Optimal. Leaf size=152 \[ -\frac {b c d^2}{2 x}-\frac {1}{2} b c^2 d^2 \text {ArcTan}(c x)-\frac {d^2 (a+b \text {ArcTan}(c x))}{2 x^2}-\frac {2 i c d^2 (a+b \text {ArcTan}(c x))}{x}-a c^2 d^2 \log (x)+2 i b c^2 d^2 \log (x)-i b c^2 d^2 \log \left (1+c^2 x^2\right )-\frac {1}{2} i b c^2 d^2 \text {PolyLog}(2,-i c x)+\frac {1}{2} i b c^2 d^2 \text {PolyLog}(2,i c x) \]
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Rubi [A]
time = 0.12, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4996, 4946,
331, 209, 272, 36, 29, 31, 4940, 2438} \begin {gather*} -\frac {d^2 (a+b \text {ArcTan}(c x))}{2 x^2}-\frac {2 i c d^2 (a+b \text {ArcTan}(c x))}{x}-a c^2 d^2 \log (x)-\frac {1}{2} b c^2 d^2 \text {ArcTan}(c x)-\frac {1}{2} i b c^2 d^2 \text {Li}_2(-i c x)+\frac {1}{2} i b c^2 d^2 \text {Li}_2(i c x)-i b c^2 d^2 \log \left (c^2 x^2+1\right )+2 i b c^2 d^2 \log (x)-\frac {b c d^2}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 209
Rule 272
Rule 331
Rule 2438
Rule 4940
Rule 4946
Rule 4996
Rubi steps
\begin {align*} \int \frac {(d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=\int \left (\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{x^3}+\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{x^2}-\frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^2 \int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx+\left (2 i c d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (c^2 d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}-a c^2 d^2 \log (x)+\frac {1}{2} \left (b c d^2\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac {1}{2} \left (i b c^2 d^2\right ) \int \frac {\log (1-i c x)}{x} \, dx+\frac {1}{2} \left (i b c^2 d^2\right ) \int \frac {\log (1+i c x)}{x} \, dx+\left (2 i b c^2 d^2\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c d^2}{2 x}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}-a c^2 d^2 \log (x)-\frac {1}{2} i b c^2 d^2 \text {Li}_2(-i c x)+\frac {1}{2} i b c^2 d^2 \text {Li}_2(i c x)+\left (i b c^2 d^2\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{2} \left (b c^3 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {b c d^2}{2 x}-\frac {1}{2} b c^2 d^2 \tan ^{-1}(c x)-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}-a c^2 d^2 \log (x)-\frac {1}{2} i b c^2 d^2 \text {Li}_2(-i c x)+\frac {1}{2} i b c^2 d^2 \text {Li}_2(i c x)+\left (i b c^2 d^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\left (i b c^4 d^2\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b c d^2}{2 x}-\frac {1}{2} b c^2 d^2 \tan ^{-1}(c x)-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}-a c^2 d^2 \log (x)+2 i b c^2 d^2 \log (x)-i b c^2 d^2 \log \left (1+c^2 x^2\right )-\frac {1}{2} i b c^2 d^2 \text {Li}_2(-i c x)+\frac {1}{2} i b c^2 d^2 \text {Li}_2(i c x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 137, normalized size = 0.90 \begin {gather*} -\frac {d^2 \left (a+4 i a c x+b c x+b \text {ArcTan}(c x)+4 i b c x \text {ArcTan}(c x)+b c^2 x^2 \text {ArcTan}(c x)+2 a c^2 x^2 \log (x)-4 i b c^2 x^2 \log (c x)+2 i b c^2 x^2 \log \left (1+c^2 x^2\right )+i b c^2 x^2 \text {PolyLog}(2,-i c x)-i b c^2 x^2 \text {PolyLog}(2,i c x)\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 206, normalized size = 1.36
