Optimal. Leaf size=159 \[ -\frac {a+b \tan ^{-1}\left (c x^2\right )}{5 x^5}+\frac {b c^{5/2} \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{10 \sqrt {2}}-\frac {b c^{5/2} \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{10 \sqrt {2}}+\frac {b c^{5/2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{5 \sqrt {2}}-\frac {b c^{5/2} \tan ^{-1}\left (\sqrt {2} \sqrt {c} x+1\right )}{5 \sqrt {2}}-\frac {2 b c}{15 x^3} \]
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Rubi [A] time = 0.10, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5033, 325, 211, 1165, 628, 1162, 617, 204} \[ -\frac {a+b \tan ^{-1}\left (c x^2\right )}{5 x^5}+\frac {b c^{5/2} \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{10 \sqrt {2}}-\frac {b c^{5/2} \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{10 \sqrt {2}}+\frac {b c^{5/2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{5 \sqrt {2}}-\frac {b c^{5/2} \tan ^{-1}\left (\sqrt {2} \sqrt {c} x+1\right )}{5 \sqrt {2}}-\frac {2 b c}{15 x^3} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 325
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 5033
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}\left (c x^2\right )}{x^6} \, dx &=-\frac {a+b \tan ^{-1}\left (c x^2\right )}{5 x^5}+\frac {1}{5} (2 b c) \int \frac {1}{x^4 \left (1+c^2 x^4\right )} \, dx\\ &=-\frac {2 b c}{15 x^3}-\frac {a+b \tan ^{-1}\left (c x^2\right )}{5 x^5}-\frac {1}{5} \left (2 b c^3\right ) \int \frac {1}{1+c^2 x^4} \, dx\\ &=-\frac {2 b c}{15 x^3}-\frac {a+b \tan ^{-1}\left (c x^2\right )}{5 x^5}-\frac {1}{5} \left (b c^3\right ) \int \frac {1-c x^2}{1+c^2 x^4} \, dx-\frac {1}{5} \left (b c^3\right ) \int \frac {1+c x^2}{1+c^2 x^4} \, dx\\ &=-\frac {2 b c}{15 x^3}-\frac {a+b \tan ^{-1}\left (c x^2\right )}{5 x^5}-\frac {1}{10} \left (b c^2\right ) \int \frac {1}{\frac {1}{c}-\frac {\sqrt {2} x}{\sqrt {c}}+x^2} \, dx-\frac {1}{10} \left (b c^2\right ) \int \frac {1}{\frac {1}{c}+\frac {\sqrt {2} x}{\sqrt {c}}+x^2} \, dx+\frac {\left (b c^{5/2}\right ) \int \frac {\frac {\sqrt {2}}{\sqrt {c}}+2 x}{-\frac {1}{c}-\frac {\sqrt {2} x}{\sqrt {c}}-x^2} \, dx}{10 \sqrt {2}}+\frac {\left (b c^{5/2}\right ) \int \frac {\frac {\sqrt {2}}{\sqrt {c}}-2 x}{-\frac {1}{c}+\frac {\sqrt {2} x}{\sqrt {c}}-x^2} \, dx}{10 \sqrt {2}}\\ &=-\frac {2 b c}{15 x^3}-\frac {a+b \tan ^{-1}\left (c x^2\right )}{5 x^5}+\frac {b c^{5/2} \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{10 \sqrt {2}}-\frac {b c^{5/2} \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{10 \sqrt {2}}-\frac {\left (b c^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {c} x\right )}{5 \sqrt {2}}+\frac {\left (b c^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {c} x\right )}{5 \sqrt {2}}\\ &=-\frac {2 b c}{15 x^3}-\frac {a+b \tan ^{-1}\left (c x^2\right )}{5 x^5}+\frac {b c^{5/2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {c} x\right )}{5 \sqrt {2}}-\frac {b c^{5/2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {c} x\right )}{5 \sqrt {2}}+\frac {b c^{5/2} \log \left (1-\sqrt {2} \sqrt {c} x+c x^2\right )}{10 \sqrt {2}}-\frac {b c^{5/2} \log \left (1+\sqrt {2} \sqrt {c} x+c x^2\right )}{10 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 177, normalized size = 1.11 \[ -\frac {a}{5 x^5}+\frac {b c^{5/2} \log \left (c x^2-\sqrt {2} \sqrt {c} x+1\right )}{10 \sqrt {2}}-\frac {b c^{5/2} \log \left (c x^2+\sqrt {2} \sqrt {c} x+1\right )}{10 \sqrt {2}}-\frac {b c^{5/2} \tan ^{-1}\left (\frac {2 \sqrt {c} x-\sqrt {2}}{\sqrt {2}}\right )}{5 \sqrt {2}}-\frac {b c^{5/2} \tan ^{-1}\left (\frac {2 \sqrt {c} x+\sqrt {2}}{\sqrt {2}}\right )}{5 \sqrt {2}}-\frac {2 b c}{15 x^3}-\frac {b \tan ^{-1}\left (c x^2\right )}{5 x^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 350, normalized size = 2.20 \[ \frac {12 \, \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {1}{4}} x^{5} \arctan \left (-\frac {b^{4} c^{10} + \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {3}{4}} b