Optimal. Leaf size=171 \[ -\frac {b c d^2}{20 x^4}-\frac {i b c^2 d^2}{6 x^3}+\frac {4 b c^3 d^2}{15 x^2}+\frac {i b c^4 d^2}{2 x}-\frac {d^2 (a+b \text {ArcTan}(c x))}{5 x^5}-\frac {i c d^2 (a+b \text {ArcTan}(c x))}{2 x^4}+\frac {c^2 d^2 (a+b \text {ArcTan}(c x))}{3 x^3}+\frac {8}{15} b c^5 d^2 \log (x)-\frac {1}{60} b c^5 d^2 \log (i-c x)-\frac {31}{60} b c^5 d^2 \log (i+c x) \]
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Rubi [A]
time = 0.11, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {45, 4992, 12,
1816} \begin {gather*} \frac {c^2 d^2 (a+b \text {ArcTan}(c x))}{3 x^3}-\frac {d^2 (a+b \text {ArcTan}(c x))}{5 x^5}-\frac {i c d^2 (a+b \text {ArcTan}(c x))}{2 x^4}+\frac {8}{15} b c^5 d^2 \log (x)-\frac {1}{60} b c^5 d^2 \log (-c x+i)-\frac {31}{60} b c^5 d^2 \log (c x+i)+\frac {i b c^4 d^2}{2 x}+\frac {4 b c^3 d^2}{15 x^2}-\frac {i b c^2 d^2}{6 x^3}-\frac {b c d^2}{20 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 1816
Rule 4992
Rubi steps
\begin {align*} \int \frac {(d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right )}{x^6} \, dx &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac {i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^4}+\frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac {d^2 \left (-6-15 i c x+10 c^2 x^2\right )}{30 x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac {i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^4}+\frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {1}{30} \left (b c d^2\right ) \int \frac {-6-15 i c x+10 c^2 x^2}{x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac {i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^4}+\frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {1}{30} \left (b c d^2\right ) \int \left (-\frac {6}{x^5}-\frac {15 i c}{x^4}+\frac {16 c^2}{x^3}+\frac {15 i c^3}{x^2}-\frac {16 c^4}{x}+\frac {c^5}{2 (-i+c x)}+\frac {31 c^5}{2 (i+c x)}\right ) \, dx\\ &=-\frac {b c d^2}{20 x^4}-\frac {i b c^2 d^2}{6 x^3}+\frac {4 b c^3 d^2}{15 x^2}+\frac {i b c^4 d^2}{2 x}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac {i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^4}+\frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {8}{15} b c^5 d^2 \log (x)-\frac {1}{60} b c^5 d^2 \log (i-c x)-\frac {31}{60} b c^5 d^2 \log (i+c x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 129, normalized size = 0.75 \begin {gather*} \frac {d^2 \left (-12 a-30 i a c x-3 b c x+20 a c^2 x^2-10 i b c^2 x^2+16 b c^3 x^3+30 i b c^4 x^4+2 b \left (-6-15 i c x+10 c^2 x^2+15 i c^5 x^5\right ) \text {ArcTan}(c x)+32 b c^5 x^5 \log (x)-16 b c^5 x^5 \log \left (1+c^2 x^2\right )\right )}{60 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 173, normalized size = 1.01
method | result | size |
derivativedivides | \(c^{5} \left (d^{2} a \left (\frac {1}{3 c^{3} x^{3}}-\frac {i}{2 c^{4} x^{4}}-\frac {1}{5 c^{5} x^{5}}\right )+\frac {d^{2} b \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {i d^{2} b \arctan \left (c x \right )}{2 c^{4} x^{4}}-\frac {d^{2} b \arctan \left (c x \right )}{5 c^{5} x^{5}}-\frac {4 b \ln \left (c^{2} x^{2}+1\right ) d^{2}}{15}+\frac {i d^{2} b \arctan \left (c x \right )}{2}-\frac {i d^{2} b}{6 c^{3} x^{3}}+\frac {i d^{2} b}{2 c x}-\frac {d^{2} b}{20 c^{4} x^{4}}+\frac {4 d^{2} b}{15 c^{2} x^{2}}+\frac {8 d^{2} b \ln \left (c x \right )}{15}\right )\) | \(173\) |
default | \(c^{5} \left (d^{2} a \left (\frac {1}{3 c^{3} x^{3}}-\frac {i}{2 c^{4} x^{4}}-\frac {1}{5 c^{5} x^{5}}\right )+\frac {d^{2} b \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {i d^{2} b \arctan \left (c x \right )}{2 c^{4} x^{4}}-\frac {d^{2} b \arctan \left (c x \right )}{5 c^{5} x^{5}}-\frac {4 b \ln \left (c^{2} x^{2}+1\right ) d^{2}}{15}+\frac {i d^{2} b \arctan \left (c x \right )}{2}-\frac {i d^{2} b}{6 c^{3} x^{3}}+\frac {i d^{2} b}{2 c x}-\frac {d^{2} b}{20 c^{4} x^{4}}+\frac {4 d^{2} b}{15 c^{2} x^{2}}+\frac {8 d^{2} b \ln \left (c x \right )}{15}\right )\) | \(173\) |
