Optimal. Leaf size=39 \[ -\frac{a+b \tan ^{-1}\left (c x^2\right )}{2 x^2}-\frac{1}{4} b c \log \left (c^2 x^4+1\right )+b c \log (x) \]
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Rubi [A] time = 0.0236941, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5033, 266, 36, 29, 31} \[ -\frac{a+b \tan ^{-1}\left (c x^2\right )}{2 x^2}-\frac{1}{4} b c \log \left (c^2 x^4+1\right )+b c \log (x) \]
Antiderivative was successfully verified.
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Rule 5033
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}\left (c x^2\right )}{x^3} \, dx &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{2 x^2}+(b c) \int \frac{1}{x \left (1+c^2 x^4\right )} \, dx\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{2 x^2}+\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^4\right )\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{2 x^2}+\frac{1}{4} (b c) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^4\right )-\frac{1}{4} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^4\right )\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{2 x^2}+b c \log (x)-\frac{1}{4} b c \log \left (1+c^2 x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0062847, size = 44, normalized size = 1.13 \[ -\frac{a}{2 x^2}-\frac{1}{4} b c \log \left (c^2 x^4+1\right )-\frac{b \tan ^{-1}\left (c x^2\right )}{2 x^2}+b c \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 39, normalized size = 1. \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{b\arctan \left ( c{x}^{2} \right ) }{2\,{x}^{2}}}-{\frac{bc\ln \left ({c}^{2}{x}^{4}+1 \right ) }{4}}+bc\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998128, size = 55, normalized size = 1.41 \begin{align*} -\frac{1}{4} \,{\left (c{\left (\log \left (c^{2} x^{4} + 1\right ) - \log \left (x^{4}\right )\right )} + \frac{2 \, \arctan \left (c x^{2}\right )}{x^{2}}\right )} b - \frac{a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.74874, size = 111, normalized size = 2.85 \begin{align*} -\frac{b c x^{2} \log \left (c^{2} x^{4} + 1\right ) - 4 \, b c x^{2} \log \left (x\right ) + 2 \, b \arctan \left (c x^{2}\right ) + 2 \, a}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 45.4078, size = 593, normalized size = 15.21 \begin{align*} \begin{cases} - \frac{a}{2 x^{2}} & \text{for}\: c = 0 \\- \frac{a - \infty i b}{2 x^{2}} & \text{for}\: c = - \frac{i}{x^{2}} \\- \frac{a + \infty i b}{2 x^{2}} & \text{for}\: c = \frac{i}{x^{2}} \\- \frac{i a c^{2} x^{4} \sqrt{\frac{1}{c^{2}}}}{2 i c^{2} x^{6} \sqrt{\frac{1}{c^{2}}} + 2 i x^{2} \sqrt{\frac{1}{c^{2}}}} - \frac{i a \sqrt{\frac{1}{c^{2}}}}{2 i c^{2} x^{6} \sqrt{\frac{1}{c^{2}}} + 2 i x^{2} \sqrt{\frac{1}{c^{2}}}} + \frac{2 i b c^{3} x^{6} \sqrt{\frac{1}{c^{2}}} \log{\left (x \right )}}{2 i c^{2} x^{6} \sqrt{\frac{1}{c^{2}}} + 2 i x^{2} \sqrt{\frac{1}{c^{2}}}} - \frac{i b c^{3} x^{6} \sqrt{\frac{1}{c^{2}}} \log{\left (x^{2} + i \sqrt{\frac{1}{c^{2}}} \right )}}{2 i c^{2} x^{6} \sqrt{\frac{1}{c^{2}}} + 2 i x^{2} \sqrt{\frac{1}{c^{2}}}} + \frac{b c^{2} x^{6} \operatorname{atan}{\left (c x^{2} \right )}}{2 i c^{2} x^{6} \sqrt{\frac{1}{c^{2}}} + 2 i x^{2} \sqrt{\frac{1}{c^{2}}}} - \frac{i b c^{2} x^{4} \sqrt{\frac{1}{c^{2}}} \operatorname{atan}{\left (c x^{2} \right )}}{2 i c^{2} x^{6} \sqrt{\frac{1}{c^{2}}} + 2 i x^{2} \sqrt{\frac{1}{c^{2}}}} + \frac{2 i b c x^{2} \sqrt{\frac{1}{c^{2}}} \log{\left (x \right )}}{2 i c^{2} x^{6} \sqrt{\frac{1}{c^{2}}} + 2 i x^{2} \sqrt{\frac{1}{c^{2}}}} - \frac{i b c x^{2} \sqrt{\frac{1}{c^{2}}} \log{\left (x^{2} + i \sqrt{\frac{1}{c^{2}}} \right )}}{2 i c^{2} x^{6} \sqrt{\frac{1}{c^{2}}} + 2 i x^{2} \sqrt{\frac{1}{c^{2}}}} + \frac{b x^{2} \operatorname{atan}{\left (c x^{2} \right )}}{2 i c^{2} x^{6} \sqrt{\frac{1}{c^{2}}} + 2 i x^{2} \sqrt{\frac{1}{c^{2}}}} - \frac{i b \sqrt{\frac{1}{c^{2}}} \operatorname{atan}{\left (c x^{2} \right )}}{2 i c^{2} x^{6} \sqrt{\frac{1}{c^{2}}} + 2 i x^{2} \sqrt{\frac{1}{c^{2}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22596, size = 81, normalized size = 2.08 \begin{align*} -\frac{b c^{3} x^{2} \log \left (c^{2} x^{4} + 1\right ) - 2 \, b c^{3} x^{2} \log \left (c x^{2}\right ) + 2 \, b c^{2} \arctan \left (c x^{2}\right ) + 2 \, a c^{2}}{4 \, c^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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