Optimal. Leaf size=57 \[ -\frac {a+b \tanh ^{-1}(c x)}{c d^2 (c x+1)}-\frac {b}{2 c d^2 (c x+1)}+\frac {b \tanh ^{-1}(c x)}{2 c d^2} \]
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Rubi [A] time = 0.05, 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 = {5926, 627, 44, 207} \[ -\frac {a+b \tanh ^{-1}(c x)}{c d^2 (c x+1)}-\frac {b}{2 c d^2 (c x+1)}+\frac {b \tanh ^{-1}(c x)}{2 c d^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 627
Rule 5926
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 = 0.07, size = 64, normalized size = 1.12 \[ \frac {-4 a-(b c x+b) \log (1-c x)+b \log (c x+1)+b c x \log (c x+1)-4 b \tanh ^{-1}(c x)-2 b}{4 c d^2 (c x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.29, size = 49, normalized size = 0.86 \[ \frac {{\left (b c x - b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right ) - 4 \, a - 2 \, b}{4 \, {\left (c^{2} d^{2} x + c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 63, normalized size = 1.11 \[ \frac {1}{4} \, c {\left (\frac {{\left (c x - 1\right )} b \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )} c^{2} d^{2}} + \frac {{\left (c x - 1\right )} {\left (2 \, a + b\right )}}{{\left (c x + 1\right )} c^{2} d^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 84, normalized size = 1.47 \[ -\frac {a}{c \,d^{2} \left (c x +1\right )}-\frac {b \arctanh \left (c x \right )}{c \,d^{2} \left (c x +1\right )}-\frac {b \ln \left (c x -1\right )}{4 c \,d^{2}}-\frac {b}{2 c \,d^{2} \left (c x +1\right )}+\frac {b \ln \left (c x +1\right )}{4 c \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 96, normalized size = 1.68 \[ -\frac {1}{4} \, {\left (c {\left (\frac {2}{c^{3} d^{2} x + c^{2} d^{2}} - \frac {\log \left (c x + 1\right )}{c^{2} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{2} d^{2}}\right )} + \frac {4 \, \operatorname {artanh}\left (c x\right )}{c^{2} d^{2} x + c d^{2}}\right )} b - \frac {a}{c^{2} d^{2} x + c d^{2}} \]
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 \[ -\frac {b\,\mathrm {atanh}\left (c\,x\right )-c\,\left (2\,a\,x+b\,x+b\,x\,\mathrm {atanh}\left (c\,x\right )\right )}{2\,x\,c^2\,d^2+2\,c\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.39, size = 121, normalized size = 2.12 \[ \begin {cases} - \frac {2 a}{2 c^{2} d^{2} x + 2 c d^{2}} + \frac {b c x \operatorname {atanh}{\left (c x \right )}}{2 c^{2} d^{2} x + 2 c d^{2}} - \frac {b \operatorname {atanh}{\left (c x \right )}}{2 c^{2} d^{2} x + 2 c d^{2}} - \frac {b}{2 c^{2} d^{2} x + 2 c d^{2}} & \text {for}\: d \neq 0 \\\tilde {\infty } \left (a x + b x \operatorname {atanh}{\left (c x \right )} + \frac {b \log {\left (x - \frac {1}{c} \right )}}{c} + \frac {b \operatorname {atanh}{\left (c x \right )}}{c}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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