Optimal. Leaf size=138 \[ \frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)-\frac {1}{12} a^5 c^3 x^3+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac {5}{4} a^3 c^3 x+\frac {3}{2} i a^2 c^3 \text {Li}_2(-i a x)-\frac {3}{2} i a^2 c^3 \text {Li}_2(i a x)+\frac {3}{4} a^2 c^3 \tan ^{-1}(a x)-\frac {c^3 \tan ^{-1}(a x)}{2 x^2}-\frac {a c^3}{2 x} \]
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Rubi [A] time = 0.15, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4948, 4852, 325, 203, 4848, 2391, 321, 302} \[ \frac {3}{2} i a^2 c^3 \text {PolyLog}(2,-i a x)-\frac {3}{2} i a^2 c^3 \text {PolyLog}(2,i a x)-\frac {1}{12} a^5 c^3 x^3+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac {5}{4} a^3 c^3 x+\frac {3}{4} a^2 c^3 \tan ^{-1}(a x)-\frac {c^3 \tan ^{-1}(a x)}{2 x^2}-\frac {a c^3}{2 x} \]
Antiderivative was successfully verified.
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Rule 203
Rule 302
Rule 321
Rule 325
Rule 2391
Rule 4848
Rule 4852
Rule 4948
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)}{x^3} \, dx &=\int \left (\frac {c^3 \tan ^{-1}(a x)}{x^3}+\frac {3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+a^6 c^3 x^3 \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int \frac {\tan ^{-1}(a x)}{x^3} \, dx+\left (3 a^2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{x} \, dx+\left (3 a^4 c^3\right ) \int x \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx\\ &=-\frac {c^3 \tan ^{-1}(a x)}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac {1}{2} \left (a c^3\right ) \int \frac {1}{x^2 \left (1+a^2 x^2\right )} \, dx+\frac {1}{2} \left (3 i a^2 c^3\right ) \int \frac {\log (1-i a x)}{x} \, dx-\frac {1}{2} \left (3 i a^2 c^3\right ) \int \frac {\log (1+i a x)}{x} \, dx-\frac {1}{2} \left (3 a^5 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (a^7 c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx\\ &=-\frac {a c^3}{2 x}-\frac {3}{2} a^3 c^3 x-\frac {c^3 \tan ^{-1}(a x)}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac {3}{2} i a^2 c^3 \text {Li}_2(-i a x)-\frac {3}{2} i a^2 c^3 \text {Li}_2(i a x)-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx+\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{4} \left (a^7 c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac {a c^3}{2 x}-\frac {5}{4} a^3 c^3 x-\frac {1}{12} a^5 c^3 x^3+a^2 c^3 \tan ^{-1}(a x)-\frac {c^3 \tan ^{-1}(a x)}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac {3}{2} i a^2 c^3 \text {Li}_2(-i a x)-\frac {3}{2} i a^2 c^3 \text {Li}_2(i a x)-\frac {1}{4} \left (a^3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx\\ &=-\frac {a c^3}{2 x}-\frac {5}{4} a^3 c^3 x-\frac {1}{12} a^5 c^3 x^3+\frac {3}{4} a^2 c^3 \tan ^{-1}(a x)-\frac {c^3 \tan ^{-1}(a x)}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)+\frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac {3}{2} i a^2 c^3 \text {Li}_2(-i a x)-\frac {3}{2} i a^2 c^3 \text {Li}_2(i a x)\\ \end {align*}
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Mathematica [C] time = 0.04, size = 154, normalized size = 1.12 \[ \frac {1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)-\frac {1}{12} a^5 c^3 x^3+\frac {3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac {5}{4} a^3 c^3 x-\frac {a c^3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-a^2 x^2\right )}{2 x}+\frac {3}{2} i a^2 c^3 \text {Li}_2(-i a x)-\frac {3}{2} i a^2 c^3 \text {Li}_2(i a x)+\frac {5}{4} a^2 c^3 \tan ^{-1}(a x)-\frac {c^3 \tan ^{-1}(a x)}{2 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )}{x^{3}}, x\right ) \]
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 = 0.05, size = 177, normalized size = 1.28 \[ \frac {a^{6} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {3 a^{4} c^{3} x^{2} \arctan \left (a x \right )}{2}+3 a^{2} c^{3} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {c^{3} \arctan \left (a x \right )}{2 x^{2}}-\frac {a^{5} c^{3} x^{3}}{12}-\frac {5 a^{3} c^{3} x}{4}-\frac {a \,c^{3}}{2 x}+\frac {3 a^{2} c^{3} \arctan \left (a x \right )}{4}+\frac {3 i a^{2} c^{3} \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}-\frac {3 i a^{2} c^{3} \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}+\frac {3 i a^{2} c^{3} \dilog \left (i a x +1\right )}{2}-\frac {3 i a^{2} c^{3} \dilog \left (-i a x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 155, normalized size = 1.12 \[ -\frac {a^{5} c^{3} x^{5} + 15 \, a^{3} c^{3} x^{3} + 9 \, \pi a^{2} c^{3} x^{2} \log \left (a^{2} x^{2} + 1\right ) - 36 \, a^{2} c^{3} x^{2} \arctan \left (a x\right ) \log \left (a x\right ) + 18 i \, a^{2} c^{3} x^{2} {\rm Li}_2\left (i \, a x + 1\right ) - 18 i \, a^{2} c^{3} x^{2} {\rm Li}_2\left (-i \, a x + 1\right ) + 6 \, a c^{3} x - 3 \, {\left (a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - 2 \, c^{3}\right )} \arctan \left (a x\right )}{12 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 152, normalized size = 1.10 \[ \left \{\begin {array}{cl} 0 & \text {\ if\ \ }a=0\\ 3\,a^4\,c^3\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {1}{2\,a^2}+\frac {x^2}{2}\right )-\frac {a^2\,c^3\,\left (3\,\mathrm {atan}\left (a\,x\right )-3\,a\,x+a^3\,x^3\right )}{12}-\frac {c^3\,\mathrm {atan}\left (a\,x\right )}{2\,x^2}-\frac {c^3\,\left (a^3\,\mathrm {atan}\left (a\,x\right )+\frac {a^2}{x}\right )}{2\,a}-\frac {3\,a^3\,c^3\,x}{2}+\frac {a^6\,c^3\,x^4\,\mathrm {atan}\left (a\,x\right )}{4}-\frac {a^2\,c^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1-a\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {a^2\,c^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1+a\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ c^{3} \left (\int \frac {\operatorname {atan}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {3 a^{2} \operatorname {atan}{\left (a x \right )}}{x}\, dx + \int 3 a^{4} x \operatorname {atan}{\left (a x \right )}\, dx + \int a^{6} x^{3} \operatorname {atan}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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