Optimal. Leaf size=139 \[ -\frac {3 b^3}{4 c (1+c x)}+\frac {3 b^3 \tanh ^{-1}(c x)}{4 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c (1+c x)}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (1+c x)} \]
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Rubi [A]
time = 0.15, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6065, 6063,
641, 46, 213, 6095} \begin {gather*} -\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c (c x+1)}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c (c x+1)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (c x+1)}-\frac {3 b^3}{4 c (c x+1)}+\frac {3 b^3 \tanh ^{-1}(c x)}{4 c} \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 )^3}{(1+c x)^2} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (1+c x)}+(3 b) \int \left (\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 (1+c x)^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{2 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (1+c x)}+\frac {1}{2} (3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^2} \, dx-\frac {1}{2} (3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{-1+c^2 x^2} \, dx\\ &=-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (1+c x)}+\left (3 b^2\right ) \int \left (\frac {a+b \tanh ^{-1}(c x)}{2 (1+c x)^2}-\frac {a+b \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (1+c x)}+\frac {1}{2} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac {1}{2} \left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx\\ &=-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c (1+c x)}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (1+c x)}+\frac {1}{2} \left (3 b^3\right ) \int \frac {1}{(1+c x) \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c (1+c x)}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (1+c x)}+\frac {1}{2} \left (3 b^3\right ) \int \frac {1}{(1-c x) (1+c x)^2} \, dx\\ &=-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c (1+c x)}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (1+c x)}+\frac {1}{2} \left (3 b^3\right ) \int \left (\frac {1}{2 (1+c x)^2}-\frac {1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {3 b^3}{4 c (1+c x)}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c (1+c x)}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (1+c x)}-\frac {1}{4} \left (3 b^3\right ) \int \frac {1}{-1+c^2 x^2} \, dx\\ &=-\frac {3 b^3}{4 c (1+c x)}+\frac {3 b^3 \tanh ^{-1}(c x)}{4 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c (1+c x)}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c (1+c x)}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{c (1+c x)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 198, normalized size = 1.42 \begin {gather*} \frac {-8 a^3-12 a^2 b-12 a b^2-6 b^3-12 b \left (2 a^2+2 a b+b^2\right ) \tanh ^{-1}(c x)+6 b^2 (2 a+b) (-1+c x) \tanh ^{-1}(c x)^2+4 b^3 (-1+c x) \tanh ^{-1}(c x)^3-3 b \left (2 a^2+2 a b+b^2\right ) (1+c x) \log (1-c x)+6 a^2 b \log (1+c x)+6 a b^2 \log (1+c x)+3 b^3 \log (1+c x)+6 a^2 b c x \log (1+c x)+6 a b^2 c x \log (1+c x)+3 b^3 c x \log (1+c x)}{8 c (1+c x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 6.84, size = 1811, normalized size = 13.03
method | result | size |
risch | \(\frac {b^{3} \left (c x -1\right ) \ln \left (c x +1\right )^{3}}{16 \left (c x +1\right ) c}+\frac {3 b^{2} \left (-b c x \ln \left (-c x +1\right )+2 c x a +b c x +b \ln \left (-c x +1\right )-2 a -b \right ) \ln \left (c x +1\right )^{2}}{16 \left (c x +1\right ) c}-\frac {3 b \left (-b^{2} c x \ln \left (-c x +1\right )^{2}+4 a b c x \ln \left (-c x +1\right )+2 b^{2} c x \ln \left (-c x +1\right )+b^{2} \ln \left (-c x +1\right )^{2}-4 b \ln \left (-c x +1\right ) a -2 b^{2} \ln \left (-c x +1\right )+8 a^{2}+8 a b +4 b^{2}\right ) \ln \left (c x +1\right )}{16 \left (c x +1\right ) c}-\frac {12 b^{3}-6 \ln \left (-c x +1\right )^{2} a \,b^{2} c x +\ln \left (-c x +1\right )^{3} b^{3} c x +6 \ln \left (-c x +1\right )^{2} a \,b^{2}-24 \ln \left (-c x +1\right ) a^{2} b +12 a^{2} b \ln \left (c x -1\right )+12 a \,b^{2} \ln \left (c x -1\right )+24 a \,b^{2}+16 a^{3}+12 \ln \left (c x -1\right ) a^{2} b c x +12 \ln \left (c x -1\right ) a \,b^{2} c x -12 \ln \left (-c x -1\right ) a^{2} b c x -12 \ln \left (-c x -1\right ) a \,b^{2} c x -12 \ln \left (-c x -1\right ) a^{2} b -12 \ln \left (-c x -1\right ) a \,b^{2}-24 a \,b^{2} \ln \left (-c x +1\right )+3 b^{3} \ln \left (-c x +1\right )^{2}-12 b^{3} \ln \left (-c x +1\right )+6 b^{3} \ln \left (c x -1\right )-6 b^{3} \ln \left (-c x -1\right )-3 b^{3} c x \ln \left (-c x +1\right )^{2}+6 \ln \left (c x -1\right ) b^{3} c x -6 \ln \left (-c x -1\right ) b^{3} c x +24 a^{2} b -\ln \left (-c x +1\right )^{3} b^{3}}{16 \left (c x +1\right ) c}\) | \(525\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1811\) |
default | \(\text {Expression too large to display}\) | \(1811\) |
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. 529 vs.
