, normalized size = 0.86 \[ -\frac {a^{5} c^{3} x^{5} + 16 \, a^{3} c^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 16 \, a^{3} c^{3} x^{3} \log \relax (x) + a c^{3} x - 2 \, {\left (a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} - 9 \, a^{2} c^{3} x^{2} - c^{3}\right )} \arctan \left (a x\right )}{6 \, x^{3}} \]
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 = 107, normalized size = 0.92 \[ \frac {a^{6} c^{3} x^{3} \arctan \left (a x \right )}{3}+3 a^{4} c^{3} x \arctan \left (a x \right )-\frac {c^{3} \arctan \left (a x \right )}{3 x^{3}}-\frac {3 a^{2} c^{3} \arctan \left (a x \right )}{x}-\frac {a^{5} c^{3} x^{2}}{6}-\frac {a \,c^{3}}{6 x^{2}}+\frac {8 a^{3} c^{3} \ln \left (a x \right )}{3}-\frac {8 a^{3} c^{3} \ln \left (a^{2} x^{2}+1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 96, normalized size = 0.83 \[ -\frac {1}{6} \, {\left (a^{4} c^{3} x^{2} + 16 \, a^{2} c^{3} \log \left (a^{2} x^{2} + 1\right ) - 16 \, a^{2} c^{3} \log \relax (x) + \frac {c^{3}}{x^{2}}\right )} a + \frac {1}{3} \, {\left (a^{6} c^{3} x^{3} + 9 \, a^{4} c^{3} x - \frac {9 \, a^{2} c^{3} x^{2} + c^{3}}{x^{3}}\right )} \arctan \left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 97, normalized size = 0.84 \[ -\frac {c^3\,\left (2\,\mathrm {atan}\left (a\,x\right )+a\,x-a^3\,x^3+a^5\,x^5+18\,a^2\,x^2\,\mathrm {atan}\left (a\,x\right )-18\,a^4\,x^4\,\mathrm {atan}\left (a\,x\right )-2\,a^6\,x^6\,\mathrm {atan}\left (a\,x\right )+16\,a^3\,x^3\,\ln \left (a^2\,x^2+1\right )-16\,a^3\,x^3\,\ln \relax (x)\right )}{6\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.17, size = 117, normalized size = 1.01 \[ \begin {cases} \frac {a^{6} c^{3} x^{3} \operatorname {atan}{\left (a x \right )}}{3} - \frac {a^{5} c^{3} x^{2}}{6} + 3 a^{4} c^{3} x \operatorname {atan}{\left (a x \right )} + \frac {8 a^{3} c^{3} \log {\relax (x )}}{3} - \frac {8 a^{3} c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3} - \frac {3 a^{2} c^{3} \operatorname {atan}{\left (a x \right )}}{x} - \frac {a c^{3}}{6 x^{2}} - \frac {c^{3} \operatorname {atan}{\left (a x \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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