Optimal. Leaf size=180 \[ -\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-i a c^3 d^3 x-3 a c^2 d^3 \log (x)-i b c^3 d^3 x \tan ^{-1}(c x)-\frac {3}{2} i b c^2 d^3 \text {Li}_2(-i c x)+\frac {3}{2} i b c^2 d^3 \text {Li}_2(i c x)-i b c^2 d^3 \log \left (c^2 x^2+1\right )+3 i b c^2 d^3 \log (x)-\frac {1}{2} b c^2 d^3 \tan ^{-1}(c x)-\frac {b c d^3}{2 x} \]
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Rubi [A] time = 0.18, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4876, 4846, 260, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391} \[ -\frac {3}{2} i b c^2 d^3 \text {PolyLog}(2,-i c x)+\frac {3}{2} i b c^2 d^3 \text {PolyLog}(2,i c x)-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-i a c^3 d^3 x-3 a c^2 d^3 \log (x)-i b c^2 d^3 \log \left (c^2 x^2+1\right )+3 i b c^2 d^3 \log (x)-\frac {1}{2} b c^2 d^3 \tan ^{-1}(c x)-i b c^3 d^3 x \tan ^{-1}(c x)-\frac {b c d^3}{2 x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 203
Rule 260
Rule 266
Rule 325
Rule 2391
Rule 4846
Rule 4848
Rule 4852
Rule 4876
Rubi steps
\begin {align*} \int \frac {(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=\int \left (-i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^3}+\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^2}-\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^3 \int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx+\left (3 i c d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (3 c^2 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx-\left (i c^3 d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=-i a c^3 d^3 x-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-3 a c^2 d^3 \log (x)+\frac {1}{2} \left (b c d^3\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac {1}{2} \left (3 i b c^2 d^3\right ) \int \frac {\log (1-i c x)}{x} \, dx+\frac {1}{2} \left (3 i b c^2 d^3\right ) \int \frac {\log (1+i c x)}{x} \, dx+\left (3 i b c^2 d^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx-\left (i b c^3 d^3\right ) \int \tan ^{-1}(c x) \, dx\\ &=-\frac {b c d^3}{2 x}-i a c^3 d^3 x-i b c^3 d^3 x \tan ^{-1}(c x)-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-3 a c^2 d^3 \log (x)-\frac {3}{2} i b c^2 d^3 \text {Li}_2(-i c x)+\frac {3}{2} i b c^2 d^3 \text {Li}_2(i c x)+\frac {1}{2} \left (3 i b c^2 d^3\right ) \operatorname {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^3\right ) \int \frac {1}{1+c^2 x^2} \, dx+\left (i b c^4 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx\\ &=-\frac {b c d^3}{2 x}-i a c^3 d^3 x-\frac {1}{2} b c^2 d^3 \tan ^{-1}(c x)-i b c^3 d^3 x \tan ^{-1}(c x)-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-3 a c^2 d^3 \log (x)+\frac {1}{2} i b c^2 d^3 \log \left (1+c^2 x^2\right )-\frac {3}{2} i b c^2 d^3 \text {Li}_2(-i c x)+\frac {3}{2} i b c^2 d^3 \text {Li}_2(i c x)+\frac {1}{2} \left (3 i b c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (3 i b c^4 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b c d^3}{2 x}-i a c^3 d^3 x-\frac {1}{2} b c^2 d^3 \tan ^{-1}(c x)-i b c^3 d^3 x \tan ^{-1}(c x)-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-3 a c^2 d^3 \log (x)+3 i b c^2 d^3 \log (x)-i b c^2 d^3 \log \left (1+c^2 x^2\right )-\frac {3}{2} i b c^2 d^3 \text {Li}_2(-i c x)+\frac {3}{2} i b c^2 d^3 \text {Li}_2(i c x)\\ \end {align*}
