Optimal. Leaf size=137 \[ \frac{4}{9 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x \tan ^{-1}(a x)}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{\tan ^{-1}(a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.141851, antiderivative size = 137, 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, 4896, 4894} \[ \frac{4}{9 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x \tan ^{-1}(a x)}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{\tan ^{-1}(a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4930
Rule 4896
Rule 4894
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac{\tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{3 a}\\ &=\frac{2}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 x \tan ^{-1}(a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{\tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{4 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 a c}\\ &=\frac{2}{27 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{4}{9 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 x \tan ^{-1}(a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{4 x \tan ^{-1}(a x)}{9 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\tan ^{-1}(a x)^2}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0815654, size = 71, normalized size = 0.52 \[ \frac{\sqrt{a^2 c x^2+c} \left (2 \left (6 a^2 x^2+7\right )+6 a x \left (2 a^2 x^2+3\right ) \tan ^{-1}(a x)-9 \tan ^{-1}(a x)^2\right )}{27 c^3 \left (a^3 x^2+a\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.294, size = 276, normalized size = 2. \begin{align*}{\frac{ \left ( 6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) \left ( i{x}^{3}{a}^{3}+3\,{a}^{2}{x}^{2}-3\,iax-1 \right ) }{216\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2+2\,i\arctan \left ( ax \right ) \right ) \left ( 1+iax \right ) }{8\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( -1+iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) }{8\,{c}^{3}{a}^{2} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( i{x}^{3}{a}^{3}-3\,{a}^{2}{x}^{2}-3\,iax+1 \right ) \left ( -6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) }{216\,{c}^{3}{a}^{2} \left ({a}^{4}{x}^{4}+2\,{a}^{2}{x}^{2}+1 \right ) }\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29556, size = 186, normalized size = 1.36 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (12 \, a^{2} x^{2} + 6 \,{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right ) - 9 \, \arctan \left (a x\right )^{2} + 14\right )}}{27 \,{\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \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}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25566, size = 131, normalized size = 0.96 \begin{align*} \frac{2 \,{\left (\frac{2 \, a x^{2}}{c} + \frac{3}{a c}\right )} x \arctan \left (a x\right )}{9 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}} - \frac{\arctan \left (a x\right )^{2}}{3 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a^{2} c} + \frac{2 \,{\left (6 \, a^{2} c x^{2} + 7 \, c\right )}}{27 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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