Optimal. Leaf size=279 \[ -\frac {i c \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 a^2 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 a^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 c x^2+c}}{3 a^2}+\frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 a^2 c}+\frac {2 i c \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{3 a^2 \sqrt {a^2 c x^2+c}}-\frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 a} \]
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Rubi [A] time = 0.18, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4930, 4878, 4890, 4886} \[ -\frac {i c \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^2 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 c x^2+c}}{3 a^2}+\frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 a^2 c}+\frac {2 i c \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{3 a^2 \sqrt {a^2 c x^2+c}}-\frac {x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 4878
Rule 4886
Rule 4890
Rule 4930
Rubi steps
\begin {align*} \int x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx &=\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 a^2 c}-\frac {2 \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx}{3 a}\\ &=\frac {\sqrt {c+a^2 c x^2}}{3 a^2}-\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a}+\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 a^2 c}-\frac {c \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}\\ &=\frac {\sqrt {c+a^2 c x^2}}{3 a^2}-\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a}+\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 a^2 c}-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a \sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {c+a^2 c x^2}}{3 a^2}-\frac {x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 a}+\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 a^2 c}+\frac {2 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 )}{3 a^2 \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 )}{3 a^2 \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 )}{3 a^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 260, normalized size = 0.93 \[ \frac {\left (a^2 x^2+1\right ) \sqrt {c \left (a^2 x^2+1\right )} \left (-\frac {4 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+\frac {4 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}-\frac {3 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}+\frac {3 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}+4 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )+2 \cos \left (2 \tan ^{-1}(a x)\right )-\tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+\tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (3 \tan ^{-1}(a x)\right )+2\right )}{12 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{2}, x\right ) \]
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 [A] time = 1.14, size = 198, normalized size = 0.71 \[ \frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right )^{2} x^{2} a^{2}-\arctan \left (a x \right ) x a +\arctan \left (a x \right )^{2}+1\right )}{3 a^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{3 a^{2} \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int \sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{2}\,{d x} \]
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 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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 \[ \int x \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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