Optimal. Leaf size=72 \[ -\frac {2 x}{c \sqrt {c+a^2 c x^2}}+\frac {2 \text {ArcTan}(a x)}{a c \sqrt {c+a^2 c x^2}}+\frac {x \text {ArcTan}(a x)^2}{c \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.03, 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 = {5018, 197}
\begin {gather*} \frac {x \text {ArcTan}(a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \text {ArcTan}(a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2 x}{c \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 5018
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 = 0.04, size = 49, normalized size = 0.68 \begin {gather*} \frac {\sqrt {c+a^2 c x^2} \left (-2 a x+2 \text {ArcTan}(a x)+a x \text {ArcTan}(a x)^2\right )}{c^2 \left (a+a^3 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.23, size = 114, normalized size = 1.58
method | result | size |
default | \(\frac {\left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) a \,c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{2 \left (a^{2} x^{2}+1\right ) a \,c^{2}}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 53, normalized size = 0.74 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.58, size = 51, normalized size = 0.71 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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