Optimal. Leaf size=157 \[ -\frac {b^2}{16 c d^3 (1+c x)^2}-\frac {3 b^2}{16 c d^3 (1+c x)}+\frac {3 b^2 \tanh ^{-1}(c x)}{16 c d^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d^3 (1+c x)^2} \]
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Rubi [A]
time = 0.13, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6065, 6063,
641, 46, 213, 6095} \begin {gather*} -\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (c x+1)}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (c x+1)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d^3 (c x+1)^2}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c d^3}-\frac {3 b^2}{16 c d^3 (c x+1)}-\frac {b^2}{16 c d^3 (c x+1)^2}+\frac {3 b^2 \tanh ^{-1}(c x)}{16 c d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 213
Rule 641
Rule 6063
Rule 6065
Rule 6095
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(d+c d x)^3} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d^3 (1+c x)^2}+\frac {b \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 d^2 (1+c x)^3}+\frac {a+b \tanh ^{-1}(c x)}{4 d^2 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{4 d^2 \left (-1+c^2 x^2\right )}\right ) \, dx}{d}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d^3 (1+c x)^2}+\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{4 d^3}-\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{4 d^3}+\frac {b \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx}{2 d^3}\\ &=-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d^3 (1+c x)^2}+\frac {b^2 \int \frac {1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx}{4 d^3}+\frac {b^2 \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{4 d^3}\\ &=-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d^3 (1+c x)^2}+\frac {b^2 \int \frac {1}{(1-c x) (1+c x)^3} \, dx}{4 d^3}+\frac {b^2 \int \frac {1}{(1-c x) (1+c x)^2} \, dx}{4 d^3}\\ &=-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d^3 (1+c x)^2}+\frac {b^2 \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{4 d^3}+\frac {b^2 \int \left (\frac {1}{2 (1+c x)^3}+\frac {1}{4 (1+c x)^2}-\frac {1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx}{4 d^3}\\ &=-\frac {b^2}{16 c d^3 (1+c x)^2}-\frac {3 b^2}{16 c d^3 (1+c x)}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d^3 (1+c x)^2}-\frac {b^2 \int \frac {1}{-1+c^2 x^2} \, dx}{16 d^3}-\frac {b^2 \int \frac {1}{-1+c^2 x^2} \, dx}{8 d^3}\\ &=-\frac {b^2}{16 c d^3 (1+c x)^2}-\frac {3 b^2}{16 c d^3 (1+c x)}+\frac {3 b^2 \tanh ^{-1}(c x)}{16 c d^3}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )}{4 c d^3 (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{8 c d^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 c d^3 (1+c x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 183, normalized size = 1.17 \begin {gather*} \frac {-8 a^2-4 a b-b^2}{16 c d^3 (1+c x)^2}-\frac {b (4 a+3 b)}{16 c d^3 (1+c x)}-\frac {b (4 a+2 b+b c x) \tanh ^{-1}(c x)}{4 c d^3 (1+c x)^2}+\frac {b^2 \left (-3+2 c x+c^2 x^2\right ) \tanh ^{-1}(c x)^2}{8 c d^3 (1+c x)^2}+\frac {\left (-4 a b-3 b^2\right ) \log (1-c x)}{32 c d^3}+\frac {\left (4 a b+3 b^2\right ) \log (1+c x)}{32 c d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(341\) vs.
\(2(143)=286\).
