Optimal. Leaf size=77 \[ -\frac {b}{8 c d^3 (1+c x)^2}-\frac {b}{8 c d^3 (1+c x)}+\frac {b \tanh ^{-1}(c x)}{8 c d^3}-\frac {a+b \tanh ^{-1}(c x)}{2 c d^3 (1+c x)^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6063, 641, 46,
213} \begin {gather*} -\frac {a+b \tanh ^{-1}(c x)}{2 c d^3 (c x+1)^2}-\frac {b}{8 c d^3 (c x+1)}-\frac {b}{8 c d^3 (c x+1)^2}+\frac {b \tanh ^{-1}(c x)}{8 c d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 213
Rule 641
Rule 6063
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{(d+c d x)^3} \, dx &=-\frac {a+b \tanh ^{-1}(c x)}{2 c d^3 (1+c x)^2}+\frac {b \int \frac {1}{(d+c d x)^2 \left (1-c^2 x^2\right )} \, dx}{2 d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{2 c d^3 (1+c x)^2}+\frac {b \int \frac {1}{\left (\frac {1}{d}-\frac {c x}{d}\right ) (d+c d x)^3} \, dx}{2 d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{2 c d^3 (1+c x)^2}+\frac {b \int \left (\frac {1}{2 d^2 (1+c x)^3}+\frac {1}{4 d^2 (1+c x)^2}-\frac {1}{4 d^2 \left (-1+c^2 x^2\right )}\right ) \, dx}{2 d}\\ &=-\frac {b}{8 c d^3 (1+c x)^2}-\frac {b}{8 c d^3 (1+c x)}-\frac {a+b \tanh ^{-1}(c x)}{2 c d^3 (1+c x)^2}-\frac {b \int \frac {1}{-1+c^2 x^2} \, dx}{8 d^3}\\ &=-\frac {b}{8 c d^3 (1+c x)^2}-\frac {b}{8 c d^3 (1+c x)}+\frac {b \tanh ^{-1}(c x)}{8 c d^3}-\frac {a+b \tanh ^{-1}(c x)}{2 c d^3 (1+c x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 86, normalized size = 1.12 \begin {gather*} \frac {-8 a-4 b-2 b c x-8 b \tanh ^{-1}(c x)-b (1+c x)^2 \log (1-c x)+b \log (1+c x)+2 b c x \log (1+c x)+b c^2 x^2 \log (1+c x)}{16 c d^3 (1+c x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 86, normalized size = 1.12
method | result | size |
derivativedivides | \(\frac {-\frac {a}{2 d^{3} \left (c x +1\right )^{2}}-\frac {b \arctanh \left (c x \right )}{2 d^{3} \left (c x +1\right )^{2}}-\frac {b}{8 d^{3} \left (c x +1\right )^{2}}-\frac {b}{8 d^{3} \left (c x +1\right )}+\frac {b \ln \left (c x +1\right )}{16 d^{3}}-\frac {b \ln \left (c x -1\right )}{16 d^{3}}}{c}\) | \(86\) |
default | \(\frac {-\frac {a}{2 d^{3} \left (c x +1\right )^{2}}-\frac {b \arctanh \left (c x \right )}{2 d^{3} \left (c x +1\right )^{2}}-\frac {b}{8 d^{3} \left (c x +1\right )^{2}}-\frac {b}{8 d^{3} \left (c x +1\right )}+\frac {b \ln \left (c x +1\right )}{16 d^{3}}-\frac {b \ln \left (c x -1\right )}{16 d^{3}}}{c}\) | \(86\) |
risch | \(-\frac {b \ln \left (c x +1\right )}{4 c \,d^{3} \left (c x +1\right )^{2}}-\frac {\ln \left (c x -1\right ) b \,c^{2} x^{2}-b \,c^{2} \ln \left (-c x -1\right ) x^{2}+2 \ln \left (c x -1\right ) b c x -2 \ln \left (-c x -1\right ) b c x +2 b c x +b \ln \left (c x -1\right )-b \ln \left (-c x -1\right )-4 b \ln \left (-c x +1\right )+8 a +4 b}{16 d^{3} \left (c x +1\right )^{2} c}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 134, normalized size = 1.74 \begin {gather*} -\frac {1}{16} \, {\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 )} b - \frac {a}{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 = 0.35, size = 75, normalized size = 0.97 \begin {gather*} -\frac {2 \, b c x - {\left (b c^{2} x^{2} + 2 \, b c x - 3 \, b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right ) + 8 \, a + 4 \, b}{16 \, {\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs.
\(2 (66) = 132\).
time = 0.77, size = 224, normalized size = 2.91 \begin {gather*} \begin {cases} - \frac {4 a}{8 c^{3} d^{3} x^{2} + 16 c^{2} d^{3} x + 8 c d^{3}} + \frac {b c^{2} x^{2} \operatorname {atanh}{\left (c x \right )}}{8 c^{3} d^{3} x^{2} + 16 c^{2} d^{3} x + 8 c d^{3}} + \frac {2 b c x \operatorname {atanh}{\left (c x \right )}}{8 c^{3} d^{3} x^{2} + 16 c^{2} d^{3} x + 8 c d^{3}} - \frac {b c x}{8 c^{3} d^{3} x^{2} + 16 c^{2} d^{3} x + 8 c d^{3}} - \frac {3 b \operatorname {atanh}{\left (c x \right )}}{8 c^{3} d^{3} x^{2} + 16 c^{2} d^{3} x + 8 c d^{3}} - \frac {2 b}{8 c^{3} d^{3} x^{2} + 16 c^{2} d^{3} x + 8 c d^{3}} & \text {for}\: c \neq 0 \\\frac {a x}{d^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 118, normalized size = 1.53 \begin {gather*} \frac {1}{32} \, c {\left (\frac {2 \, {\left (c x - 1\right )}^{2} {\left (\frac {2 \, {\left (c x + 1\right )} b}{c x - 1} - b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )}^{2} c^{2} d^{3}} + \frac {{\left (c x - 1\right )}^{2} {\left (\frac {8 \, {\left (c x + 1\right )} a}{c x - 1} - 4 \, a + \frac {4 \, {\left (c x + 1\right )} b}{c x - 1} - b\right )}}{{\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 = 1.09, size = 123, normalized size = 1.60 \begin {gather*} \frac {c^2\,\left (\frac {a\,x^2}{2}+\frac {b\,x^2}{4}-\frac {b\,x^2\,\ln \left (c^2\,x^2-1\right )}{16}+\frac {b\,x^2\,\ln \left (c\,x+1\right )}{8}\right )-\frac {b\,\ln \left (c^2\,x^2-1\right )}{16}-\frac {b\,\mathrm {atanh}\left (c\,x\right )}{2}+\frac {b\,\ln \left (c\,x+1\right )}{8}+c\,\left (a\,x+\frac {3\,b\,x}{8}+\frac {b\,x\,\ln \left (c\,x+1\right )}{4}-\frac {b\,x\,\ln \left (c^2\,x^2-1\right )}{8}\right )}{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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