Optimal. Leaf size=169 \[ \frac{d x^2 \left (35 a^4 c^2+21 a^2 c d+5 d^2\right )}{70 a^5}+\frac{\left (35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2+5 d^3\right ) \log \left (1-a^2 x^2\right )}{70 a^7}+\frac{d^2 x^4 \left (21 a^2 c+5 d\right )}{140 a^3}+c^2 d x^3 \coth ^{-1}(a x)+c^3 x \coth ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac{d^3 x^6}{42 a}+\frac{1}{7} d^3 x^7 \coth ^{-1}(a x) \]
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Rubi [A] time = 0.1268, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {194, 5977, 1810, 260} \[ \frac{d x^2 \left (35 a^4 c^2+21 a^2 c d+5 d^2\right )}{70 a^5}+\frac{\left (35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2+5 d^3\right ) \log \left (1-a^2 x^2\right )}{70 a^7}+\frac{d^2 x^4 \left (21 a^2 c+5 d\right )}{140 a^3}+c^2 d x^3 \coth ^{-1}(a x)+c^3 x \coth ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac{d^3 x^6}{42 a}+\frac{1}{7} d^3 x^7 \coth ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 194
Rule 5977
Rule 1810
Rule 260
Rubi steps
\begin{align*} \int \left (c+d x^2\right )^3 \coth ^{-1}(a x) \, dx &=c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac{1}{7} d^3 x^7 \coth ^{-1}(a x)-a \int \frac{c^3 x+c^2 d x^3+\frac{3}{5} c d^2 x^5+\frac{d^3 x^7}{7}}{1-a^2 x^2} \, dx\\ &=c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac{1}{7} d^3 x^7 \coth ^{-1}(a x)-a \int \left (-\frac{d \left (35 a^4 c^2+21 a^2 c d+5 d^2\right ) x}{35 a^6}-\frac{d^2 \left (21 a^2 c+5 d\right ) x^3}{35 a^4}-\frac{d^3 x^5}{7 a^2}+\frac{\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) x}{35 a^6 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac{d \left (35 a^4 c^2+21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac{d^2 \left (21 a^2 c+5 d\right ) x^4}{140 a^3}+\frac{d^3 x^6}{42 a}+c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac{1}{7} d^3 x^7 \coth ^{-1}(a x)-\frac{\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \int \frac{x}{1-a^2 x^2} \, dx}{35 a^5}\\ &=\frac{d \left (35 a^4 c^2+21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac{d^2 \left (21 a^2 c+5 d\right ) x^4}{140 a^3}+\frac{d^3 x^6}{42 a}+c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac{1}{7} d^3 x^7 \coth ^{-1}(a x)+\frac{\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \log \left (1-a^2 x^2\right )}{70 a^7}\\ \end{align*}
Mathematica [A] time = 0.0786468, size = 150, normalized size = 0.89 \[ \frac{a^2 d x^2 \left (a^4 \left (210 c^2+63 c d x^2+10 d^2 x^4\right )+3 a^2 d \left (42 c+5 d x^2\right )+30 d^2\right )+6 \left (35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2+5 d^3\right ) \log \left (1-a^2 x^2\right )+12 a^7 x \coth ^{-1}(a x) \left (35 c^2 d x^2+35 c^3+21 c d^2 x^4+5 d^3 x^6\right )}{420 a^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 233, normalized size = 1.4 \begin{align*}{\frac{{d}^{3}{x}^{7}{\rm arccoth} \left (ax\right )}{7}}+{\frac{3\,c{d}^{2}{x}^{5}{\rm arccoth} \left (ax\right )}{5}}+{c}^{2}d{x}^{3}{\rm arccoth} \left (ax\right )+{c}^{3}x{\rm arccoth} \left (ax\right )+{\frac{3\,c{d}^{2}{x}^{2}}{10\,{a}^{3}}}+{\frac{3\,c{d}^{2}{x}^{4}}{20\,a}}+{\frac{{c}^{2}{x}^{2}d}{2\,a}}+{\frac{{d}^{3}{x}^{6}}{42\,a}}+{\frac{{d}^{3}{x}^{2}}{14\,{a}^{5}}}+{\frac{\ln \left ( ax-1 \right ){c}^{3}}{2\,a}}+{\frac{\ln \left ( ax-1 \right ){c}^{2}d}{2\,{a}^{3}}}+{\frac{3\,\ln \left ( ax-1 \right ) c{d}^{2}}{10\,{a}^{5}}}+{\frac{\ln \left ( ax-1 \right ){d}^{3}}{14\,{a}^{7}}}+{\frac{\ln \left ( ax+1 \right ){c}^{3}}{2\,a}}+{\frac{\ln \left ( ax+1 \right ){c}^{2}d}{2\,{a}^{3}}}+{\frac{3\,\ln \left ( ax+1 \right ) c{d}^{2}}{10\,{a}^{5}}}+{\frac{\ln \left ( ax+1 \right ){d}^{3}}{14\,{a}^{7}}}+{\frac{{x}^{4}{d}^{3}}{28\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974128, size = 267, normalized size = 1.58 \begin{align*} \frac{1}{420} \, a{\left (\frac{10 \, a^{4} d^{3} x^{6} + 3 \,{\left (21 \, a^{4} c d^{2} + 5 \, a^{2} d^{3}\right )} x^{4} + 6 \,{\left (35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} x^{2}}{a^{6}} + \frac{6 \,{\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \log \left (a x + 1\right )}{a^{8}} + \frac{6 \,{\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \log \left (a x - 1\right )}{a^{8}}\right )} + \frac{1}{35} \,{\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \operatorname{arcoth}\left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62707, size = 382, normalized size = 2.26 \begin{align*} \frac{10 \, a^{6} d^{3} x^{6} + 3 \,{\left (21 \, a^{6} c d^{2} + 5 \, a^{4} d^{3}\right )} x^{4} + 6 \,{\left (35 \, a^{6} c^{2} d + 21 \, a^{4} c d^{2} + 5 \, a^{2} d^{3}\right )} x^{2} + 6 \,{\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \log \left (a^{2} x^{2} - 1\right ) + 6 \,{\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \log \left (\frac{a x + 1}{a x - 1}\right )}{420 \, a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.31205, size = 282, normalized size = 1.67 \begin{align*} \begin{cases} c^{3} x \operatorname{acoth}{\left (a x \right )} + c^{2} d x^{3} \operatorname{acoth}{\left (a x \right )} + \frac{3 c d^{2} x^{5} \operatorname{acoth}{\left (a x \right )}}{5} + \frac{d^{3} x^{7} \operatorname{acoth}{\left (a x \right )}}{7} + \frac{c^{3} \log{\left (x - \frac{1}{a} \right )}}{a} + \frac{c^{3} \operatorname{acoth}{\left (a x \right )}}{a} + \frac{c^{2} d x^{2}}{2 a} + \frac{3 c d^{2} x^{4}}{20 a} + \frac{d^{3} x^{6}}{42 a} + \frac{c^{2} d \log{\left (x - \frac{1}{a} \right )}}{a^{3}} + \frac{c^{2} d \operatorname{acoth}{\left (a x \right )}}{a^{3}} + \frac{3 c d^{2} x^{2}}{10 a^{3}} + \frac{d^{3} x^{4}}{28 a^{3}} + \frac{3 c d^{2} \log{\left (x - \frac{1}{a} \right )}}{5 a^{5}} + \frac{3 c d^{2} \operatorname{acoth}{\left (a x \right )}}{5 a^{5}} + \frac{d^{3} x^{2}}{14 a^{5}} + \frac{d^{3} \log{\left (x - \frac{1}{a} \right )}}{7 a^{7}} + \frac{d^{3} \operatorname{acoth}{\left (a x \right )}}{7 a^{7}} & \text{for}\: a \neq 0 \\\frac{i \pi \left (c^{3} x + c^{2} d x^{3} + \frac{3 c d^{2} x^{5}}{5} + \frac{d^{3} x^{7}}{7}\right )}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{2} + c\right )}^{3} \operatorname{arcoth}\left (a x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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