Optimal. Leaf size=21 \[ \text {Int}\left ((a+b x)^2 \sqrt {\cot ^{-1}(a+b x)},x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx &=\int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx\\ \end {align*}
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Mathematica [A]
time = 6.42, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.60, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{2} \sqrt {\mathrm {arccot}\left (b x +a \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right )^{2} \sqrt {\operatorname {acot}{\left (a + b x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \sqrt {\mathrm {acot}\left (a+b\,x\right )}\,{\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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