Optimal. Leaf size=169 \[ \frac {d \left (35 a^4 c^2+21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac {d^2 \left (21 a^2 c+5 d\right ) x^4}{140 a^3}+\frac {d^3 x^6}{42 a}+c^3 x \coth ^{-1}(a x)+c^2 d x^3 \coth ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \coth ^{-1}(a x)+\frac {1}{7} d^3 x^7 \coth ^{-1}(a x)+\frac {\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \log \left (1-a^2 x^2\right )}{70 a^7} \]
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Rubi [A]
time = 0.09, antiderivative size = 169, normalized size of antiderivative = 1.00, 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 = {200, 6124,
1824, 266} \begin {gather*} \frac {d^2 x^4 \left (21 a^2 c+5 d\right )}{140 a^3}+\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^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}+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 {d^3 x^6}{42 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 200
Rule 266
Rule 1824
Rule 6124
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*}
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Mathematica [A]
time = 0.05, size = 150, normalized size = 0.89 \begin {gather*} \frac {a^2 d x^2 \left (30 d^2+3 a^2 d \left (42 c+5 d x^2\right )+a^4 \left (210 c^2+63 c d x^2+10 d^2 x^4\right )\right )+12 a^7 x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right ) \coth ^{-1}(a x)+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 (1-a^2 x^2\right )}{420 a^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 211, normalized size = 1.25
method | result | size |
derivativedivides | \(\frac {\mathrm {arccoth}\left (a x \right ) c^{3} a x +a \,\mathrm {arccoth}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\mathrm {arccoth}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\mathrm {arccoth}\left (a x \right ) d^{3} x^{7}}{7}+\frac {-\frac {\left (-35 a^{6} c^{3}-35 a^{4} c^{2} d -21 a^{2} c \,d^{2}-5 d^{3}\right ) \ln \left (a x +1\right )}{2}+\frac {\left (35 a^{6} c^{3}+35 a^{4} c^{2} d +21 a^{2} c \,d^{2}+5 d^{3}\right ) \ln \left (a x -1\right )}{2}+\frac {5 d^{3} a^{4} x^{4}}{4}+\frac {5 d^{3} a^{2} x^{2}}{2}+\frac {21 c \,a^{4} d^{2} x^{2}}{2}+\frac {21 c \,a^{6} d^{2} x^{4}}{4}+\frac {35 c^{2} a^{6} d \,x^{2}}{2}+\frac {5 d^{3} a^{6} x^{6}}{6}}{35 a^{6}}}{a}\) | \(211\) |
default | \(\frac {\mathrm {arccoth}\left (a x \right ) c^{3} a x +a \,\mathrm {arccoth}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\mathrm {arccoth}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\mathrm {arccoth}\left (a x \right ) d^{3} x^{7}}{7}+\frac {-\frac {\left (-35 a^{6} c^{3}-35 a^{4} c^{2} d -21 a^{2} c \,d^{2}-5 d^{3}\right ) \ln \left (a x +1\right )}{2}+\frac {\left (35 a^{6} c^{3}+35 a^{4} c^{2} d +21 a^{2} c \,d^{2}+5 d^{3}\right ) \ln \left (a x -1\right )}{2}+\frac {5 d^{3} a^{4} x^{4}}{4}+\frac {5 d^{3} a^{2} x^{2}}{2}+\frac {21 c \,a^{4} d^{2} x^{2}}{2}+\frac {21 c \,a^{6} d^{2} x^{4}}{4}+\frac {35 c^{2} a^{6} d \,x^{2}}{2}+\frac {5 d^{3} a^{6} x^{6}}{6}}{35 a^{6}}}{a}\) | \(211\) |
risch | \(\left (\frac {1}{14} d^{3} x^{7}+\frac {3}{10} c \,d^{2} x^{5}+\frac {1}{2} c^{2} d \,x^{3}+\frac {1}{2} c^{3} x \right ) \ln \left (a x +1\right )-\frac {d^{3} x^{7} \ln \left (a x -1\right )}{14}-\frac {3 c \,d^{2} x^{5} \ln \left (a x -1\right )}{10}+\frac {d^{3} x^{6}}{42 a}-\frac {c^{2} d \,x^{3} \ln \left (a x -1\right )}{2}+\frac {3 c \,d^{2} x^{4}}{20 a}-\frac {c^{3} x \ln \left (a x -1\right )}{2}+\frac {c^{2} d \,x^{2}}{2 a}+\frac {d^{3} x^{4}}{28 a^{3}}+\frac {\ln \left (a^{2} x^{2}-1\right ) c^{3}}{2 a}+\frac {3 c \,d^{2} x^{2}}{10 a^{3}}+\frac {\ln \left (a^{2} x^{2}-1\right ) c^{2} d}{2 a^{3}}+\frac {d^{3} x^{2}}{14 a^{5}}+\frac {3 \ln \left (a^{2} x^{2}-1\right ) c \,d^{2}}{10 a^{5}}+\frac {\ln \left (a^{2} x^{2}-1\right ) d^{3}}{14 a^{7}}\) | \(241\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 198, normalized size = 1.17 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 177, normalized size = 1.05 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.59, size = 282, normalized size = 1.67 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 932 vs.