time = 0.35, size = 342, normalized size = 2.18
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}}{2 d^{3} \left (c x +1\right )^{2}}-\frac {b^{2} \arctanh \left (c x \right )^{2}}{2 d^{3} \left (c x +1\right )^{2}}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{8 d^{3}}-\frac {b^{2} \arctanh \left (c x \right )}{4 d^{3} \left (c x +1\right )^{2}}-\frac {b^{2} \arctanh \left (c x \right )}{4 d^{3} \left (c x +1\right )}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{8 d^{3}}+\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16 d^{3}}-\frac {b^{2} \ln \left (c x -1\right )^{2}}{32 d^{3}}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16 d^{3}}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{16 d^{3}}-\frac {b^{2} \ln \left (c x +1\right )^{2}}{32 d^{3}}-\frac {3 b^{2} \ln \left (c x -1\right )}{32 d^{3}}-\frac {b^{2}}{16 d^{3} \left (c x +1\right )^{2}}-\frac {3 b^{2}}{16 d^{3} \left (c x +1\right )}+\frac {3 b^{2} \ln \left (c x +1\right )}{32 d^{3}}-\frac {a b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )^{2}}-\frac {a b \ln \left (c x -1\right )}{8 d^{3}}-\frac {a b}{4 d^{3} \left (c x +1\right )^{2}}-\frac {a b}{4 d^{3} \left (c x +1\right )}+\frac {a b \ln \left (c x +1\right )}{8 d^{3}}}{c}\) | \(342\) |
default | \(\frac {-\frac {a^{2}}{2 d^{3} \left (c x +1\right )^{2}}-\frac {b^{2} \arctanh \left (c x \right )^{2}}{2 d^{3} \left (c x +1\right )^{2}}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{8 d^{3}}-\frac {b^{2} \arctanh \left (c x \right )}{4 d^{3} \left (c x +1\right )^{2}}-\frac {b^{2} \arctanh \left (c x \right )}{4 d^{3} \left (c x +1\right )}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{8 d^{3}}+\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16 d^{3}}-\frac {b^{2} \ln \left (c x -1\right )^{2}}{32 d^{3}}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{16 d^{3}}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{16 d^{3}}-\frac {b^{2} \ln \left (c x +1\right )^{2}}{32 d^{3}}-\frac {3 b^{2} \ln \left (c x -1\right )}{32 d^{3}}-\frac {b^{2}}{16 d^{3} \left (c x +1\right )^{2}}-\frac {3 b^{2}}{16 d^{3} \left (c x +1\right )}+\frac {3 b^{2} \ln \left (c x +1\right )}{32 d^{3}}-\frac {a b \arctanh \left (c x \right )}{d^{3} \left (c x +1\right )^{2}}-\frac {a b \ln \left (c x -1\right )}{8 d^{3}}-\frac {a b}{4 d^{3} \left (c x +1\right )^{2}}-\frac {a b}{4 d^{3} \left (c x +1\right )}+\frac {a b \ln \left (c x +1\right )}{8 d^{3}}}{c}\) | \(342\) |
risch | \(\frac {b^{2} \left (c^{2} x^{2}+2 c x -3\right ) \ln \left (c x +1\right )^{2}}{32 d^{3} \left (c x +1\right )^{2} c}-\frac {b \left (b \,x^{2} \ln \left (-c x +1\right ) c^{2}+2 b c x \ln \left (-c x +1\right )+2 b c x -3 b \ln \left (-c x +1\right )+8 a +4 b \right ) \ln \left (c x +1\right )}{16 d^{3} \left (c x +1\right )^{2} c}-\frac {-b^{2} c^{2} x^{2} \ln \left (-c x +1\right )^{2}+4 \ln \left (c x -1\right ) a b \,c^{2} x^{2}+3 \ln \left (c x -1\right ) b^{2} c^{2} x^{2}-4 b \,c^{2} \ln \left (-c x -1\right ) x^{2} a -3 b^{2} c^{2} \ln \left (-c x -1\right ) x^{2}-2 b^{2} c x \ln \left (-c x +1\right )^{2}+8 \ln \left (c x -1\right ) a b c x +6 \ln \left (c x -1\right ) b^{2} c x -8 \ln \left (-c x -1\right ) a b c x -6 \ln \left (-c x -1\right ) b^{2} c x -4 b^{2} c x \ln \left (-c x +1\right )+8 a b c x +6 b^{2} c x +3 b^{2} \ln \left (-c x +1\right )^{2}+4 a b \ln \left (c x -1\right )+3 b^{2} \ln \left (c x -1\right )-4 \ln \left (-c x -1\right ) a b -3 b^{2} \ln \left (-c x -1\right )-16 b \ln \left (-c x +1\right ) a -8 b^{2} \ln \left (-c x +1\right )+16 a^{2}+16 a b +8 b^{2}}{32 d^{3} \left (c x +1\right )^{2} c}\) | \(406\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 399 vs.
