Optimal. Leaf size=300 \[ \frac {a^3 x}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2}}{2 c^2 x}-\frac {a^2 \text {ArcTan}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{2 c^2 x^2}+\frac {3 a^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {3 i a^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 c \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 c \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.44, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5086, 5082,
270, 5078, 5074, 5050, 197} \begin {gather*} -\frac {\text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{2 c^2 x^2}-\frac {a^2 \text {ArcTan}(a x)}{c \sqrt {a^2 c x^2+c}}+\frac {3 a^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {a \sqrt {a^2 c x^2+c}}{2 c^2 x}-\frac {3 i a^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 c \sqrt {a^2 c x^2+c}}+\frac {3 i a^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 c \sqrt {a^2 c x^2+c}}+\frac {a^3 x}{c \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 270
Rule 5050
Rule 5074
Rule 5078
Rule 5082
Rule 5086
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{2 c^2 x^2}+a^4 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac {a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{2 c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx}{2 c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=-\frac {a \sqrt {c+a^2 c x^2}}{2 c^2 x}-\frac {a^2 \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{2 c^2 x^2}+a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx}{2 c \sqrt {c+a^2 c x^2}}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {a^3 x}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2}}{2 c^2 x}-\frac {a^2 \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{2 c^2 x^2}+\frac {3 a^2 \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 )}{c \sqrt {c+a^2 c x^2}}-\frac {3 i a^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 c \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.93, size = 258, normalized size = 0.86 \begin {gather*} -\frac {a^2 \left (-8 a x+8 \text {ArcTan}(a x)+a x \csc ^2\left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sqrt {1+a^2 x^2} \text {ArcTan}(a x) \csc ^2\left (\frac {1}{2} \text {ArcTan}(a x)\right )+12 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \log \left (1-e^{i \text {ArcTan}(a x)}\right )-12 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \log \left (1+e^{i \text {ArcTan}(a x)}\right )+12 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-e^{i \text {ArcTan}(a x)}\right )-12 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,e^{i \text {ArcTan}(a x)}\right )-\sqrt {1+a^2 x^2} \text {ArcTan}(a x) \sec ^2\left (\frac {1}{2} \text {ArcTan}(a x)\right )+2 \sqrt {1+a^2 x^2} \tan \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )}{8 c \sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 273, normalized size = 0.91
method | result | size |
default | \(-\frac {a^{2} \left (\arctan \left (a x \right )+i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right ) a^{2}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\left (a x +\arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c^{2} x^{2}}+\frac {3 i a^{2} \left (i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, c^{2}}\) | \(273\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atan}{\left (a x \right )}}{x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \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 = -0.00 \begin {gather*} \int \frac {\mathrm {atan}\left (a\,x\right )}{x^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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