\(2 (157) = 314\).
time = 0.41, size = 932, normalized size = 5.51 \begin {gather*} \frac {1}{105} \, a {\left (\frac {3 \, {\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \log \left (\frac {{\left | a x + 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{8}} - \frac {3 \, {\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \log \left ({\left | \frac {a x + 1}{a x - 1} - 1 \right |}\right )}{a^{8}} + \frac {2 \, {\left (\frac {3 \, {\left (35 \, a^{4} c^{2} d + 42 \, a^{2} c d^{2} + 15 \, d^{3}\right )} {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} - \frac {6 \, {\left (70 \, a^{4} c^{2} d + 63 \, a^{2} c d^{2} + 15 \, d^{3}\right )} {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {2 \, {\left (315 \, a^{4} c^{2} d + 252 \, a^{2} c d^{2} + 85 \, d^{3}\right )} {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {6 \, {\left (70 \, a^{4} c^{2} d + 63 \, a^{2} c d^{2} + 15 \, d^{3}\right )} {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {3 \, {\left (35 \, a^{4} c^{2} d + 42 \, a^{2} c d^{2} + 15 \, d^{3}\right )} {\left (a x + 1\right )}}{a x - 1}\right )}}{a^{8} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{6}} + \frac {3 \, {\left (\frac {35 \, {\left (a x + 1\right )}^{6} a^{6} c^{3}}{{\left (a x - 1\right )}^{6}} - \frac {210 \, {\left (a x + 1\right )}^{5} a^{6} c^{3}}{{\left (a x - 1\right )}^{5}} + \frac {525 \, {\left (a x + 1\right )}^{4} a^{6} c^{3}}{{\left (a x - 1\right )}^{4}} - \frac {700 \, {\left (a x + 1\right )}^{3} a^{6} c^{3}}{{\left (a x - 1\right )}^{3}} + \frac {525 \, {\left (a x + 1\right )}^{2} a^{6} c^{3}}{{\left (a x - 1\right )}^{2}} - \frac {210 \, {\left (a x + 1\right )} a^{6} c^{3}}{a x - 1} + 35 \, a^{6} c^{3} + \frac {105 \, {\left (a x + 1\right )}^{6} a^{4} c^{2} d}{{\left (a x - 1\right )}^{6}} - \frac {420 \, {\left (a x + 1\right )}^{5} a^{4} c^{2} d}{{\left (a x - 1\right )}^{5}} + \frac {665 \, {\left (a x + 1\right )}^{4} a^{4} c^{2} d}{{\left (a x - 1\right )}^{4}} - \frac {560 \, {\left (a x + 1\right )}^{3} a^{4} c^{2} d}{{\left (a x - 1\right )}^{3}} + \frac {315 \, {\left (a x + 1\right )}^{2} a^{4} c^{2} d}{{\left (a x - 1\right )}^{2}} - \frac {140 \, {\left (a x + 1\right )} a^{4} c^{2} d}{a x - 1} + 35 \, a^{4} c^{2} d + \frac {105 \, {\left (a x + 1\right )}^{6} a^{2} c d^{2}}{{\left (a x - 1\right )}^{6}} - \frac {210 \, {\left (a x + 1\right )}^{5} a^{2} c d^{2}}{{\left (a x - 1\right )}^{5}} + \frac {315 \, {\left (a x + 1\right )}^{4} a^{2} c d^{2}}{{\left (a x - 1\right )}^{4}} - \frac {420 \, {\left (a x + 1\right )}^{3} a^{2} c d^{2}}{{\left (a x - 1\right )}^{3}} + \frac {231 \, {\left (a x + 1\right )}^{2} a^{2} c d^{2}}{{\left (a x - 1\right )}^{2}} - \frac {42 \, {\left (a x + 1\right )} a^{2} c d^{2}}{a x - 1} + 21 \, a^{2} c d^{2} + \frac {35 \, {\left (a x + 1\right )}^{6} d^{3}}{{\left (a x - 1\right )}^{6}} + \frac {175 \, {\left (a x + 1\right )}^{4} d^{3}}{{\left (a x - 1\right )}^{4}} + \frac {105 \, {\left (a x + 1\right )}^{2} d^{3}}{{\left (a x - 1\right )}^{2}} + 5 \, d^{3}\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{a^{8} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{7}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.52, size = 190, normalized size = 1.12 \begin {gather*} c^3\,x\,\mathrm {acoth}\left (a\,x\right )+\frac {d^3\,x^7\,\mathrm {acoth}\left (a\,x\right )}{7}+\frac {c^3\,\ln \left (a^2\,x^2-1\right )}{2\,a}+\frac {d^3\,\ln \left (a^2\,x^2-1\right )}{14\,a^7}+\frac {d^3\,x^6}{42\,a}+\frac {d^3\,x^4}{28\,a^3}+\frac {d^3\,x^2}{14\,a^5}+\frac {c^2\,d\,\ln \left (a^2\,x^2-1\right )}{2\,a^3}+\frac {3\,c\,d^2\,\ln \left (a^2\,x^2-1\right )}{10\,a^5}+\frac {c^2\,d\,x^2}{2\,a}+\frac {3\,c\,d^2\,x^4}{20\,a}+\frac {3\,c\,d^2\,x^2}{10\,a^3}+c^2\,d\,x^3\,\mathrm {acoth}\left (a\,x\right )+\frac {3\,c\,d^2\,x^5\,\mathrm {acoth}\left (a\,x\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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