Optimal. Leaf size=240 \[ \frac {i a^2 c \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {i a^2 c \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {a \sqrt {a^2 c x^2+c}}{2 x}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac {a^2 c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.35, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4946, 4962, 264, 4958, 4954} \[ \frac {i a^2 c \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {i a^2 c \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {a \sqrt {a^2 c x^2+c}}{2 x}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac {a^2 c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 4946
Rule 4954
Rule 4958
Rule 4962
Rubi steps
\begin {align*} \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x^2}-c \int \frac {\tan ^{-1}(a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx+(a c) \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {a \sqrt {c+a^2 c x^2}}{x}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}-\frac {1}{2} (a c) \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{2} \left (a^2 c\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {a \sqrt {c+a^2 c x^2}}{2 x}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}+\frac {\left (a^2 c \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {a \sqrt {c+a^2 c x^2}}{2 x}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}-\frac {a^2 c \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^2 c \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {i a^2 c \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 1.12, size = 165, normalized size = 0.69 \[ \frac {a^2 \sqrt {c \left (a^2 x^2+1\right )} \left (4 i \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )-4 i \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )-2 \tan \left (\frac {1}{2} \tan ^{-1}(a x)\right )+4 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-4 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )-2 \cot \left (\frac {1}{2} \tan ^{-1}(a x)\right )-\tan ^{-1}(a x) \csc ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )+\tan ^{-1}(a x) \sec ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )}{8 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )}{x^{3}}, x\right ) \]
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 = 1.06, size = 169, normalized size = 0.70 \[ -\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +\arctan \left (a x \right )\right )}{2 x^{2}}+\frac {i a^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )}{x^{3}}\,{d x} \]
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 \[ \int \frac {\mathrm {atan}\left (a\,x\right )\,\sqrt {c\,a^2\,x^2+c}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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