Optimal. Leaf size=109 \[ \frac {d x^2 \left (10 a^2 c-3 d\right )}{30 a^3}+\frac {\left (15 a^4 c^2-10 a^2 c d+3 d^2\right ) \log \left (a^2 x^2+1\right )}{30 a^5}+c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)+\frac {d^2 x^4}{20 a} \]
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Rubi [A] time = 0.13, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 = {194, 4913, 1594, 1247, 698} \[ \frac {\left (15 a^4 c^2-10 a^2 c d+3 d^2\right ) \log \left (a^2 x^2+1\right )}{30 a^5}+\frac {d x^2 \left (10 a^2 c-3 d\right )}{30 a^3}+c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {d^2 x^4}{20 a}+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 194
Rule 698
Rule 1247
Rule 1594
Rule 4913
Rubi steps
\begin {align*} \int \left (c+d x^2\right )^2 \cot ^{-1}(a x) \, dx &=c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)+a \int \frac {c^2 x+\frac {2}{3} c d x^3+\frac {d^2 x^5}{5}}{1+a^2 x^2} \, dx\\ &=c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)+a \int \frac {x \left (c^2+\frac {2}{3} c d x^2+\frac {d^2 x^4}{5}\right )}{1+a^2 x^2} \, dx\\ &=c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {c^2+\frac {2 c d x}{3}+\frac {d^2 x^2}{5}}{1+a^2 x} \, dx,x,x^2\right )\\ &=c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)+\frac {1}{2} a \operatorname {Subst}\left (\int \left (\frac {\left (10 a^2 c-3 d\right ) d}{15 a^4}+\frac {d^2 x}{5 a^2}+\frac {15 a^4 c^2-10 a^2 c d+3 d^2}{15 a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {\left (10 a^2 c-3 d\right ) d x^2}{30 a^3}+\frac {d^2 x^4}{20 a}+c^2 x \cot ^{-1}(a x)+\frac {2}{3} c d x^3 \cot ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cot ^{-1}(a x)+\frac {\left (15 a^4 c^2-10 a^2 c d+3 d^2\right ) \log \left (1+a^2 x^2\right )}{30 a^5}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 97, normalized size = 0.89 \[ \frac {4 a^5 x \cot ^{-1}(a x) \left (15 c^2+10 c d x^2+3 d^2 x^4\right )+a^2 d x^2 \left (a^2 \left (20 c+3 d x^2\right )-6 d\right )+\left (30 a^4 c^2-20 a^2 c d+6 d^2\right ) \log \left (a^2 x^2+1\right )}{60 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.52, size = 108, normalized size = 0.99 \[ \frac {3 \, a^{4} d^{2} x^{4} + 2 \, {\left (10 \, a^{4} c d - 3 \, a^{2} d^{2}\right )} x^{2} + 4 \, {\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \operatorname {arccot}\left (a x\right ) + 2 \, {\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a^{2} x^{2} + 1\right )}{60 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 171, normalized size = 1.57 \[ \frac {1}{60} \, {\left (\frac {4 \, {\left (3 \, d^{2} + \frac {10 \, c d}{x^{2}} + \frac {15 \, c^{2}}{x^{4}}\right )} x^{5} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {{\left (3 \, d^{2} + \frac {20 \, c d}{x^{2}} + \frac {45 \, c^{2}}{x^{4}} - \frac {6 \, d^{2}}{a^{2} x^{2}} - \frac {30 \, c d}{a^{2} x^{4}} + \frac {9 \, d^{2}}{a^{4} x^{4}}\right )} x^{4}}{a^{2}} + \frac {2 \, {\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{6}} - \frac {2 \, {\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{6}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 119, normalized size = 1.09 \[ \frac {d^{2} x^{5} \mathrm {arccot}\left (a x \right )}{5}+\frac {2 c d \,x^{3} \mathrm {arccot}\left (a x \right )}{3}+c^{2} x \,\mathrm {arccot}\left (a x \right )+\frac {d c \,x^{2}}{3 a}+\frac {d^{2} x^{4}}{20 a}-\frac {x^{2} d^{2}}{10 a^{3}}+\frac {c^{2} \ln \left (a^{2} x^{2}+1\right )}{2 a}-\frac {\ln \left (a^{2} x^{2}+1\right ) c d}{3 a^{3}}+\frac {\ln \left (a^{2} x^{2}+1\right ) d^{2}}{10 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 103, normalized size = 0.94 \[ \frac {1}{60} \, a {\left (\frac {3 \, a^{2} d^{2} x^{4} + 2 \, {\left (10 \, a^{2} c d - 3 \, d^{2}\right )} x^{2}}{a^{4}} + \frac {2 \, {\left (15 \, a^{4} c^{2} - 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )} + \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \operatorname {arccot}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 116, normalized size = 1.06 \[ \frac {a^4\,\left (\frac {c^2\,\ln \left (a^2\,x^2+1\right )}{2}+\frac {d^2\,x^4}{20}+\frac {c\,d\,x^2}{3}\right )-a^2\,\left (\frac {d^2\,x^2}{10}+\frac {c\,d\,\ln \left (a^2\,x^2+1\right )}{3}\right )+\frac {d^2\,\ln \left (a^2\,x^2+1\right )}{10}}{a^5}+c^2\,x\,\mathrm {acot}\left (a\,x\right )+\frac {d^2\,x^5\,\mathrm {acot}\left (a\,x\right )}{5}+\frac {2\,c\,d\,x^3\,\mathrm {acot}\left (a\,x\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.36, size = 151, normalized size = 1.39 \[ \begin {cases} c^{2} x \operatorname {acot}{\left (a x \right )} + \frac {2 c d x^{3} \operatorname {acot}{\left (a x \right )}}{3} + \frac {d^{2} x^{5} \operatorname {acot}{\left (a x \right )}}{5} + \frac {c^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a} + \frac {c d x^{2}}{3 a} + \frac {d^{2} x^{4}}{20 a} - \frac {c d \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 a^{3}} - \frac {d^{2} x^{2}}{10 a^{3}} + \frac {d^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{10 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{2} x + \frac {2 c d x^{3}}{3} + \frac {d^{2} x^{5}}{5}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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