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {d^{2} a}{2 c^{2} x^{2}}-\frac {2 i d^{2} a}{c x}-d^{2} a \ln \left (c x \right )-\frac {d^{2} b \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {2 i d^{2} b \arctan \left (c x \right )}{c x}-d^{2} b \ln \left (c x \right ) \arctan \left (c x \right )-\frac {i d^{2} b \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i d^{2} b \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i d^{2} b \dilog \left (i c x +1\right )}{2}+\frac {i d^{2} b \dilog \left (-i c x +1\right )}{2}-i d^{2} b \ln \left (c^{2} x^{2}+1\right )-\frac {b \,d^{2} \arctan \left (c x \right )}{2}-\frac {d^{2} b}{2 c x}+2 i d^{2} b \ln \left (c x \right )\right )\) | \(206\) |
default | \(c^{2} \left (-\frac {d^{2} a}{2 c^{2} x^{2}}-\frac {2 i d^{2} a}{c x}-d^{2} a \ln \left (c x \right )-\frac {d^{2} b \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {2 i d^{2} b \arctan \left (c x \right )}{c x}-d^{2} b \ln \left (c x \right ) \arctan \left (c x \right )-\frac {i d^{2} b \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i d^{2} b \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i d^{2} b \dilog \left (i c x +1\right )}{2}+\frac {i d^{2} b \dilog \left (-i c x +1\right )}{2}-i d^{2} b \ln \left (c^{2} x^{2}+1\right )-\frac {b \,d^{2} \arctan \left (c x \right )}{2}-\frac {d^{2} b}{2 c x}+2 i d^{2} b \ln \left (c x \right )\right )\) | \(206\) |
risch | \(-\frac {2 i d^{2} c a}{x}-\frac {i b \,d^{2} c^{2} \dilog \left (i c x +1\right )}{2}+\frac {i b \,d^{2} \ln \left (i c x +1\right )}{4 x^{2}}-\frac {b \,c^{2} d^{2} \arctan \left (c x \right )}{2}+\frac {d^{2} c b \ln \left (-i c x +1\right )}{x}-\frac {b c \,d^{2}}{2 x}+\frac {3 i b \,d^{2} c^{2} \ln \left (i c x \right )}{4}-d^{2} c^{2} a \ln \left (-i c x \right )-i b \,c^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )-\frac {d^{2} a}{2 x^{2}}-\frac {i d^{2} b \ln \left (-i c x +1\right )}{4 x^{2}}+\frac {i d^{2} c^{2} b \dilog \left (-i c x +1\right )}{2}+\frac {5 i d^{2} c^{2} b \ln \left (-i c x \right )}{4}-\frac {b \,d^{2} c \ln \left (i c x +1\right )}{x}\) | \(219\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - d^{2} \left (\int \left (- \frac {a}{x^{3}}\right )\, dx + \int \frac {a c^{2}}{x}\, dx + \int \left (- \frac {b \operatorname {atan}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \left (- \frac {2 i a c}{x^{2}}\right )\, dx + \int \frac {b c^{2} \operatorname {atan}{\left (c x \right )}}{x}\, dx + \int \left (- \frac {2 i b c \operatorname {atan}{\left (c x \right )}}{x^{2}}\right )\, dx\right ) \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 = 0.74, size = 161, normalized size = 1.06 \begin {gather*} \left \{\begin {array}{cl} -\frac {a\,d^2}{2\,x^2} & \text {\ if\ \ }c=0\\ b\,d^2\,\left (c^2\,\ln \left (x\right )-\frac {c^2\,\ln \left (c^2\,x^2+1\right )}{2}\right )\,2{}\mathrm {i}+\frac {b\,c^2\,d^2\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\frac {b\,c^2\,d^2\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\frac {b\,d^2\,\left (c^3\,\mathrm {atan}\left (c\,x\right )+\frac {c^2}{x}\right )}{2\,c}-\frac {a\,d^2\,\left (2\,c^2\,x^2\,\ln \left (x\right )+1+c\,x\,4{}\mathrm {i}\right )}{2\,x^2}-\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{2\,x^2}-\frac {b\,c\,d^2\,\mathrm {atan}\left (c\,x\right )\,2{}\mathrm {i}}{x} & \text {\ if\ \ }c\neq 0 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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