risch | \(-\frac {i d^{2} b \left (10 c^{2} x^{2}-15 i c x -6\right ) \ln \left (i c x +1\right )}{60 x^{5}}-\frac {d^{2} \left (c^{5} b \ln \left (c x -i\right ) x^{5}+31 c^{5} b \ln \left (-c x -i\right ) x^{5}-32 c^{5} b \ln \left (-x \right ) x^{5}-30 i b \,c^{4} x^{4}-10 i b \,c^{2} x^{2} \ln \left (-i c x +1\right )-16 b \,c^{3} x^{3}+10 i b \,c^{2} x^{2}-20 a \,c^{2} x^{2}+30 i a c x -15 b c x \ln \left (-i c x +1\right )+6 i b \ln \left (-i c x +1\right )+3 x b c +12 a \right )}{60 x^{5}}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 183, normalized size = 1.07 \begin {gather*} -\frac {1}{6} \, {\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^{2} d^{2} + \frac {1}{6} i \, {\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 c d^{2} - \frac {1}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b d^{2} + \frac {a c^{2} d^{2}}{3 \, x^{3}} - \frac {i \, a c d^{2}}{2 \, x^{4}} - \frac {a d^{2}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.44, size = 167, normalized size = 0.98 \begin {gather*} \frac {32 \, b c^{5} d^{2} x^{5} \log \left (x\right ) - 31 \, b c^{5} d^{2} x^{5} \log \left (\frac {c x + i}{c}\right ) - b c^{5} d^{2} x^{5} \log \left (\frac {c x - i}{c}\right ) + 30 i \, b c^{4} d^{2} x^{4} + 16 \, b c^{3} d^{2} x^{3} + 10 \, {\left (2 \, a - i \, b\right )} c^{2} d^{2} x^{2} - 3 \, {\left (10 i \, a + b\right )} c d^{2} x - 12 \, a d^{2} + {\left (10 i \, b c^{2} d^{2} x^{2} + 15 \, b c d^{2} x - 6 i \, b d^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{60 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 15.02, size = 287, normalized size = 1.68 \begin {gather*} \frac {8 b c^{5} d^{2} \log {\left (10395 b^{2} c^{11} d^{4} x \right )}}{15} - \frac {b c^{5} d^{2} \log {\left (10395 b^{2} c^{11} d^{4} x - 10395 i b^{2} c^{10} d^{4} \right )}}{60} - \frac {31 b c^{5} d^{2} \log {\left (10395 b^{2} c^{11} d^{4} x + 10395 i b^{2} c^{10} d^{4} \right )}}{60} + \frac {\left (- 10 i b c^{2} d^{2} x^{2} - 15 b c d^{2} x + 6 i b d^{2}\right ) \log {\left (i c x + 1 \right )}}{60 x^{5}} + \frac {\left (10 i b c^{2} d^{2} x^{2} + 15 b c d^{2} x - 6 i b d^{2}\right ) \log {\left (- i c x + 1 \right )}}{60 x^{5}} - \frac {12 a d^{2} - 30 i b c^{4} d^{2} x^{4} - 16 b c^{3} d^{2} x^{3} + x^{2} \left (- 20 a c^{2} d^{2} + 10 i b c^{2} d^{2}\right ) + x \left (30 i a c d^{2} + 3 b c d^{2}\right )}{60 x^{5}} \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 = 0.92, size = 244, normalized size = 1.43 \begin {gather*} \frac {8\,b\,c^5\,d^2\,\ln \left (x\right )}{15}-\frac {4\,b\,c^5\,d^2\,\ln \left (c^2\,x^2+1\right )}{15}-\frac {\frac {a\,d^2}{5}+\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{5}-\frac {4\,b\,c^5\,d^2\,x^5}{15}-\frac {b\,c^6\,d^2\,x^6\,1{}\mathrm {i}}{2}-\frac {c^4\,d^2\,x^4\,\left (a+b\,1{}\mathrm {i}\right )}{3}+\frac {c\,d^2\,x\,\left (b+a\,10{}\mathrm {i}\right )}{20}-\frac {c^2\,d^2\,x^2\,\left (4\,a-b\,5{}\mathrm {i}\right )}{30}+\frac {c^3\,d^2\,x^3\,\left (-13\,b+a\,30{}\mathrm {i}\right )}{60}-\frac {2\,b\,c^2\,d^2\,x^2\,\mathrm {atan}\left (c\,x\right )}{15}+\frac {b\,c^3\,d^2\,x^3\,\mathrm {atan}\left (c\,x\right )\,1{}\mathrm {i}}{2}-\frac {b\,c^4\,d^2\,x^4\,\mathrm {atan}\left (c\,x\right )}{3}+\frac {b\,c\,d^2\,x\,\mathrm {atan}\left (c\,x\right )\,1{}\mathrm {i}}{2}}{c^2\,x^7+x^5}+\frac {b\,c^8\,d^2\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {c^2}}\right )\,1{}\mathrm {i}}{2\,{\left (c^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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