Optimal. Leaf size=55 \[ -\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} b c^3 \log \left (c^2 x^4+1\right )-\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4} \]
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Rubi [A] time = 0.0322249, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5033, 266, 44} \[ -\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} b c^3 \log \left (c^2 x^4+1\right )-\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}\left (c x^2\right )}{x^7} \, dx &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{3} (b c) \int \frac{1}{x^5 \left (1+c^2 x^4\right )} \, dx\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^4\right )\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c^2}{x}+\frac{c^4}{1+c^2 x}\right ) \, dx,x,x^4\right )\\ &=-\frac{b c}{12 x^4}-\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}-\frac{1}{3} b c^3 \log (x)+\frac{1}{12} b c^3 \log \left (1+c^2 x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0137974, size = 60, normalized size = 1.09 \[ -\frac{a}{6 x^6}+\frac{1}{12} b c^3 \log \left (c^2 x^4+1\right )-\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4}-\frac{b \tan ^{-1}\left (c x^2\right )}{6 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 51, normalized size = 0.9 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\frac{b\arctan \left ( c{x}^{2} \right ) }{6\,{x}^{6}}}+{\frac{b{c}^{3}\ln \left ({c}^{2}{x}^{4}+1 \right ) }{12}}-{\frac{bc}{12\,{x}^{4}}}-{\frac{b{c}^{3}\ln \left ( x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00167, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{12} \,{\left ({\left (c^{2} \log \left (c^{2} x^{4} + 1\right ) - c^{2} \log \left (x^{4}\right ) - \frac{1}{x^{4}}\right )} c - \frac{2 \, \arctan \left (c x^{2}\right )}{x^{6}}\right )} b - \frac{a}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.82218, size = 130, normalized size = 2.36 \begin{align*} \frac{b c^{3} x^{6} \log \left (c^{2} x^{4} + 1\right ) - 4 \, b c^{3} x^{6} \log \left (x\right ) - b c x^{2} - 2 \, b \arctan \left (c x^{2}\right ) - 2 \, a}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 140.139, size = 784, normalized size = 14.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14157, size = 93, normalized size = 1.69 \begin{align*} \frac{b c^{7} x^{6} \log \left (c^{2} x^{4} + 1\right ) - 2 \, b c^{7} x^{6} \log \left (c x^{2}\right ) - b c^{5} x^{2} - 2 \, b c^{4} \arctan \left (c x^{2}\right ) - 2 \, a c^{4}}{12 \, c^{4} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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