Optimal. Leaf size=84 \[ -\frac {a \sqrt {a^2 c x^2+c}}{6 x^2}-\frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{3 c x^3}-\frac {1}{6} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4944, 266, 47, 63, 208} \[ -\frac {a \sqrt {a^2 c x^2+c}}{6 x^2}-\frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{3 c x^3}-\frac {1}{6} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 4944
Rubi steps
\begin {align*} \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x^4} \, dx &=-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 c x^3}+\frac {1}{3} a \int \frac {\sqrt {c+a^2 c x^2}}{x^3} \, dx\\ &=-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 c x^3}+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {\sqrt {c+a^2 c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {a \sqrt {c+a^2 c x^2}}{6 x^2}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 c x^3}+\frac {1}{12} \left (a^3 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {a \sqrt {c+a^2 c x^2}}{6 x^2}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 c x^3}+\frac {1}{6} a \operatorname {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 )\\ &=-\frac {a \sqrt {c+a^2 c x^2}}{6 x^2}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 c x^3}-\frac {1}{6} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 105, normalized size = 1.25 \[ \frac {a^3 \sqrt {c} x^3 \log (x)-a x \left (\sqrt {a^2 c x^2+c}+a^2 \sqrt {c} x^2 \log \left (\sqrt {c} \sqrt {a^2 c x^2+c}+c\right )\right )-2 \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 84, normalized size = 1.00 \[ \frac {a^{3} \sqrt {c} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {a^{2} c x^{2} + c} {\left (a x + 2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )}}{12 \, x^{3}} \]
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 [C] time = 1.22, size = 153, normalized size = 1.82 \[ -\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right ) x^{2} a^{2}+a x +2 \arctan \left (a x \right )\right )}{6 x^{3}}+\frac {a^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )}{6 \sqrt {a^{2} x^{2}+1}}-\frac {a^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{6 \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 73, normalized size = 0.87 \[ -\frac {1}{6} \, {\left ({\left (a^{2} \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) - \sqrt {a^{2} x^{2} + 1} a^{2} + \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{x^{2}}\right )} a + \frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \arctan \left (a x\right )}{x^{3}}\right )} \sqrt {c} \]
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 \[ \int \frac {\mathrm {atan}\left (a\,x\right )\,\sqrt {c\,a^2\,x^2+c}}{x^4} \,d x \]
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 \[ \int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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