Optimal. Leaf size=72 \[ -\frac {2 x}{c \sqrt {a^2 c x^2+c}}+\frac {x \tan ^{-1}(a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \tan ^{-1}(a x)}{a c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, 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, 191} \[ -\frac {2 x}{c \sqrt {a^2 c x^2+c}}+\frac {x \tan ^{-1}(a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \tan ^{-1}(a x)}{a c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 4898
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=\frac {2 \tan ^{-1}(a x)}{a c \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 x}{c \sqrt {c+a^2 c x^2}}+\frac {2 \tan ^{-1}(a x)}{a c \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 49, normalized size = 0.68 \[ \frac {\sqrt {a^2 c x^2+c} \left (-2 a x+a x \tan ^{-1}(a x)^2+2 \tan ^{-1}(a x)\right )}{c^2 \left (a^3 x^2+a\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 51, normalized size = 0.71 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (a x \arctan \left (a x\right )^{2} - 2 \, a x + 2 \, \arctan \left (a x\right )\right )}}{a^{3} c^{2} x^{2} + a c^{2}} \]
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 = 0.45, size = 114, normalized size = 1.58 \[ \frac {\left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2} a}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{2 \left (a^{2} x^{2}+1\right ) c^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 53, normalized size = 0.74 \[ \frac {x \arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c} c} - \frac {2 \, {\left (a x - \arctan \left (a x\right )\right )}}{\sqrt {a^{2} x^{2} + 1} a c^{\frac {3}{2}}} \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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