Optimal. Leaf size=298 \[ \frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{8 a^2}+\frac {1}{4} x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {i c \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a^3 c}+\frac {\sqrt {a^2 c x^2+c}}{8 a^3}+\frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4 a^3 \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.27, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4946, 4952, 261, 4890, 4886, 266, 43} \[ -\frac {i c \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a^3 c}+\frac {\sqrt {a^2 c x^2+c}}{8 a^3}+\frac {1}{4} x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{8 a^2}+\frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4 a^3 \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 4886
Rule 4890
Rule 4946
Rule 4952
Rubi steps
\begin {align*} \int x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx &=\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{4} c \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{4} (a c) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}-\frac {c \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}-\frac {1}{8} (a c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c+a^2 c x^2}}{8 a^3}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{8} (a c) \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {c+a^2 c x^2}}{8 a^3}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{12 a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 3.01, size = 278, normalized size = 0.93 \[ \frac {\sqrt {c \left (a^2 x^2+1\right )} \left (-\frac {1}{4} \left (a^2 x^2+1\right )^2 \left (-\frac {2}{\sqrt {a^2 x^2+1}}+3 \tan ^{-1}(a x) \left (-\frac {14 a x}{\sqrt {a^2 x^2+1}}+3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+2 \sin \left (3 \tan ^{-1}(a x)\right )+4 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (2 \tan ^{-1}(a x)\right )+\left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (4 \tan ^{-1}(a x)\right )\right )-6 \cos \left (3 \tan ^{-1}(a x)\right )\right )-6 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+6 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )\right )}{48 a^3 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right ), x\right ) \]
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 [A] time = 1.30, size = 199, normalized size = 0.67 \[ \frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (6 \arctan \left (a x \right ) x^{3} a^{3}-2 a^{2} x^{2}+3 \arctan \left (a x \right ) x a +1\right )}{24 a^{3}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8 a^{3} \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\mathrm {atan}\left (a\,x\right )\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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 \[ \int x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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