Integrand size = 11, antiderivative size = 60 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^3} \, dx=\frac {c+d x}{4 d \left (1+(c+d x)^2\right )^2}+\frac {3 (c+d x)}{8 d \left (1+(c+d x)^2\right )}+\frac {3 \arctan (c+d x)}{8 d} \]
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Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {253, 205, 209} \[ \int \frac {1}{\left (1+(c+d x)^2\right )^3} \, dx=\frac {3 \arctan (c+d x)}{8 d}+\frac {3 (c+d x)}{8 d \left ((c+d x)^2+1\right )}+\frac {c+d x}{4 d \left ((c+d x)^2+1\right )^2} \]
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Rule 205
Rule 209
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 \text {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 \text {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 \tan ^{-1}(c+d x)}{8 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^3} \, dx=\frac {\frac {2 (c+d x)}{\left (1+(c+d x)^2\right )^2}+\frac {3 (c+d x)}{1+(c+d x)^2}+3 \arctan (c+d x)}{8 d} \]
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Time = 0.94 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18
method | result | size |
risch | \(\frac {\frac {3 d^{2} x^{3}}{8}+\frac {9 x^{2} c d}{8}+\left (\frac {9 c^{2}}{8}+\frac {5}{8}\right ) x +\frac {c \left (3 c^{2}+5\right )}{8 d}}{\left (d^{2} x^{2}+2 c d x +c^{2}+1\right )^{2}}+\frac {3 \arctan \left (d x +c \right )}{8 d}\) | \(71\) |
default | \(\frac {2 d^{2} x +2 c d}{8 d^{2} \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )^{2}}+\frac {\frac {3}{8} d^{2} x +\frac {3}{8} c d}{d^{2} \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}+\frac {3 \arctan \left (\frac {2 d^{2} x +2 c d}{2 d}\right )}{8 d}\) | \(94\) |
parallelrisch | \(-\frac {-3 i \ln \left (d x +c +i\right ) c^{4} d^{3}+6 i \ln \left (d x +c -i\right ) c^{2} d^{3}-6 i \ln \left (d x +c +i\right ) c^{2} d^{3}+3 i \ln \left (d x +c -i\right ) x^{4} d^{7}-3 i \ln \left (d x +c +i\right ) x^{4} d^{7}+6 i \ln \left (d x +c -i\right ) x^{2} d^{5}+3 i \ln \left (d x +c -i\right ) c^{4} d^{3}-6 i \ln \left (d x +c +i\right ) x^{2} d^{5}-6 d^{6} x^{3}-18 x^{2} c \,d^{5}-18 x \,c^{2} d^{4}+3 i \ln \left (d x +c -i\right ) d^{3}-3 i \ln \left (d x +c +i\right ) d^{3}+12 i \ln \left (d x +c -i\right ) x c \,d^{4}-12 i \ln \left (d x +c +i\right ) x c \,d^{4}-6 c^{3} d^{3}-10 d^{3} c +12 i \ln \left (d x +c -i\right ) x^{3} c \,d^{6}-12 i \ln \left (d x +c +i\right ) x^{3} c \,d^{6}+18 i \ln \left (d x +c -i\right ) x^{2} c^{2} d^{5}-18 i \ln \left (d x +c +i\right ) x^{2} c^{2} d^{5}+12 i \ln \left (d x +c -i\right ) x \,c^{3} d^{4}-12 i \ln \left (d x +c +i\right ) x \,c^{3} d^{4}-10 x \,d^{4}}{16 d^{4} \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )^{2}}\) | \(380\) |
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (54) = 108\).
Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.55 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^3} \, dx=\frac {3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 3 \, c^{3} + {\left (9 \, c^{2} + 5\right )} d x + 3 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} x^{2} + c^{4} + 4 \, {\left (c^{3} + c\right )} d x + 2 \, c^{2} + 1\right )} \arctan \left (d x + c\right ) + 5 \, c}{8 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{3} x^{2} + 4 \, {\left (c^{3} + c\right )} d^{2} x + {\left (c^{4} + 2 \, c^{2} + 1\right )} d\right )}} \]
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Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.43 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^3} \, dx=\frac {3 c^{3} + 9 c d^{2} x^{2} + 5 c + 3 d^{3} x^{3} + x \left (9 c^{2} d + 5 d\right )}{8 c^{4} d + 16 c^{2} d + 32 c d^{4} x^{3} + 8 d^{5} x^{4} + 8 d + x^{2} \cdot \left (48 c^{2} d^{3} + 16 d^{3}\right ) + x \left (32 c^{3} d^{2} + 32 c d^{2}\right )} + \frac {- \frac {3 i \log {\left (x + \frac {3 c - 3 i}{3 d} \right )}}{16} + \frac {3 i \log {\left (x + \frac {3 c + 3 i}{3 d} \right )}}{16}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (54) = 108\).
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.92 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^3} \, dx=\frac {3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 3 \, c^{3} + {\left (9 \, c^{2} + 5\right )} d x + 5 \, c}{8 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{3} x^{2} + 4 \, {\left (c^{3} + c\right )} d^{2} x + {\left (c^{4} + 2 \, c^{2} + 1\right )} d\right )}} + \frac {3 \, \arctan \left (\frac {d^{2} x + c d}{d}\right )}{8 \, d} \]
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none
Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^3} \, dx=\frac {3 \, \arctan \left (d x + c\right )}{8 \, d} + \frac {3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 9 \, c^{2} d x + 3 \, c^{3} + 5 \, d x + 5 \, c}{8 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{2} d} \]
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Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\left (1+(c+d x)^2\right )^3} \, dx=\frac {3\,\mathrm {atan}\left (c+d\,x\right )}{8\,d}+\frac {x\,\left (\frac {9\,c^2}{8}+\frac {5}{8}\right )+\frac {3\,c^3+5\,c}{8\,d}+\frac {3\,d^2\,x^3}{8}+\frac {9\,c\,d\,x^2}{8}}{x^2\,\left (6\,c^2\,d^2+2\,d^2\right )+2\,c^2+c^4+x\,\left (4\,d\,c^3+4\,d\,c\right )+d^4\,x^4+4\,c\,d^3\,x^3+1} \]
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