Optimal. Leaf size=124 \[ -\frac {b c d}{12 x^3}-\frac {i b c^2 d}{6 x^2}+\frac {b c^3 d}{4 x}-\frac {d (a+b \text {ArcTan}(c x))}{4 x^4}-\frac {i c d (a+b \text {ArcTan}(c x))}{3 x^3}-\frac {1}{3} i b c^4 d \log (x)+\frac {1}{24} i b c^4 d \log (i-c x)+\frac {7}{24} i b c^4 d \log (i+c x) \]
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Rubi [A]
time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {45, 4992, 12,
815} \begin {gather*} -\frac {d (a+b \text {ArcTan}(c x))}{4 x^4}-\frac {i c d (a+b \text {ArcTan}(c x))}{3 x^3}-\frac {1}{3} i b c^4 d \log (x)+\frac {1}{24} i b c^4 d \log (-c x+i)+\frac {7}{24} i b c^4 d \log (c x+i)+\frac {b c^3 d}{4 x}-\frac {i b c^2 d}{6 x^2}-\frac {b c d}{12 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 815
Rule 4992
Rubi steps
\begin {align*} \int \frac {(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {i c d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac {d (-3-4 i c x)}{12 x^4 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {i c d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {1}{12} (b c d) \int \frac {-3-4 i c x}{x^4 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {i c d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {1}{12} (b c d) \int \left (-\frac {3}{x^4}-\frac {4 i c}{x^3}+\frac {3 c^2}{x^2}+\frac {4 i c^3}{x}-\frac {i c^4}{2 (-i+c x)}-\frac {7 i c^4}{2 (i+c x)}\right ) \, dx\\ &=-\frac {b c d}{12 x^3}-\frac {i b c^2 d}{6 x^2}+\frac {b c^3 d}{4 x}-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {i c d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {1}{3} i b c^4 d \log (x)+\frac {1}{24} i b c^4 d \log (i-c x)+\frac {7}{24} i b c^4 d \log (i+c x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 100, normalized size = 0.81 \begin {gather*} \frac {d \left (-3 a-4 i a c x-b c x-2 i b c^2 x^2+3 b c^3 x^3+b \left (-3-4 i c x+3 c^4 x^4\right ) \text {ArcTan}(c x)-4 i b c^4 x^4 \log (x)+2 i b c^4 x^4 \log \left (1+c^2 x^2\right )\right )}{12 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 119, normalized size = 0.96
method | result | size |
derivativedivides | \(c^{4} \left (d a \left (-\frac {1}{4 c^{4} x^{4}}-\frac {i}{3 c^{3} x^{3}}\right )-\frac {d b \arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i d b \arctan \left (c x \right )}{3 c^{3} x^{3}}+\frac {i d b \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {d b \arctan \left (c x \right )}{4}-\frac {i d b}{6 c^{2} x^{2}}-\frac {i d b \ln \left (c x \right )}{3}-\frac {d b}{12 c^{3} x^{3}}+\frac {d b}{4 c x}\right )\) | \(119\) |
default | \(c^{4} \left (d a \left (-\frac {1}{4 c^{4} x^{4}}-\frac {i}{3 c^{3} x^{3}}\right )-\frac {d b \arctan \left (c x \right )}{4 c^{4} x^{4}}-\frac {i d b \arctan \left (c x \right )}{3 c^{3} x^{3}}+\frac {i d b \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {d b \arctan \left (c x \right )}{4}-\frac {i d b}{6 c^{2} x^{2}}-\frac {i d b \ln \left (c x \right )}{3}-\frac {d b}{12 c^{3} x^{3}}+\frac {d b}{4 c x}\right )\) | \(119\) |
risch | \(-\frac {\left (4 d b c x -3 i b d \right ) \ln \left (i c x +1\right )}{24 x^{4}}-\frac {i d \left (8 c^{4} b \ln \left (-45 c x \right ) x^{4}-7 c^{4} b \ln \left (-15 c x -15 i\right ) x^{4}-c^{4} b \ln \left (9 c x -9 i\right ) x^{4}+6 i b \,c^{3} x^{3}+4 b \,c^{2} x^{2}+8 a c x +4 i b c x \ln \left (-i c x +1\right )+3 b \ln \left (-i c x +1\right )-2 i b c x -6 i a \right )}{24 x^{4}}\) | \(142\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 102, normalized size = 0.82 \begin {gather*} \frac {1}{6} i \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c d + \frac {1}{12} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b d - \frac {i \, a c d}{3 \, x^{3}} - \frac {a d}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.19, size = 119, normalized size = 0.96 \begin {gather*} \frac {-8 i \, b c^{4} d x^{4} \log \left (x\right ) + 7 i \, b c^{4} d x^{4} \log \left (\frac {c x + i}{c}\right ) + i \, b c^{4} d x^{4} \log \left (\frac {c x - i}{c}\right ) + 6 \, b c^{3} d x^{3} - 4 i \, b c^{2} d x^{2} - 2 \, {\left (4 i \, a + b\right )} c d x - 6 \, a d + {\left (4 \, b c d x - 3 i \, b d\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{24 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.02, size = 214, normalized size = 1.73 \begin {gather*} - \frac {i b c^{4} d \log {\left (135 b^{2} c^{9} d^{2} x \right )}}{3} + \frac {i b c^{4} d \log {\left (135 b^{2} c^{9} d^{2} x - 135 i b^{2} c^{8} d^{2} \right )}}{24} + \frac {7 i b c^{4} d \log {\left (135 b^{2} c^{9} d^{2} x + 135 i b^{2} c^{8} d^{2} \right )}}{24} + \frac {\left (- 4 b c d x + 3 i b d\right ) \log {\left (i c x + 1 \right )}}{24 x^{4}} + \frac {\left (4 b c d x - 3 i b d\right ) \log {\left (- i c x + 1 \right )}}{24 x^{4}} + \frac {- 3 a d + 3 b c^{3} d x^{3} - 2 i b c^{2} d x^{2} + x \left (- 4 i a c d - b c d\right )}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.58, size = 116, normalized size = 0.94 \begin {gather*} \frac {d\,\left (\frac {3\,b\,c^7\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {c^2}}\right )}{{\left (c^2\right )}^{3/2}}+b\,c^4\,\ln \left (c^2\,x^2+1\right )\,2{}\mathrm {i}-b\,c^4\,\ln \left (x\right )\,4{}\mathrm {i}\right )}{12}-\frac {\frac {d\,\left (3\,a+3\,b\,\mathrm {atan}\left (c\,x\right )\right )}{12}+\frac {d\,x\,\left (a\,c\,4{}\mathrm {i}+b\,c+b\,c\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i}\right )}{12}-\frac {b\,c^3\,d\,x^3}{4}+\frac {b\,c^2\,d\,x^2\,1{}\mathrm {i}}{6}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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