\(2 (143) = 286\).
time = 0.29, size = 399, normalized size = 2.54 \begin {gather*} -\frac {1}{8} \, {\left (c {\left (\frac {2 \, {\left (c x + 2\right )}}{c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}} - \frac {\log \left (c x + 1\right )}{c^{2} d^{3}} + \frac {\log \left (c x - 1\right )}{c^{2} d^{3}}\right )} + \frac {8 \, \operatorname {artanh}\left (c x\right )}{c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}}\right )} a b - \frac {1}{32} \, {\left (4 \, c {\left (\frac {2 \, {\left (c x + 2\right )}}{c^{4} d^{3} x^{2} + 2 \, c^{3} d^{3} x + c^{2} d^{3}} - \frac {\log \left (c x + 1\right )}{c^{2} d^{3}} + \frac {\log \left (c x - 1\right )}{c^{2} d^{3}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {{\left ({\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 6 \, c x - {\left (3 \, c^{2} x^{2} + 6 \, c x + 2 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 3\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{2} x^{2} + 2 \, c x + 1\right )} \log \left (c x - 1\right ) + 8\right )} c^{2}}{c^{5} d^{3} x^{2} + 2 \, c^{4} d^{3} x + c^{3} d^{3}}\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x\right )^{2}}{2 \, {\left (c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}\right )}} - \frac {a^{2}}{2 \, {\left (c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 156, normalized size = 0.99 \begin {gather*} -\frac {2 \, {\left (4 \, a b + 3 \, b^{2}\right )} c x - {\left (b^{2} c^{2} x^{2} + 2 \, b^{2} c x - 3 \, b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} + 16 \, a^{2} + 16 \, a b + 8 \, b^{2} - {\left ({\left (4 \, a b + 3 \, b^{2}\right )} c^{2} x^{2} + 2 \, {\left (4 \, a b + b^{2}\right )} c x - 12 \, a b - 5 \, b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{32 \, {\left (c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 232, normalized size = 1.48 \begin {gather*} \frac {1}{64} \, c {\left (\frac {2 \, {\left (\frac {2 \, {\left (c x + 1\right )} b^{2}}{c x - 1} - b^{2}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )}^{2} c^{2} d^{3}} + \frac {2 \, {\left (\frac {8 \, {\left (c x + 1\right )} a b}{c x - 1} - 4 \, a b + \frac {4 \, {\left (c x + 1\right )} b^{2}}{c x - 1} - b^{2}\right )} {\left (c x - 1\right )}^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{2} c^{2} d^{3}} + \frac {{\left (\frac {16 \, {\left (c x + 1\right )} a^{2}}{c x - 1} - 8 \, a^{2} + \frac {16 \, {\left (c x + 1\right )} a b}{c x - 1} - 4 \, a b + \frac {8 \, {\left (c x + 1\right )} b^{2}}{c x - 1} - b^{2}\right )} {\left (c x - 1\right )}^{2}}{{\left (c x + 1\right )}^{2} c^{2} d^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.16, size = 373, normalized size = 2.38 \begin {gather*} \frac {11\,b^2\,\ln \left (1-c\,x\right )-11\,b^2\,\ln \left (c\,x+1\right )-16\,a\,b-3\,b^2\,{\ln \left (c\,x+1\right )}^2-3\,b^2\,{\ln \left (1-c\,x\right )}^2+12\,b^2\,\mathrm {atanh}\left (c\,x\right )-16\,a^2-8\,b^2-16\,a\,b\,\ln \left (c\,x+1\right )+16\,a\,b\,\ln \left (1-c\,x\right )+6\,b^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )+8\,a\,b\,\mathrm {atanh}\left (c\,x\right )-6\,b^2\,c\,x-10\,b^2\,c\,x\,\ln \left (c\,x+1\right )+10\,b^2\,c\,x\,\ln \left (1-c\,x\right )+b^2\,c^2\,x^2\,{\ln \left (c\,x+1\right )}^2+b^2\,c^2\,x^2\,{\ln \left (1-c\,x\right )}^2+12\,b^2\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+2\,b^2\,c\,x\,{\ln \left (c\,x+1\right )}^2+2\,b^2\,c\,x\,{\ln \left (1-c\,x\right )}^2+24\,b^2\,c\,x\,\mathrm {atanh}\left (c\,x\right )-3\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )+3\,b^2\,c^2\,x^2\,\ln \left (1-c\,x\right )-8\,a\,b\,c\,x-4\,b^2\,c\,x\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )+8\,a\,b\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )+16\,a\,b\,c\,x\,\mathrm {atanh}\left (c\,x\right )-2\,b^2\,c^2\,x^2\,\ln \left (c\,x+1\right )\,\ln \left (1-c\,x\right )}{32\,c\,d^3\,{\left (c\,x+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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