Optimal. Leaf size=86 \[ -\frac {x \sqrt {a^2 c x^2+c}}{6 a}+\frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{3 a^2 c}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{6 a^2} \]
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Rubi [A] time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4930, 195, 217, 206} \[ -\frac {x \sqrt {a^2 c x^2+c}}{6 a}+\frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{3 a^2 c}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{6 a^2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 4930
Rubi steps
\begin {align*} \int x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx &=\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 a^2 c}-\frac {\int \sqrt {c+a^2 c x^2} \, dx}{3 a}\\ &=-\frac {x \sqrt {c+a^2 c x^2}}{6 a}+\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 a^2 c}-\frac {c \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{6 a}\\ &=-\frac {x \sqrt {c+a^2 c x^2}}{6 a}+\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 a^2 c}-\frac {c \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}\\ &=-\frac {x \sqrt {c+a^2 c x^2}}{6 a}+\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 a^2 c}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^2}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 86, normalized size = 1.00 \[ -\frac {a x \sqrt {a^2 c x^2+c}+\sqrt {c} \log \left (\sqrt {c} \sqrt {a^2 c x^2+c}+a c x\right )-2 \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{6 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 77, normalized size = 0.90 \[ -\frac {2 \, \sqrt {a^{2} c x^{2} + c} {\left (a x - 2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} - \sqrt {c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right )}{12 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.93, size = 156, normalized size = 1.81 \[ \frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right ) x^{2} a^{2}-a x +2 \arctan \left (a x \right )\right )}{6 a^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )}{6 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )}{6 a^{2} \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 260, normalized size = 3.02 \[ \frac {4 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c} \arctan \left (a x\right ) - 2 \, {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} {\left (a x \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, -a^{2} x^{2} + 3\right )\right ) + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, -a^{2} x^{2} + 3\right )\right )\right )} \sqrt {c} + \sqrt {c} {\left (\arctan \left ({\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right ) + 2, a x + {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right )\right ) + \arctan \left ({\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right ) - 2, -a x + {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right )\right )\right )}}{12 \, a^{2}} \]
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 \[ \int x\,\mathrm {atan}\left (a\,x\right )\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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 \[ \int x \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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