Optimal. Leaf size=59 \[ \frac {\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{a^2 c}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^2 \sqrt {c}} \]
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Rubi [A]
time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5050, 223, 212}
\begin {gather*} \frac {\text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 5050
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a}\\ &=\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^2 c}-\frac {\text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{a}\\ &=\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^2 c}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^2 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 60, normalized size = 1.02 \begin {gather*} \frac {\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)-\sqrt {c} \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{a^2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.39, size = 144, normalized size = 2.44
method | result | size |
default | \(\frac {\arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{a^{2} c}+\frac {\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{2} c}-\frac {\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{2} c}\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 61, normalized size = 1.03 \begin {gather*} \frac {2 \, \sqrt {a^{2} x^{2} + 1} \arctan \left (a x\right ) - \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) + \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{2 \, a^{2} \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.55, size = 64, normalized size = 1.08 \begin {gather*} \frac {2 \, \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right ) + \sqrt {c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right )}{2 \, a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.02 \begin {gather*} \int \frac {x\,\mathrm {atan}\left (a\,x\right )}{\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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