Optimal. Leaf size=57 \[ \frac {\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)+\frac {d x^2}{6 a} \]
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Rubi [A] time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5977, 1593, 444, 43} \[ \frac {\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+c x \coth ^{-1}(a x)+\frac {d x^2}{6 a}+\frac {1}{3} d x^3 \coth ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 1593
Rule 5977
Rubi steps
\begin {align*} \int \left (c+d x^2\right ) \coth ^{-1}(a x) \, dx &=c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)-a \int \frac {c x+\frac {d x^3}{3}}{1-a^2 x^2} \, dx\\ &=c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)-a \int \frac {x \left (c+\frac {d x^2}{3}\right )}{1-a^2 x^2} \, dx\\ &=c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)-\frac {1}{2} a \operatorname {Subst}\left (\int \frac {c+\frac {d x}{3}}{1-a^2 x} \, dx,x,x^2\right )\\ &=c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)-\frac {1}{2} a \operatorname {Subst}\left (\int \left (-\frac {d}{3 a^2}+\frac {-3 a^2 c-d}{3 a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {d x^2}{6 a}+c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)+\frac {\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{6 a^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 69, normalized size = 1.21 \[ \frac {c \log \left (1-a^2 x^2\right )}{2 a}+\frac {d \log \left (1-a^2 x^2\right )}{6 a^3}+c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)+\frac {d x^2}{6 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 64, normalized size = 1.12 \[ \frac {a^{2} d x^{2} + {\left (3 \, a^{2} c + d\right )} \log \left (a^{2} x^{2} - 1\right ) + {\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{6 \, a^{3}} \]
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 \[ \int {\left (d x^{2} + c\right )} \operatorname {arcoth}\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 1.33 \[ \frac {d \,x^{3} \mathrm {arccoth}\left (a x \right )}{3}+c x \,\mathrm {arccoth}\left (a x \right )+\frac {d \,x^{2}}{6 a}+\frac {\ln \left (a x -1\right ) c}{2 a}+\frac {\ln \left (a x -1\right ) d}{6 a^{3}}+\frac {c \ln \left (a x +1\right )}{2 a}+\frac {\ln \left (a x +1\right ) d}{6 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 65, normalized size = 1.14 \[ \frac {1}{6} \, a {\left (\frac {d x^{2}}{a^{2}} + \frac {{\left (3 \, a^{2} c + d\right )} \log \left (a x + 1\right )}{a^{4}} + \frac {{\left (3 \, a^{2} c + d\right )} \log \left (a x - 1\right )}{a^{4}}\right )} + \frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \operatorname {arcoth}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 60, normalized size = 1.05 \[ \frac {\frac {d\,\ln \left (a^2\,x^2-1\right )}{6}+a^2\,\left (\frac {c\,\ln \left (a^2\,x^2-1\right )}{2}+\frac {d\,x^2}{6}\right )}{a^3}+\frac {d\,x^3\,\mathrm {acoth}\left (a\,x\right )}{3}+c\,x\,\mathrm {acoth}\left (a\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 87, normalized size = 1.53 \[ \begin {cases} c x \operatorname {acoth}{\left (a x \right )} + \frac {d x^{3} \operatorname {acoth}{\left (a x \right )}}{3} + \frac {c \log {\left (x - \frac {1}{a} \right )}}{a} + \frac {c \operatorname {acoth}{\left (a x \right )}}{a} + \frac {d x^{2}}{6 a} + \frac {d \log {\left (x - \frac {1}{a} \right )}}{3 a^{3}} + \frac {d \operatorname {acoth}{\left (a x \right )}}{3 a^{3}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c x + \frac {d x^{3}}{3}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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