Optimal. Leaf size=110 \[ \frac {d x^2 \left (10 a^2 c+3 d\right )}{30 a^3}+\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \log \left (1-a^2 x^2\right )}{30 a^5}+c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-1}(a x)+\frac {d^2 x^4}{20 a} \]
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Rubi [A] time = 0.13, antiderivative size = 110, 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, 5977, 1594, 1247, 698} \[ \frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \log \left (1-a^2 x^2\right )}{30 a^5}+\frac {d x^2 \left (10 a^2 c+3 d\right )}{30 a^3}+c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {d^2 x^4}{20 a}+\frac {1}{5} d^2 x^5 \coth ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 194
Rule 698
Rule 1247
Rule 1594
Rule 5977
Rubi steps
\begin {align*} \int \left (c+d x^2\right )^2 \coth ^{-1}(a x) \, dx &=c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-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 \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-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 \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-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 \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-1}(a x)-\frac {1}{2} a \operatorname {Subst}\left (\int \left (-\frac {d \left (10 a^2 c+3 d\right )}{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 {d \left (10 a^2 c+3 d\right ) x^2}{30 a^3}+\frac {d^2 x^4}{20 a}+c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-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 = 0.05, size = 98, normalized size = 0.89 \[ \frac {4 a^5 x \coth ^{-1}(a x) \left (15 c^2+10 c d x^2+3 d^2 x^4\right )+a^2 d x^2 \left (a^2 \left (20 c+3 d x^2\right )+6 d\right )+\left (30 a^4 c^2+20 a^2 c d+6 d^2\right ) \log \left (1-a^2 x^2\right )}{60 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 118, normalized size = 1.07 \[ \frac {3 \, a^{4} d^{2} x^{4} + 2 \, {\left (10 \, a^{4} c d + 3 \, a^{2} d^{2}\right )} x^{2} + 2 \, {\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a^{2} x^{2} - 1\right ) + 2 \, {\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{60 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{2} + c\right )}^{2} \operatorname {arcoth}\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 148, normalized size = 1.35 \[ \frac {d^{2} x^{5} \mathrm {arccoth}\left (a x \right )}{5}+\frac {2 c d \,x^{3} \mathrm {arccoth}\left (a x \right )}{3}+c^{2} x \,\mathrm {arccoth}\left (a x \right )+\frac {d^{2} x^{4}}{20 a}+\frac {c d \,x^{2}}{3 a}+\frac {x^{2} d^{2}}{10 a^{3}}+\frac {\ln \left (a x -1\right ) c^{2}}{2 a}+\frac {\ln \left (a x -1\right ) c d}{3 a^{3}}+\frac {\ln \left (a x -1\right ) d^{2}}{10 a^{5}}+\frac {c^{2} \ln \left (a x +1\right )}{2 a}+\frac {\ln \left (a x +1\right ) c d}{3 a^{3}}+\frac {\ln \left (a x +1\right ) d^{2}}{10 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 131, normalized size = 1.19 \[ \frac {1}{60} \, a {\left (\frac {3 \, a^{2} d^{2} x^{4} + 2 \, {\left (10 \, a^{2} c d + 3 \, d^{2}\right )} x^{2}}{a^{4}} + \frac {2 \, {\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a x + 1\right )}{a^{6}} + \frac {2 \, {\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a x - 1\right )}{a^{6}}\right )} + \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \operatorname {arcoth}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 115, normalized size = 1.05 \[ \frac {a^4\,\left (\frac {c^2\,\ln \left (a^2\,x^2-1\right )}{2}+\frac {d^2\,x^4}{20}+\frac {c\,d\,x^2}{3}\right )+a^2\,\left (\frac {d^2\,x^2}{10}+\frac {c\,d\,\ln \left (a^2\,x^2-1\right )}{3}\right )+\frac {d^2\,\ln \left (a^2\,x^2-1\right )}{10}}{a^5}+c^2\,x\,\mathrm {acoth}\left (a\,x\right )+\frac {d^2\,x^5\,\mathrm {acoth}\left (a\,x\right )}{5}+\frac {2\,c\,d\,x^3\,\mathrm {acoth}\left (a\,x\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.13, size = 182, normalized size = 1.65 \[ \begin {cases} c^{2} x \operatorname {acoth}{\left (a x \right )} + \frac {2 c d x^{3} \operatorname {acoth}{\left (a x \right )}}{3} + \frac {d^{2} x^{5} \operatorname {acoth}{\left (a x \right )}}{5} + \frac {c^{2} \log {\left (x - \frac {1}{a} \right )}}{a} + \frac {c^{2} \operatorname {acoth}{\left (a x \right )}}{a} + \frac {c d x^{2}}{3 a} + \frac {d^{2} x^{4}}{20 a} + \frac {2 c d \log {\left (x - \frac {1}{a} \right )}}{3 a^{3}} + \frac {2 c d \operatorname {acoth}{\left (a x \right )}}{3 a^{3}} + \frac {d^{2} x^{2}}{10 a^{3}} + \frac {d^{2} \log {\left (x - \frac {1}{a} \right )}}{5 a^{5}} + \frac {d^{2} \operatorname {acoth}{\left (a x \right )}}{5 a^{5}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c^{2} x + \frac {2 c d x^{3}}{3} + \frac {d^{2} x^{5}}{5}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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