Optimal. Leaf size=60 \[ \frac {3 (c+d x)}{8 d \left ((c+d x)^2+1\right )}+\frac {c+d x}{4 d \left ((c+d x)^2+1\right )^2}+\frac {3 \tan ^{-1}(c+d x)}{8 d} \]
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Rubi [A] time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {247, 199, 203} \[ \frac {3 (c+d x)}{8 d \left ((c+d x)^2+1\right )}+\frac {c+d x}{4 d \left ((c+d x)^2+1\right )^2}+\frac {3 \tan ^{-1}(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 199
Rule 203
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 \tan ^{-1}(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 0.87 \[ \frac {\frac {3 (c+d x)}{(c+d x)^2+1}+\frac {2 (c+d x)}{\left ((c+d x)^2+1\right )^2}+3 \tan ^{-1}(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 153, normalized size = 2.55 \[ \frac {3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 3 \, c^{3} + {\left (9 \, c^{2} + 5\right )} d x + 3 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} x^{2} + c^{4} + 4 \, {\left (c^{3} + c\right )} d x + 2 \, c^{2} + 1\right )} \arctan \left (d x + c\right ) + 5 \, c}{8 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{3} x^{2} + 4 \, {\left (c^{3} + c\right )} d^{2} x + {\left (c^{4} + 2 \, c^{2} + 1\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 73, normalized size = 1.22 \[ \frac {3 \, \arctan \left (d x + c\right )}{8 \, d} + \frac {3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 9 \, c^{2} d x + 3 \, c^{3} + 5 \, d x + 5 \, c}{8 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 1.57 \[ \frac {3 \arctan \left (\frac {2 d^{2} x +2 c d}{2 d}\right )}{8 d}+\frac {2 d^{2} x +2 c d}{8 \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )^{2} d^{2}}+\frac {\frac {3}{8} d^{2} x +\frac {3}{8} c d}{\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.51, size = 115, normalized size = 1.92 \[ \frac {3 \, d^{3} x^{3} + 9 \, c d^{2} x^{2} + 3 \, c^{3} + {\left (9 \, c^{2} + 5\right )} d x + 5 \, c}{8 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{3} x^{2} + 4 \, {\left (c^{3} + c\right )} d^{2} x + {\left (c^{4} + 2 \, c^{2} + 1\right )} d\right )}} + \frac {3 \, \arctan \left (\frac {d^{2} x + c d}{d}\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 111, normalized size = 1.85 \[ \frac {3\,\mathrm {atan}\left (c+d\,x\right )}{8\,d}+\frac {x\,\left (\frac {9\,c^2}{8}+\frac {5}{8}\right )+\frac {3\,c^3+5\,c}{8\,d}+\frac {3\,d^2\,x^3}{8}+\frac {9\,c\,d\,x^2}{8}}{x^2\,\left (6\,c^2\,d^2+2\,d^2\right )+2\,c^2+c^4+x\,\left (4\,d\,c^3+4\,d\,c\right )+d^4\,x^4+4\,c\,d^3\,x^3+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.93, size = 146, normalized size = 2.43 \[ \frac {3 c^{3} + 9 c d^{2} x^{2} + 5 c + 3 d^{3} x^{3} + x \left (9 c^{2} d + 5 d\right )}{8 c^{4} d + 16 c^{2} d + 32 c d^{4} x^{3} + 8 d^{5} x^{4} + 8 d + x^{2} \left (48 c^{2} d^{3} + 16 d^{3}\right ) + x \left (32 c^{3} d^{2} + 32 c d^{2}\right )} + \frac {- \frac {3 i \log {\left (x + \frac {3 c - 3 i}{3 d} \right )}}{16} + \frac {3 i \log {\left (x + \frac {3 c + 3 i}{3 d} \right )}}{16}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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