c^{3} x - \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {3}{4}} \sqrt {b^{2} c^{6} x^{2} + \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {1}{4}} b c^{3} x + \sqrt {b^{4} c^{10}}}}{b^{4} c^{10}}\right ) + 12 \, \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {1}{4}} x^{5} \arctan \left (\frac {b^{4} c^{10} - \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {3}{4}} b c^{3} x + \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {3}{4}} \sqrt {b^{2} c^{6} x^{2} - \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {1}{4}} b c^{3} x + \sqrt {b^{4} c^{10}}}}{b^{4} c^{10}}\right ) - 3 \, \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {1}{4}} x^{5} \log \left (b^{2} c^{6} x^{2} + \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {1}{4}} b c^{3} x + \sqrt {b^{4} c^{10}}\right ) + 3 \, \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {1}{4}} x^{5} \log \left (b^{2} c^{6} x^{2} - \sqrt {2} \left (b^{4} c^{10}\right )^{\frac {1}{4}} b c^{3} x + \sqrt {b^{4} c^{10}}\right ) - 8 \, b c x^{2} - 12 \, b \arctan \left (c x^{2}\right ) - 12 \, a}{60 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 150, normalized size = 0.94 \[ -\frac {1}{20} \, b c^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{\sqrt {{\left | c \right |}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{\sqrt {{\left | c \right |}}} + \frac {\sqrt {2} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{\sqrt {{\left | c \right |}}} - \frac {\sqrt {2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{\sqrt {{\left | c \right |}}}\right )} - \frac {2 \, b c x^{2} + 3 \, b \arctan \left (c x^{2}\right ) + 3 \, a}{15 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 138, normalized size = 0.87 \[ -\frac {a}{5 x^{5}}-\frac {b \arctan \left (c \,x^{2}\right )}{5 x^{5}}-\frac {2 b c}{15 x^{3}}-\frac {b \,c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )}{10}-\frac {b \,c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )}{20}-\frac {b \,c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 138, normalized size = 0.87 \[ -\frac {1}{60} \, {\left ({\left (6 \, \sqrt {2} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right ) + 6 \, \sqrt {2} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right ) + 3 \, \sqrt {2} c^{\frac {3}{2}} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right ) - 3 \, \sqrt {2} c^{\frac {3}{2}} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right ) + \frac {8}{x^{3}}\right )} c + \frac {12 \, \arctan \left (c x^{2}\right )}{x^{5}}\right )} b - \frac {a}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 63, normalized size = 0.40 \[ -\frac {\frac {2\,b\,c\,x^2}{3}+a}{5\,x^5}-\frac {b\,\mathrm {atan}\left (c\,x^2\right )}{5\,x^5}+\frac {{\left (-1\right )}^{1/4}\,b\,c^{5/2}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\right )\,1{}\mathrm {i}}{5}+\frac {{\left (-1\right )}^{1/4}\,b\,c^{5/2}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\,1{}\mathrm {i}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 63.23, size = 185, normalized size = 1.16 \[ \begin {cases} - \frac {a}{5 x^{5}} + \frac {\sqrt [4]{-1} b c^{3} \sqrt [4]{\frac {1}{c^{2}}} \log {\left (x - \sqrt [4]{-1} \sqrt [4]{\frac {1}{c^{2}}} \right )}}{5} - \frac {\sqrt [4]{-1} b c^{3} \sqrt [4]{\frac {1}{c^{2}}} \log {\left (x^{2} + i \sqrt {\frac {1}{c^{2}}} \right )}}{10} + \frac {\sqrt [4]{-1} b c^{3} \sqrt [4]{\frac {1}{c^{2}}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} x}{\sqrt [4]{\frac {1}{c^{2}}}} \right )}}{5} - \frac {\left (-1\right )^{\frac {3}{4}} b c^{2} \operatorname {atan}{\left (c x^{2} \right )}}{5 \sqrt [4]{\frac {1}{c^{2}}}} - \frac {2 b c}{15 x^{3}} - \frac {b \operatorname {atan}{\left (c x^{2} \right )}}{5 x^{5}} & \text {for}\: c \neq 0 \\- \frac {a}{5 x^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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