Optimal. Leaf size=57 \[ -\frac {b}{2 c d^2 (1+c x)}+\frac {b \tanh ^{-1}(c x)}{2 c d^2}-\frac {a+b \tanh ^{-1}(c x)}{c d^2 (1+c x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, 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)}{c d^2 (c x+1)}-\frac {b}{2 c d^2 (c x+1)}+\frac {b \tanh ^{-1}(c x)}{2 c d^2} \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)^2} \, dx &=-\frac {a+b \tanh ^{-1}(c x)}{c d^2 (1+c x)}+\frac {b \int \frac {1}{(d+c d x) \left (1-c^2 x^2\right )} \, dx}{d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{c d^2 (1+c x)}+\frac {b \int \frac {1}{\left (\frac {1}{d}-\frac {c x}{d}\right ) (d+c d x)^2} \, dx}{d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{c d^2 (1+c x)}+\frac {b \int \left (\frac {1}{2 d (1+c x)^2}-\frac {1}{2 d \left (-1+c^2 x^2\right )}\right ) \, dx}{d}\\ &=-\frac {b}{2 c d^2 (1+c x)}-\frac {a+b \tanh ^{-1}(c x)}{c d^2 (1+c x)}-\frac {b \int \frac {1}{-1+c^2 x^2} \, dx}{2 d^2}\\ &=-\frac {b}{2 c d^2 (1+c x)}+\frac {b \tanh ^{-1}(c x)}{2 c d^2}-\frac {a+b \tanh ^{-1}(c x)}{c d^2 (1+c x)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 64, normalized size = 1.12 \begin {gather*} \frac {-4 a-2 b-4 b \tanh ^{-1}(c x)-(b+b c x) \log (1-c x)+b \log (1+c x)+b c x \log (1+c x)}{4 c d^2 (1+c x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 73, normalized size = 1.28
method | result | size |
derivativedivides | \(\frac {-\frac {a}{d^{2} \left (c x +1\right )}-\frac {b \arctanh \left (c x \right )}{d^{2} \left (c x +1\right )}-\frac {b}{2 d^{2} \left (c x +1\right )}+\frac {b \ln \left (c x +1\right )}{4 d^{2}}-\frac {b \ln \left (c x -1\right )}{4 d^{2}}}{c}\) | \(73\) |
default | \(\frac {-\frac {a}{d^{2} \left (c x +1\right )}-\frac {b \arctanh \left (c x \right )}{d^{2} \left (c x +1\right )}-\frac {b}{2 d^{2} \left (c x +1\right )}+\frac {b \ln \left (c x +1\right )}{4 d^{2}}-\frac {b \ln \left (c x -1\right )}{4 d^{2}}}{c}\) | \(73\) |
risch | \(-\frac {b \ln \left (c x +1\right )}{2 c \,d^{2} \left (c x +1\right )}-\frac {\ln \left (c x -1\right ) b c x -\ln \left (-c x -1\right ) b c x +b \ln \left (c x -1\right )-b \ln \left (-c x -1\right )-2 b \ln \left (-c x +1\right )+4 a +2 b}{4 d^{2} \left (c x +1\right ) c}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 96, normalized size = 1.68 \begin {gather*} -\frac {1}{4} \, {\left (c {\left (\frac {2}{c^{3} d^{2} x + c^{2} d^{2}} - \frac {\log \left (c x + 1\right )}{c^{2} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{2} d^{2}}\right )} + \frac {4 \, \operatorname {artanh}\left (c x\right )}{c^{2} d^{2} x + c d^{2}}\right )} b - \frac {a}{c^{2} d^{2} x + c d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 49, normalized size = 0.86 \begin {gather*} \frac {{\left (b c x - b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right ) - 4 \, a - 2 \, b}{4 \, {\left (c^{2} d^{2} x + c d^{2}\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. 95 vs.
\(2 (46) = 92\).
time = 0.71, size = 95, normalized size = 1.67 \begin {gather*} \begin {cases} - \frac {2 a}{2 c^{2} d^{2} x + 2 c d^{2}} + \frac {b c x \operatorname {atanh}{\left (c x \right )}}{2 c^{2} d^{2} x + 2 c d^{2}} - \frac {b \operatorname {atanh}{\left (c x \right )}}{2 c^{2} d^{2} x + 2 c d^{2}} - \frac {b}{2 c^{2} d^{2} x + 2 c d^{2}} & \text {for}\: c \neq 0 \\\frac {a x}{d^{2}} & \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 = 63, normalized size = 1.11 \begin {gather*} \frac {1}{4} \, c {\left (\frac {{\left (c x - 1\right )} b \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )} c^{2} d^{2}} + \frac {{\left (c x - 1\right )} {\left (2 \, a + b\right )}}{{\left (c x + 1\right )} c^{2} d^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.07, size = 45, normalized size = 0.79 \begin {gather*} -\frac {b\,\mathrm {atanh}\left (c\,x\right )-c\,\left (2\,a\,x+b\,x+b\,x\,\mathrm {atanh}\left (c\,x\right )\right )}{2\,x\,c^2\,d^2+2\,c\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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