Optimal. Leaf size=242 \[ -\frac {\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{x}-\frac {2 i a c \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {i a c \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a c \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5070, 5064,
272, 65, 214, 5010, 5006} \begin {gather*} -\frac {2 i a c \sqrt {a^2 x^2+1} \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \text {ArcTan}(a x)}{\sqrt {a^2 c x^2+c}}-\frac {\text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{x}+\frac {i a c \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {i a c \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 5006
Rule 5010
Rule 5064
Rule 5070
Rubi steps
\begin {align*} \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x^2} \, dx &=c \int \frac {\tan ^{-1}(a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x}+(a c) \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^2 c \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac {2 i a 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 )}{\sqrt {c+a^2 c x^2}}+\frac {i a c \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a c \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {1}{2} (a c) \text {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac {2 i a 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 )}{\sqrt {c+a^2 c x^2}}+\frac {i a c \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a c \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c+a^2 c x^2}\right )}{a}\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac {2 i a 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 )}{\sqrt {c+a^2 c x^2}}-a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {i a c \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a c \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 163, normalized size = 0.67 \begin {gather*} -\frac {a \sqrt {c \left (1+a^2 x^2\right )} \left (\frac {\sqrt {1+a^2 x^2} \text {ArcTan}(a x)}{a x}-\text {ArcTan}(a x) \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+\text {ArcTan}(a x) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+\log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-\log \left (\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-i \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )+i \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )\right )}{\sqrt {1+a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 221, normalized size = 0.91
method | result | size |
default | \(-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )}{x}+\frac {i a \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-i \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )+\dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{\sqrt {a^{2} x^{2}+1}}\) | \(221\) |
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 {\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 )\,\sqrt {c\,a^2\,x^2+c}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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