Optimal. Leaf size=64 \[ \frac {3 (c+d x)}{8 d \left (1-(c+d x)^2\right )}+\frac {c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac {3 \tanh ^{-1}(c+d x)}{8 d} \]
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Rubi [A] time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 199, 206} \[ \frac {3 (c+d x)}{8 d \left (1-(c+d x)^2\right )}+\frac {c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac {3 \tanh ^{-1}(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\left (1-(c+d x)^2\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac {3 (c+d x)}{8 d \left (1-(c+d x)^2\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{8 d}\\ &=\frac {c+d x}{4 d \left (1-(c+d x)^2\right )^2}+\frac {3 (c+d x)}{8 d \left (1-(c+d x)^2\right )}+\frac {3 \tanh ^{-1}(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 1.02 \[ \frac {-\frac {6 (c+d x)}{(c+d x)^2-1}+\frac {4 (c+d x)}{\left ((c+d x)^2-1\right )^2}-3 \log (-c-d x+1)+3 \log (c+d x+1)}{16 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 220, normalized size = 3.44 \[ -\frac {6 \, d^{3} x^{3} + 18 \, c d^{2} x^{2} + 6 \, c^{3} + 2 \, {\left (9 \, c^{2} - 5\right )} d x - 3 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} - 1\right )} d^{2} x^{2} + c^{4} + 4 \, {\left (c^{3} - c\right )} d x - 2 \, c^{2} + 1\right )} \log \left (d x + c + 1\right ) + 3 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} - 1\right )} d^{2} x^{2} + c^{4} + 4 \, {\left (c^{3} - c\right )} d x - 2 \, c^{2} + 1\right )} \log \left (d x + c - 1\right ) - 10 \, c}{16 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \, {\left (3 \, c^{2} - 1\right )} d^{3} x^{2} + 4 \, {\left (c^{3} - c\right )} d^{2} x + {\left (c^{4} - 2 \, c^{2} + 1\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 88, normalized size = 1.38 \[ \frac {3 \, \log \left ({\left | d x + c + 1 \right |}\right )}{16 \, d} - \frac {3 \, \log \left ({\left | d x + c - 1 \right |}\right )}{16 \, d} - \frac {3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 9 \, c^{2} d x + 3 \, c^{3} - 5 \, d x - 5 \, c}{8 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )}^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 78, normalized size = 1.22 \[ -\frac {3 \ln \left (d x +c -1\right )}{16 d}+\frac {3 \ln \left (d x +c +1\right )}{16 d}+\frac {1}{16 \left (d x +c -1\right )^{2} d}-\frac {3}{16 \left (d x +c -1\right ) d}-\frac {1}{16 \left (d x +c +1\right )^{2} d}-\frac {3}{16 \left (d x +c +1\right ) d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 122, normalized size = 1.91 \[ -\frac {3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 3 \, c^{3} + {\left (9 \, c^{2} - 5\right )} d x - 5 \, c}{8 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \, {\left (3 \, c^{2} - 1\right )} d^{3} x^{2} + 4 \, {\left (c^{3} - c\right )} d^{2} x + {\left (c^{4} - 2 \, c^{2} + 1\right )} d\right )}} + \frac {3 \, \log \left (d x + c + 1\right )}{16 \, d} - \frac {3 \, \log \left (d x + c - 1\right )}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 114, normalized size = 1.78 \[ \frac {3\,\mathrm {atanh}\left (c+d\,x\right )}{8\,d}-\frac {x\,\left (\frac {9\,c^2}{8}-\frac {5}{8}\right )-\frac {5\,c-3\,c^3}{8\,d}+\frac {3\,d^2\,x^3}{8}+\frac {9\,c\,d\,x^2}{8}}{c^4-2\,c^2-x^2\,\left (2\,d^2-6\,c^2\,d^2\right )-x\,\left (4\,c\,d-4\,c^3\,d\right )+d^4\,x^4+4\,c\,d^3\,x^3+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.03, size = 141, normalized size = 2.20 \[ - \frac {3 c^{3} + 9 c d^{2} x^{2} - 5 c + 3 d^{3} x^{3} + x \left (9 c^{2} d - 5 d\right )}{8 c^{4} d - 16 c^{2} d + 32 c d^{4} x^{3} + 8 d^{5} x^{4} + 8 d + x^{2} \left (48 c^{2} d^{3} - 16 d^{3}\right ) + x \left (32 c^{3} d^{2} - 32 c d^{2}\right )} - \frac {\frac {3 \log {\left (x + \frac {3 c - 3}{3 d} \right )}}{16} - \frac {3 \log {\left (x + \frac {3 c + 3}{3 d} \right )}}{16}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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