Optimal. Leaf size=142 \[ -\frac{7 a}{16 c^3 \left (a^2 x^2+1\right )}-\frac{a}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac{a \log \left (a^2 x^2+1\right )}{2 c^3}-\frac{7 a^2 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}-\frac{a^2 x \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{a \log (x)}{c^3}-\frac{15 a \tan ^{-1}(a x)^2}{16 c^3}-\frac{\tan ^{-1}(a x)}{c^3 x} \]
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Rubi [A] time = 0.262566, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {4966, 4918, 4852, 266, 36, 29, 31, 4884, 4892, 261, 4896} \[ -\frac{7 a}{16 c^3 \left (a^2 x^2+1\right )}-\frac{a}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac{a \log \left (a^2 x^2+1\right )}{2 c^3}-\frac{7 a^2 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}-\frac{a^2 x \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{a \log (x)}{c^3}-\frac{15 a \tan ^{-1}(a x)^2}{16 c^3}-\frac{\tan ^{-1}(a x)}{c^3 x} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4918
Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4884
Rule 4892
Rule 261
Rule 4896
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac{a}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{a^2 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{\int \frac{\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac{\left (3 a^2\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-\frac{a^2 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac{a}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{a^2 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac{7 a \tan ^{-1}(a x)^2}{16 c^3}+\frac{\int \frac{\tan ^{-1}(a x)}{x^2} \, dx}{c^3}-\frac{a^2 \int \frac{\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{c^2}+\frac{\left (3 a^3\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}+\frac{a^3 \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\\ &=-\frac{a}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a}{16 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac{15 a \tan ^{-1}(a x)^2}{16 c^3}+\frac{a \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx}{c^3}\\ &=-\frac{a}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a}{16 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac{15 a \tan ^{-1}(a x)^2}{16 c^3}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^3}\\ &=-\frac{a}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a}{16 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac{15 a \tan ^{-1}(a x)^2}{16 c^3}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 c^3}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^3}\\ &=-\frac{a}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a}{16 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)}{c^3 x}-\frac{a^2 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{7 a^2 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac{15 a \tan ^{-1}(a x)^2}{16 c^3}+\frac{a \log (x)}{c^3}-\frac{a \log \left (1+a^2 x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.0939301, size = 118, normalized size = 0.83 \[ \frac{a x \left (-7 a^2 x^2+16 \left (a^2 x^2+1\right )^2 \log (x)-8 \left (a^2 x^2+1\right )^2 \log \left (a^2 x^2+1\right )-8\right )-15 a x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2-2 \left (15 a^4 x^4+25 a^2 x^2+8\right ) \tan ^{-1}(a x)}{16 c^3 x \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 135, normalized size = 1. \begin{align*} -{\frac{7\,\arctan \left ( ax \right ){a}^{4}{x}^{3}}{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{9\,{a}^{2}x\arctan \left ( ax \right ) }{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{15\,a \left ( \arctan \left ( ax \right ) \right ) ^{2}}{16\,{c}^{3}}}-{\frac{\arctan \left ( ax \right ) }{{c}^{3}x}}-{\frac{a}{16\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}-{\frac{a\ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,{c}^{3}}}-{\frac{7\,a}{16\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}+{\frac{a\ln \left ( ax \right ) }{{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62712, size = 244, normalized size = 1.72 \begin{align*} -\frac{1}{8} \,{\left (\frac{15 \, a^{4} x^{4} + 25 \, a^{2} x^{2} + 8}{a^{4} c^{3} x^{5} + 2 \, a^{2} c^{3} x^{3} + c^{3} x} + \frac{15 \, a \arctan \left (a x\right )}{c^{3}}\right )} \arctan \left (a x\right ) - \frac{{\left (7 \, a^{2} x^{2} - 15 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 8 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \log \left (a^{2} x^{2} + 1\right ) - 16 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \log \left (x\right ) + 8\right )} a}{16 \,{\left (a^{4} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{2} + c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73396, size = 333, normalized size = 2.35 \begin{align*} -\frac{7 \, a^{3} x^{3} + 15 \,{\left (a^{5} x^{5} + 2 \, a^{3} x^{3} + a x\right )} \arctan \left (a x\right )^{2} + 8 \, a x + 2 \,{\left (15 \, a^{4} x^{4} + 25 \, a^{2} x^{2} + 8\right )} \arctan \left (a x\right ) + 8 \,{\left (a^{5} x^{5} + 2 \, a^{3} x^{3} + a x\right )} \log \left (a^{2} x^{2} + 1\right ) - 16 \,{\left (a^{5} x^{5} + 2 \, a^{3} x^{3} + a x\right )} \log \left (x\right )}{16 \,{\left (a^{4} c^{3} x^{5} + 2 \, a^{2} c^{3} x^{3} + c^{3} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.04583, size = 602, normalized size = 4.24 \begin{align*} \frac{16 a^{5} x^{5} \log{\left (x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{8 a^{5} x^{5} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{15 a^{5} x^{5} \operatorname{atan}^{2}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{30 a^{4} x^{4} \operatorname{atan}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} + \frac{32 a^{3} x^{3} \log{\left (x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{16 a^{3} x^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{30 a^{3} x^{3} \operatorname{atan}^{2}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{7 a^{3} x^{3}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{50 a^{2} x^{2} \operatorname{atan}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} + \frac{16 a x \log{\left (x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{8 a x \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{15 a x \operatorname{atan}^{2}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{8 a x}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac{16 \operatorname{atan}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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