Optimal. Leaf size=132 \[ x \cot ^{-1}\left (a x^2\right )-\frac {\text {ArcTan}\left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}+\frac {\text {ArcTan}\left (1+\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}+\frac {\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2} \sqrt {a}}-\frac {\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2} \sqrt {a}} \]
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Rubi [A]
time = 0.06, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {4931, 303,
1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}+\frac {\text {ArcTan}\left (\sqrt {2} \sqrt {a} x+1\right )}{\sqrt {2} \sqrt {a}}+\frac {\log \left (a x^2-\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2} \sqrt {a}}-\frac {\log \left (a x^2+\sqrt {2} \sqrt {a} x+1\right )}{2 \sqrt {2} \sqrt {a}}+x \cot ^{-1}\left (a x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 4931
Rubi steps
\begin {align*} \int \cot ^{-1}\left (a x^2\right ) \, dx &=x \cot ^{-1}\left (a x^2\right )+(2 a) \int \frac {x^2}{1+a^2 x^4} \, dx\\ &=x \cot ^{-1}\left (a x^2\right )-\int \frac {1-a x^2}{1+a^2 x^4} \, dx+\int \frac {1+a x^2}{1+a^2 x^4} \, dx\\ &=x \cot ^{-1}\left (a x^2\right )+\frac {\int \frac {1}{\frac {1}{a}-\frac {\sqrt {2} x}{\sqrt {a}}+x^2} \, dx}{2 a}+\frac {\int \frac {1}{\frac {1}{a}+\frac {\sqrt {2} x}{\sqrt {a}}+x^2} \, dx}{2 a}+\frac {\int \frac {\frac {\sqrt {2}}{\sqrt {a}}+2 x}{-\frac {1}{a}-\frac {\sqrt {2} x}{\sqrt {a}}-x^2} \, dx}{2 \sqrt {2} \sqrt {a}}+\frac {\int \frac {\frac {\sqrt {2}}{\sqrt {a}}-2 x}{-\frac {1}{a}+\frac {\sqrt {2} x}{\sqrt {a}}-x^2} \, dx}{2 \sqrt {2} \sqrt {a}}\\ &=x \cot ^{-1}\left (a x^2\right )+\frac {\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2} \sqrt {a}}-\frac {\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2} \sqrt {a}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}\\ &=x \cot ^{-1}\left (a x^2\right )-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}+\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {a} x\right )}{\sqrt {2} \sqrt {a}}+\frac {\log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2} \sqrt {a}}-\frac {\log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )}{2 \sqrt {2} \sqrt {a}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 102, normalized size = 0.77 \begin {gather*} x \cot ^{-1}\left (a x^2\right )+\frac {-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 )}{2 \sqrt {2} \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 97, normalized size = 0.73
method | result | size |
default | \(x \,\mathrm {arccot}\left (a \,x^{2}\right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 120, 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 )}{a^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}}\right )} + x \operatorname {arccot}\left (a x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.05, size = 189, normalized size = 1.43 \begin {gather*} x \arctan \left (\frac {1}{a x^{2}}\right ) - \sqrt {2} \frac {1}{a^{2}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} a \frac {1}{a^{2}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {\sqrt {2} a \frac {1}{a^{2}}^{\frac {3}{4}} x + x^{2} + \sqrt {\frac {1}{a^{2}}}} a \frac {1}{a^{2}}^{\frac {1}{4}} - 1\right ) - \sqrt {2} \frac {1}{a^{2}}^{\frac {1}{4}} \arctan \left (-\sqrt {2} a \frac {1}{a^{2}}^{\frac {1}{4}} x + \sqrt {2} \sqrt {-\sqrt {2} a \frac {1}{a^{2}}^{\frac {3}{4}} x + x^{2} + \sqrt {\frac {1}{a^{2}}}} a \frac {1}{a^{2}}^{\frac {1}{4}} + 1\right ) - \frac {1}{4} \, \sqrt {2} \frac {1}{a^{2}}^{\frac {1}{4}} \log \left (\sqrt {2} a \frac {1}{a^{2}}^{\frac {3}{4}} x + x^{2} + \sqrt {\frac {1}{a^{2}}}\right ) + \frac {1}{4} \, \sqrt {2} \frac {1}{a^{2}}^{\frac {1}{4}} \log \left (-\sqrt {2} a \frac {1}{a^{2}}^{\frac {3}{4}} x + x^{2} + \sqrt {\frac {1}{a^{2}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 4.37, size = 615, normalized size = 4.66 \begin {gather*} \begin {cases} \infty i x & \text {for}\: a = - \frac {i}{x^{2}} \\- \infty i x & \text {for}\: a = \frac {i}{x^{2}} \\\frac {\pi x}{2} & \text {for}\: a = 0 \\\frac {2 a^{5} x^{5} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} \operatorname {acot}{\left (a x^{2} \right )}}{2 a^{5} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} + 2 a^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}}} + \frac {2 a^{4} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{2}} \log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )}}{2 a^{5} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} + 2 a^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}}} - \frac {a^{4} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{2}} \log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{2 a^{5} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} + 2 a^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}}} + \frac {2 a^{4} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )}}{2 a^{5} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} + 2 a^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}}} + \frac {2 a^{3} x \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} \operatorname {acot}{\left (a x^{2} \right )}}{2 a^{5} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} + 2 a^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}}} + \frac {2 a^{2} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{2}} \log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )}}{2 a^{5} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} + 2 a^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}}} - \frac {a^{2} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{2}} \log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{2 a^{5} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} + 2 a^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}}} + \frac {2 a^{2} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )}}{2 a^{5} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} + 2 a^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}}} + \frac {2 a x^{4} \operatorname {acot}{\left (a x^{2} \right )}}{2 a^{5} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} + 2 a^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}}} + \frac {2 \operatorname {acot}{\left (a x^{2} \right )}}{2 a^{6} x^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}} + 2 a^{4} \left (- \frac {1}{a^{2}}\right )^{\frac {7}{4}}} & \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 = 144, normalized size = 1.09 \begin {gather*} \frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | a \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2}} + \frac {2 \, \sqrt {2} \sqrt {{\left | a \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{a^{2}} - \frac {\sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2}} + \frac {\sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{a^{2}}\right )} + x \arctan \left (\frac {1}{a x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 42, normalized size = 0.32 \begin {gather*} x\,\mathrm {acot}\left (a\,x^2\right )+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{\sqrt {a}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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