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Mathematica [A] time = 0.13, size = 164, normalized size = 0.91 \[ -\frac {i d^3 \left (2 a c^3 x^3-6 i a c^2 x^2 \log (x)+6 a c x-i a+2 b c^3 x^3 \tan ^{-1}(c x)+3 b c^2 x^2 \text {Li}_2(-i c x)-3 b c^2 x^2 \text {Li}_2(i c x)-6 b c^2 x^2 \log (c x)+2 b c^2 x^2 \log \left (c^2 x^2+1\right )-i b c^2 x^2 \tan ^{-1}(c x)-i b c x+6 b c x \tan ^{-1}(c x)-i b \tan ^{-1}(c x)\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {-2 i \, a c^{3} d^{3} x^{3} - 6 \, a c^{2} d^{3} x^{2} + 6 i \, a c d^{3} x + 2 \, a d^{3} + {\left (b c^{3} d^{3} x^{3} - 3 i \, b c^{2} d^{3} x^{2} - 3 \, b c d^{3} x + i \, b d^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{2 \, x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 243, normalized size = 1.35 \[ -\frac {3 i c^{2} d^{3} b \dilog \left (i c x +1\right )}{2}-3 c^{2} d^{3} a \ln \left (c x \right )-i a \,c^{3} d^{3} x -\frac {d^{3} a}{2 x^{2}}+3 i c^{2} d^{3} b \ln \left (c x \right )-3 c^{2} d^{3} b \ln \left (c x \right ) \arctan \left (c x \right )-\frac {3 i c^{2} d^{3} b \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {d^{3} b \arctan \left (c x \right )}{2 x^{2}}+\frac {3 i c^{2} d^{3} b \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-i b \,c^{3} d^{3} x \arctan \left (c x \right )-\frac {3 i c \,d^{3} b \arctan \left (c x \right )}{x}-\frac {3 i c \,d^{3} a}{x}-\frac {b c \,d^{3}}{2 x}+\frac {3 i c^{2} d^{3} b \dilog \left (-i c x +1\right )}{2}-i b \,c^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )-\frac {b \,c^{2} d^{3} \arctan \left (c x \right )}{2} \]
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 \[ -i \, a c^{3} d^{3} x - \frac {1}{2} i \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b c^{2} d^{3} - 3 \, b c^{2} d^{3} \int \frac {\arctan \left (c x\right )}{x}\,{d x} - 3 \, a c^{2} d^{3} \log \relax (x) - \frac {3}{2} i \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b c d^{3} - \frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b d^{3} - \frac {3 i \, a c d^{3}}{x} - \frac {a d^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 205, normalized size = 1.14 \[ \left \{\begin {array}{cl} -\frac {a\,d^3}{2\,x^2} & \text {\ if\ \ }c=0\\ b\,d^3\,\left (c^2\,\ln \relax (x)-\frac {c^2\,\ln \left (c^2\,x^2+1\right )}{2}\right )\,3{}\mathrm {i}+\frac {b\,c^2\,d^3\,\ln \left (c^2\,x^2+1\right )\,1{}\mathrm {i}}{2}+\frac {b\,c^2\,d^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}-\frac {b\,c^2\,d^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}-\frac {b\,d^3\,\left (c^3\,\mathrm {atan}\left (c\,x\right )+\frac {c^2}{x}\right )}{2\,c}-\frac {a\,d^3\,\left (6\,c^2\,x^2\,\ln \relax (x)+1+c\,x\,6{}\mathrm {i}+c^3\,x^3\,2{}\mathrm {i}\right )}{2\,x^2}-\frac {b\,d^3\,\mathrm {atan}\left (c\,x\right )}{2\,x^2}-\frac {b\,c\,d^3\,\mathrm {atan}\left (c\,x\right )\,3{}\mathrm {i}}{x}-b\,c^3\,d^3\,x\,\mathrm {atan}\left (c\,x\right )\,1{}\mathrm {i} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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