Optimal. Leaf size=107 \[ -\frac{6 x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{3 x \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}+\frac{6 \tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.13112, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4930, 4898, 191} \[ -\frac{6 x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{3 x \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}+\frac{6 \tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4930
Rule 4898
Rule 191
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{3 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a}\\ &=\frac{6 \tan ^{-1}(a x)}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{3 x \tan ^{-1}(a x)^2}{a c \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{c+a^2 c x^2}}-\frac{6 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a}\\ &=-\frac{6 x}{a c \sqrt{c+a^2 c x^2}}+\frac{6 \tan ^{-1}(a x)}{a^2 c \sqrt{c+a^2 c x^2}}+\frac{3 x \tan ^{-1}(a x)^2}{a c \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0934493, size = 61, normalized size = 0.57 \[ \frac{\sqrt{a^2 c x^2+c} \left (-6 a x-\tan ^{-1}(a x)^3+3 a x \tan ^{-1}(a x)^2+6 \tan ^{-1}(a x)\right )}{a^2 c^2 \left (a^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.275, size = 134, normalized size = 1.3 \begin{align*} -{\frac{ \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}-6\,\arctan \left ( ax \right ) +3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}-6\,i \right ) \left ( 1+iax \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( -1+iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}-6\,\arctan \left ( ax \right ) -3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}+6\,i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.28509, size = 132, normalized size = 1.23 \begin{align*} \sqrt{c}{\left (\frac{3 \, x \arctan \left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1} a c^{2}} - \frac{\arctan \left (a x\right )^{3}}{\sqrt{a^{2} x^{2} + 1} a^{2} c^{2}} - \frac{6 \,{\left (\frac{x}{\sqrt{a^{2} x^{2} + 1}} - \frac{\arctan \left (a x\right )}{\sqrt{a^{2} x^{2} + 1} a}\right )}}{a c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75106, size = 144, normalized size = 1.35 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (3 \, a x \arctan \left (a x\right )^{2} - \arctan \left (a x\right )^{3} - 6 \, a x + 6 \, \arctan \left (a x\right )\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29754, size = 134, normalized size = 1.25 \begin{align*} \frac{3 \, x \arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c} a c} - \frac{\arctan \left (a x\right )^{3}}{\sqrt{a^{2} c x^{2} + c} a^{2} c} - \frac{6 \, x}{\sqrt{a^{2} c x^{2} + c} a c} + \frac{6 \, \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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