Optimal. Leaf size=103 \[ -\frac {a}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \text {ArcTan}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{c^2 x}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5086, 5064,
272, 65, 214, 5014} \begin {gather*} -\frac {\text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{c^2 x}-\frac {a^2 x \text {ArcTan}(a x)}{c \sqrt {a^2 c x^2+c}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {a}{c \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 5014
Rule 5064
Rule 5086
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=-\frac {a}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}+\frac {a \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=-\frac {a}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}+\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )}{2 c}\\ &=-\frac {a}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}+\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 c^2}\\ &=-\frac {a}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 122, normalized size = 1.18 \begin {gather*} -\frac {a \sqrt {c \left (1+a^2 x^2\right )}}{c^2 \left (1+a^2 x^2\right )}-\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (1+2 a^2 x^2\right ) \text {ArcTan}(a x)}{c^2 x \left (1+a^2 x^2\right )}+\frac {a \log (x)}{c^{3/2}}-\frac {a \log \left (c+\sqrt {c} \sqrt {c \left (1+a^2 x^2\right )}\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.29, size = 231, normalized size = 2.24
method | result | size |
default | \(-\frac {a \left (\arctan \left (a x \right )+i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )-i\right ) a}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{c^{2} x}-\frac {a \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{2}}+\frac {a \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{2}}\) | \(231\) |
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 [A]
time = 2.18, size = 104, normalized size = 1.01 \begin {gather*} \frac {{\left (a^{3} x^{3} + a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {a^{2} c x^{2} + c} {\left (a x + {\left (2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )}}{2 \, {\left (a^{2} c^{2} x^{3} + c^{2} x\right )}} \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^{2} \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 = -0.01 \begin {gather*} \int \frac {\mathrm {atan}\left (a\,x\right )}{x^2\,{\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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