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 = 0.14, antiderivative size = 137, normalized size of antiderivative = 1.00, 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 4894
Rule 4896
Rule 4930
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*}
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Mathematica [A] time = 0.09, 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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fricas [A] time = 0.71, size = 82, normalized size = 0.60 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (12 \, a^{2} x^{2} + 6 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right ) - 9 \, \arctan \left (a x\right )^{2} + 14\right )}}{27 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} \]
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 [C] time = 1.06, size = 276, normalized size = 2.01 \[ \frac {\left (6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right ) \left (i x^{3} a^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} c^{3} a^{2}}-\frac {\left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\left (-6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i x^{3} a^{3}-3 a^{2} x^{2}-3 i a x +1\right )}{216 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) c^{3} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x \operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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