Optimal. Leaf size=118 \[ -\frac {a \sqrt {a^2 c x^2+c}}{6 c x^2}+\frac {2 a^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x^3}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{6 \sqrt {c}} \]
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Rubi [A] time = 0.20, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4962, 266, 51, 63, 208, 4944} \[ -\frac {a \sqrt {a^2 c x^2+c}}{6 c x^2}+\frac {2 a^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x^3}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{6 \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 4944
Rule 4962
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{x^4 \sqrt {c+a^2 c x^2}} \, dx &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x^3}+\frac {1}{3} a \int \frac {1}{x^3 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (2 a^2\right ) \int \frac {\tan ^{-1}(a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x}+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{3} \left (2 a^3\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {a \sqrt {c+a^2 c x^2}}{6 c x^2}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x}-\frac {1}{12} a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{3} a^3 \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 c x^2}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x}-\frac {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 )}{6 c}-\frac {(2 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 )}{3 c}\\ &=-\frac {a \sqrt {c+a^2 c x^2}}{6 c x^2}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 c x}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{6 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 110, normalized size = 0.93 \[ \frac {-5 a^3 \sqrt {c} x^3 \log (x)-a x \sqrt {a^2 c x^2+c}+2 \left (2 a^2 x^2-1\right ) \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+5 a^3 \sqrt {c} x^3 \log \left (\sqrt {c} \sqrt {a^2 c x^2+c}+c\right )}{6 c x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 89, normalized size = 0.75 \[ \frac {5 \, a^{3} \sqrt {c} x^{3} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {a^{2} c x^{2} + c} {\left (a x - 2 \, {\left (2 \, a^{2} x^{2} - 1\right )} \arctan \left (a x\right )\right )}}{12 \, c x^{3}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.23, size = 163, normalized size = 1.38 \[ \frac {\left (4 \arctan \left (a x \right ) x^{2} a^{2}-a x -2 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 c \,x^{3}}+\frac {5 a^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 \sqrt {a^{2} x^{2}+1}\, c}-\frac {5 a^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 \sqrt {a^{2} x^{2}+1}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 81, normalized size = 0.69 \[ \frac {{\left (5 \, a^{2} \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) - \frac {\sqrt {a^{2} x^{2} + 1}}{x^{2}}\right )} a + 2 \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} + 1} a^{2}}{x} - \frac {\sqrt {a^{2} x^{2} + 1}}{x^{3}}\right )} \arctan \left (a x\right )}{6 \, \sqrt {c}} \]
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 \[ \int \frac {\mathrm {atan}\left (a\,x\right )}{x^4\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
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 \[ \int \frac {\operatorname {atan}{\left (a x \right )}}{x^{4} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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