Optimal. Leaf size=100 \[ -\frac{6}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}+\frac{3 \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}-\frac{6 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0690659, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4898, 4894} \[ -\frac{6}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}+\frac{3 \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}-\frac{6 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4898
Rule 4894
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=\frac{3 \tan ^{-1}(a x)^2}{a c \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}-6 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\\ &=-\frac{6}{a c \sqrt{c+a^2 c x^2}}-\frac{6 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}+\frac{3 \tan ^{-1}(a x)^2}{a c \sqrt{c+a^2 c x^2}}+\frac{x \tan ^{-1}(a x)^3}{c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0674057, size = 56, normalized size = 0.56 \[ \frac{\sqrt{a^2 c x^2+c} \left (a x \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2-6 a x \tan ^{-1}(a x)-6\right )}{c^2 \left (a^3 x^2+a\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.236, size = 132, 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 ( ax-i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}a}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( ax+i \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}\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 = 2.55137, size = 134, normalized size = 1.34 \begin{align*} \frac{x \arctan \left (a x\right )^{3}}{\sqrt{a^{2} c x^{2} + c} c} - \frac{3 \, a{\left (\frac{2 \, x \arctan \left (a x\right )}{\sqrt{a^{2} x^{2} + 1} a c} - \frac{\arctan \left (a x\right )^{2}}{\sqrt{a^{2} x^{2} + 1} a^{2} c} + \frac{2}{\sqrt{a^{2} x^{2} + 1} a^{2} c}\right )}}{\sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78762, size = 142, normalized size = 1.42 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x \arctan \left (a x\right )^{3} - 6 \, a x \arctan \left (a x\right ) + 3 \, \arctan \left (a x\right )^{2} - 6\right )}}{a^{3} c^{2} x^{2} + a 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{\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 = 1.27381, size = 134, normalized size = 1.34 \begin{align*} \frac{x \arctan \left (a x\right )^{3}}{\sqrt{a^{2} c x^{2} + c} c} - 3 \, a{\left (\frac{2 \, x \arctan \left (a x\right )}{\sqrt{a^{2} c x^{2} + c} a c} - \frac{\arctan \left (a x\right )^{2}}{\sqrt{a^{2} c x^{2} + c} a^{2} c} + \frac{2}{\sqrt{a^{2} c x^{2} + c} a^{2} c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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