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 = 0.13, antiderivative size = 107, 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, 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 191
Rule 4898
Rule 4930
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*}
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Mathematica [A] time = 0.10, 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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fricas [A] time = 0.42, size = 62, normalized size = 0.58 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (3 \, a x \arctan \left (a x\right )^{2} - \arctan \left (a x\right )^{3} - 6 \, a x + 6 \, \arctan \left (a x\right )\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \]
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 = 1.12, size = 134, normalized size = 1.25 \[ -\frac {\left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2} a^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{2 \left (a^{2} x^{2}+1\right ) c^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 98, normalized size = 0.92 \[ \sqrt {c} {\left (\frac {3 \, x \arctan \left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} a c^{2}} - \frac {\arctan \left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1} a^{2} c^{2}} - \frac {6 \, {\left (\frac {x}{\sqrt {a^{2} x^{2} + 1}} - \frac {\arctan \left (a x\right )}{\sqrt {a^{2} x^{2} + 1} a}\right )}}{a c^{2}}\right )} \]
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 \[ \int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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