Optimal. Leaf size=135 \[ -\frac {\cot ^{-1}\left (a x^2\right )}{x}+\frac {\sqrt {a} \text {ArcTan}\left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2}}-\frac {\sqrt {a} \text {ArcTan}\left (1+\sqrt {2} \sqrt {a} x\right )}{\sqrt {2}}+\frac {\sqrt {a} \log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2}}-\frac {\sqrt {a} \log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4947, 217,
1179, 642, 1176, 631, 210} \begin {gather*} \frac {\sqrt {a} \text {ArcTan}\left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2}}-\frac {\sqrt {a} \text {ArcTan}\left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2}}+\frac {\sqrt {a} \log \left (a x^2-\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2}}-\frac {\sqrt {a} \log \left (a x^2+\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2}}-\frac {\cot ^{-1}\left (a x^2\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 4947
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}\left (a x^2\right )}{x^2} \, dx &=-\frac {\cot ^{-1}\left (a x^2\right )}{x}-(2 a) \int \frac {1}{1+a^2 x^4} \, dx\\ &=-\frac {\cot ^{-1}\left (a x^2\right )}{x}-a \int \frac {1-a x^2}{1+a^2 x^4} \, dx-a \int \frac {1+a x^2}{1+a^2 x^4} \, dx\\ &=-\frac {\cot ^{-1}\left (a x^2\right )}{x}-\frac {1}{2} \int \frac {1}{\frac {1}{a}-\frac {\sqrt {2} x}{\sqrt {a}}+x^2} \, dx-\frac {1}{2} \int \frac {1}{\frac {1}{a}+\frac {\sqrt {2} x}{\sqrt {a}}+x^2} \, dx+\frac {\sqrt {a} \int \frac {\frac {\sqrt {2}}{\sqrt {a}}+2 x}{-\frac {1}{a}-\frac {\sqrt {2} x}{\sqrt {a}}-x^2} \, dx}{2 \sqrt {2}}+\frac {\sqrt {a} \int \frac {\frac {\sqrt {2}}{\sqrt {a}}-2 x}{-\frac {1}{a}+\frac {\sqrt {2} x}{\sqrt {a}}-x^2} \, dx}{2 \sqrt {2}}\\ &=-\frac {\cot ^{-1}\left (a x^2\right )}{x}+\frac {\sqrt {a} \log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2}}-\frac {\sqrt {a} \log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2}}-\frac {\sqrt {a} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2}}+\frac {\sqrt {a} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {a} x\right )}{\sqrt {2}}\\ &=-\frac {\cot ^{-1}\left (a x^2\right )}{x}+\frac {\sqrt {a} \tan ^{-1}\left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2}}-\frac {\sqrt {a} \tan ^{-1}\left (1+\sqrt {2} \sqrt {a} x\right )}{\sqrt {2}}+\frac {\sqrt {a} \log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2}}-\frac {\sqrt {a} \log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 105, normalized size = 0.78 \begin {gather*} -\frac {\cot ^{-1}\left (a x^2\right )}{x}+\frac {\sqrt {a} \left (2 \text {ArcTan}\left (1-\sqrt {2} \sqrt {a} x\right )-2 \text {ArcTan}\left (1+\sqrt {2} \sqrt {a} x\right )+\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )-\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 98, normalized size = 0.73
method | result | size |
default | \(-\frac {\mathrm {arccot}\left (a \,x^{2}\right )}{x}-\frac {a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 123, normalized size = 0.91 \begin {gather*} -\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{\sqrt {a}} + \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}} - \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{\sqrt {a}}\right )} - \frac {\operatorname {arccot}\left (a x^{2}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 227 vs.
\(2 (99) = 198\).
time = 2.92, size = 227, normalized size = 1.68 \begin {gather*} \frac {4 \, \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {\sqrt {2} {\left (a^{2}\right )}^{\frac {3}{4}} a x + a^{2} - \sqrt {2} \sqrt {a^{2} x^{2} + \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} a x + \sqrt {a^{2}}} {\left (a^{2}\right )}^{\frac {3}{4}}}{a^{2}}\right ) + 4 \, \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {\sqrt {2} {\left (a^{2}\right )}^{\frac {3}{4}} a x - a^{2} - \sqrt {2} \sqrt {a^{2} x^{2} - \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} a x + \sqrt {a^{2}}} {\left (a^{2}\right )}^{\frac {3}{4}}}{a^{2}}\right ) - \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} x \log \left (a^{2} x^{2} + \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} a x + \sqrt {a^{2}}\right ) + \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} x \log \left (a^{2} x^{2} - \sqrt {2} {\left (a^{2}\right )}^{\frac {1}{4}} a x + \sqrt {a^{2}}\right ) - 4 \, \arctan \left (\frac {1}{a x^{2}}\right )}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.03, size = 110, normalized size = 0.81 \begin {gather*} \begin {cases} a \sqrt [4]{- \frac {1}{a^{2}}} \log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )} - \frac {a \sqrt [4]{- \frac {1}{a^{2}}} \log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{2} - a \sqrt [4]{- \frac {1}{a^{2}}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )} - \frac {\operatorname {acot}{\left (a x^{2} \right )}}{\sqrt [4]{- \frac {1}{a^{2}}}} - \frac {\operatorname {acot}{\left (a x^{2} \right )}}{x} & \text {for}\: a \neq 0 \\- \frac {\pi }{2 x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 135, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}} + \frac {\sqrt {2} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}} - \frac {\sqrt {2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{\sqrt {{\left | a \right |}}}\right )} - \frac {\arctan \left (\frac {1}{a x^{2}}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.68, size = 44, normalized size = 0.33 \begin {gather*} -\frac {\mathrm {acot}\left (a\,x^2\right )}{x}+{\left (-1\right )}^{1/4}\,\sqrt {a}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )\,1{}\mathrm {i}+{\left (-1\right )}^{1/4}\,\sqrt {a}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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