\(2 (127) = 254\).
time = 0.28, size = 529, normalized size = 3.81 \begin {gather*} -\frac {b^{3} \operatorname {artanh}\left (c x\right )^{3}}{c^{2} x + c} - \frac {3}{4} \, {\left (c {\left (\frac {2}{c^{3} x + c^{2}} - \frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} + \frac {4 \, \operatorname {artanh}\left (c x\right )}{c^{2} x + c}\right )} a^{2} b - \frac {3}{8} \, {\left (4 \, c {\left (\frac {2}{c^{3} x + c^{2}} - \frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {{\left ({\left (c x + 1\right )} \log \left (c x + 1\right )^{2} + {\left (c x + 1\right )} \log \left (c x - 1\right )^{2} - 2 \, {\left (c x + {\left (c x + 1\right )} \log \left (c x - 1\right ) + 1\right )} \log \left (c x + 1\right ) + 2 \, {\left (c x + 1\right )} \log \left (c x - 1\right ) + 4\right )} c^{2}}{c^{4} x + c^{3}}\right )} a b^{2} - \frac {1}{16} \, {\left (12 \, c {\left (\frac {2}{c^{3} x + c^{2}} - \frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )} \operatorname {artanh}\left (c x\right )^{2} - {\left (\frac {{\left ({\left (c x + 1\right )} \log \left (c x + 1\right )^{3} - {\left (c x + 1\right )} \log \left (c x - 1\right )^{3} - 3 \, {\left (c x + {\left (c x + 1\right )} \log \left (c x - 1\right ) + 1\right )} \log \left (c x + 1\right )^{2} - 3 \, {\left (c x + 1\right )} \log \left (c x - 1\right )^{2} + 3 \, {\left ({\left (c x + 1\right )} \log \left (c x - 1\right )^{2} + 2 \, c x + 2 \, {\left (c x + 1\right )} \log \left (c x - 1\right ) + 2\right )} \log \left (c x + 1\right ) - 6 \, {\left (c x + 1\right )} \log \left (c x - 1\right ) - 12\right )} c^{2}}{c^{5} x + c^{4}} - \frac {6 \, {\left ({\left (c x + 1\right )} \log \left (c x + 1\right )^{2} + {\left (c x + 1\right )} \log \left (c x - 1\right )^{2} - 2 \, {\left (c x + {\left (c x + 1\right )} \log \left (c x - 1\right ) + 1\right )} \log \left (c x + 1\right ) + 2 \, {\left (c x + 1\right )} \log \left (c x - 1\right ) + 4\right )} c \operatorname {artanh}\left (c x\right )}{c^{4} x + c^{3}}\right )} c\right )} b^{3} - \frac {3 \, a b^{2} \operatorname {artanh}\left (c x\right )^{2}}{c^{2} x + c} - \frac {a^{3}}{c^{2} x + c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 160, normalized size = 1.15 \begin {gather*} \frac {{\left (b^{3} c x - b^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3} - 16 \, a^{3} - 24 \, a^{2} b - 24 \, a b^{2} - 12 \, b^{3} - 3 \, {\left (2 \, a b^{2} + b^{3} - {\left (2 \, a b^{2} + b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 6 \, {\left (2 \, a^{2} b + 2 \, a b^{2} + b^{3} - {\left (2 \, a^{2} b + 2 \, a b^{2} + b^{3}\right )} c x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{16 \, {\left (c^{2} x + c\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*} \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{\left (c x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 172, normalized size = 1.24 \begin {gather*} \frac {1}{16} \, {\left (\frac {{\left (c x - 1\right )} b^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )^{3}}{{\left (c x + 1\right )} c^{2}} + \frac {3 \, {\left (2 \, a b^{2} + b^{3}\right )} {\left (c x - 1\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )} c^{2}} + \frac {6 \, {\left (2 \, a^{2} b + 2 \, a b^{2} + b^{3}\right )} {\left (c x - 1\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )} c^{2}} + \frac {2 \, {\left (4 \, a^{3} + 6 \, a^{2} b + 6 \, a b^{2} + 3 \, b^{3}\right )} {\left (c x - 1\right )}}{{\left (c x + 1\right )} c^{2}}\right )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.30, size = 582, normalized size = 4.19 \begin {gather*} \ln \left (1-c\,x\right )\,\left (\ln \left (c\,x+1\right )\,\left (\frac {3\,b^3\,x+\frac {3\,b^2\,\left (4\,a+b\right )}{c}}{8\,c\,x+8}-\frac {3\,\left (b^3+a\,b^2\right )}{4\,c}+\frac {6\,b^3}{c\,\left (8\,c\,x+8\right )}\right )+\frac {3\,b^3\,x+\frac {3\,b^2\,\left (4\,a+b\right )}{c}}{8\,c\,x+8}-{\ln \left (c\,x+1\right )}^2\,\left (\frac {3\,b^3}{16\,c}-\frac {3\,b^3}{c\,\left (8\,c\,x+8\right )}\right )-\frac {3\,b^3\,x+\frac {3\,\left (-4\,a^2\,b+4\,a\,b^2+5\,b^3\right )}{c}}{8\,c\,x+8}+\frac {6\,b^3}{c\,\left (8\,c\,x+8\right )}+\frac {3\,\left (8\,c\,x+24\right )\,\left (b^3+a\,b^2\right )}{4\,c\,\left (8\,c\,x+8\right )}\right )-{\ln \left (1-c\,x\right )}^2\,\left (\frac {3\,b^3}{c\,\left (8\,c\,x+8\right )}-\frac {3\,\left (b^3+a\,b^2\right )}{8\,c}-\ln \left (c\,x+1\right )\,\left (\frac {3\,b^3}{16\,c}-\frac {3\,b^3}{c\,\left (8\,c\,x+8\right )}\right )+\frac {3\,b^3\,\left (8\,c\,x+24\right )}{16\,c\,\left (8\,c\,x+8\right )}+\frac {3\,b^2\,\left (2\,a-b\right )}{c\,\left (8\,c\,x+8\right )}\right )-{\ln \left (c\,x+1\right )}^2\,\left (\frac {\frac {3\,b^3\,x}{16\,c}+\frac {3\,b^2\,\left (4\,a+3\,b\right )}{16\,c^2}}{x+\frac {1}{c}}-\frac {3\,b^2\,\left (a+b\right )}{8\,c}\right )-{\ln \left (1-c\,x\right )}^3\,\left (\frac {b^3}{16\,c}-\frac {b^3}{c\,\left (8\,c\,x+8\right )}\right )+{\ln \left (c\,x+1\right )}^3\,\left (\frac {b^3}{16\,c}-\frac {b^3}{8\,c^2\,\left (x+\frac {1}{c}\right )}\right )-\frac {\ln \left (c\,x+1\right )\,\left (\frac {3\,b\,\left (2\,a^2+3\,a\,b+2\,b^2\right )}{4\,c^2}+\frac {3\,b^2\,x\,\left (a+b\right )}{4\,c}\right )}{x+\frac {1}{c}}-\frac {4\,a^3+6\,a^2\,b+6\,a\,b^2+3\,b^3}{2\,c\,\left (2\,c\,x+2\right )}-\frac {b\,\mathrm {atan}\left (c\,x\,1{}\mathrm {i}\right )\,\left (2\,a^2+4\,a\,b+3\,b^2\right )\,3{}\mathrm {i}}{4\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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