Optimal. Leaf size=173 \[ \frac {1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-4 i a c^3 d^4 x-6 a c^2 d^4 \log (x)-\frac {1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \tan ^{-1}(c x)-3 i b c^2 d^4 \text {Li}_2(-i c x)+3 i b c^2 d^4 \text {Li}_2(i c x)+4 i b c^2 d^4 \log (x)-\frac {b c d^4}{2 x} \]
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Rubi [A] time = 0.20, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {4876, 4846, 260, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391, 321} \[ -3 i b c^2 d^4 \text {PolyLog}(2,-i c x)+3 i b c^2 d^4 \text {PolyLog}(2,i c x)+\frac {1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-4 i a c^3 d^4 x-6 a c^2 d^4 \log (x)-\frac {1}{2} b c^3 d^4 x+4 i b c^2 d^4 \log (x)-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac {b c d^4}{2 x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 203
Rule 260
Rule 266
Rule 321
Rule 325
Rule 2391
Rule 4846
Rule 4848
Rule 4852
Rule 4876
Rubi steps
\begin {align*} \int \frac {(d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=\int \left (-4 i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^3}+\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}-\frac {6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+c^4 d^4 x \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d^4 \int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx+\left (4 i c d^4\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (6 c^2 d^4\right ) \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx-\left (4 i c^3 d^4\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (c^4 d^4\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=-4 i a c^3 d^4 x-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-6 a c^2 d^4 \log (x)+\frac {1}{2} \left (b c d^4\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\left (3 i b c^2 d^4\right ) \int \frac {\log (1-i c x)}{x} \, dx+\left (3 i b c^2 d^4\right ) \int \frac {\log (1+i c x)}{x} \, dx+\left (4 i b c^2 d^4\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx-\left (4 i b c^3 d^4\right ) \int \tan ^{-1}(c x) \, dx-\frac {1}{2} \left (b c^5 d^4\right ) \int \frac {x^2}{1+c^2 x^2} \, dx\\ &=-\frac {b c d^4}{2 x}-4 i a c^3 d^4 x-\frac {1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-6 a c^2 d^4 \log (x)-3 i b c^2 d^4 \text {Li}_2(-i c x)+3 i b c^2 d^4 \text {Li}_2(i c x)+\left (2 i b c^2 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\left (4 i b c^4 d^4\right ) \int \frac {x}{1+c^2 x^2} \, dx\\ &=-\frac {b c d^4}{2 x}-4 i a c^3 d^4 x-\frac {1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-6 a c^2 d^4 \log (x)+2 i b c^2 d^4 \log \left (1+c^2 x^2\right )-3 i b c^2 d^4 \text {Li}_2(-i c x)+3 i b c^2 d^4 \text {Li}_2(i c x)+\left (2 i b c^2 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\left (2 i b c^4 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b c d^4}{2 x}-4 i a c^3 d^4 x-\frac {1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-6 a c^2 d^4 \log (x)+4 i b c^2 d^4 \log (x)-3 i b c^2 d^4 \text {Li}_2(-i c x)+3 i b c^2 d^4 \text {Li}_2(i c x)\\ \end {align*}
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Mathematica [A] time = 0.15, size = 163, normalized size = 0.94 \[ \frac {d^4 \left (a c^4 x^4-8 i a c^3 x^3-12 a c^2 x^2 \log (x)-8 i a c x-a+b c^4 x^4 \tan ^{-1}(c x)-b c^3 x^3-8 i b c^3 x^3 \tan ^{-1}(c x)-6 i b c^2 x^2 \text {Li}_2(-i c x)+6 i b c^2 x^2 \text {Li}_2(i c x)+8 i b c^2 x^2 \log (c x)-b c x-8 i b c x \tan ^{-1}(c x)-b \tan ^{-1}(c x)\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, a c^{4} d^{4} x^{4} - 8 i \, a c^{3} d^{4} x^{3} - 12 \, a c^{2} d^{4} x^{2} + 8 i \, a c d^{4} x + 2 \, a d^{4} + {\left (i \, b c^{4} d^{4} x^{4} + 4 \, b c^{3} d^{4} x^{3} - 6 i \, b c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + i \, b d^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{2 \, x^{3}}, x\right ) \]
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 = 0.07, size = 248, normalized size = 1.43 \[ -4 i b \,c^{3} d^{4} x \arctan \left (c x \right )+\frac {c^{4} d^{4} a \,x^{2}}{2}-6 c^{2} d^{4} a \ln \left (c x \right )-4 i a \,c^{3} d^{4} x -\frac {d^{4} a}{2 x^{2}}-\frac {4 i c \,d^{4} b \arctan \left (c x \right )}{x}+\frac {c^{4} d^{4} b \arctan \left (c x \right ) x^{2}}{2}-6 c^{2} d^{4} b \ln \left (c x \right ) \arctan \left (c x \right )+4 i c^{2} d^{4} b \ln \left (c x \right )-\frac {d^{4} b \arctan \left (c x \right )}{2 x^{2}}-\frac {b \,c^{3} d^{4} x}{2}-3 i c^{2} d^{4} b \dilog \left (i c x +1\right )-\frac {b c \,d^{4}}{2 x}+3 i c^{2} d^{4} b \ln \left (c x \right ) \ln \left (-i c x +1\right )-3 i c^{2} d^{4} b \ln \left (c x \right ) \ln \left (i c x +1\right )-\frac {4 i c \,d^{4} a}{x}+3 i c^{2} d^{4} b \dilog \left (-i c x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 251, normalized size = 1.45 \[ \frac {1}{2} \, a c^{4} d^{4} x^{2} - 4 i \, a c^{3} d^{4} x - \frac {1}{2} \, b c^{3} d^{4} x + \frac {3}{2} \, \pi b c^{2} d^{4} \log \left (c^{2} x^{2} + 1\right ) - 6 \, b c^{2} d^{4} \arctan \left (c x\right ) \log \left (c x\right ) - 2 i \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b c^{2} d^{4} + 3 i \, b c^{2} d^{4} {\rm Li}_2\left (i \, c x + 1\right ) - 3 i \, b c^{2} d^{4} {\rm Li}_2\left (-i \, c x + 1\right ) - 6 \, a c^{2} d^{4} \log \relax (x) - 2 i \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b c d^{4} - \frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b d^{4} - \frac {4 i \, a c d^{4}}{x} - \frac {a d^{4}}{2 \, x^{2}} + \frac {1}{2} \, {\left (b c^{4} d^{4} x^{2} + b c^{2} d^{4}\right )} \arctan \left (c x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 258, normalized size = 1.49 \[ \left \{\begin {array}{cl} -\frac {a\,d^4}{2\,x^2} & \text {\ if\ \ }c=0\\ \frac {a\,c^4\,d^4\,x^2}{2}-\frac {\frac {a\,d^4}{2}+a\,c\,d^4\,x\,4{}\mathrm {i}}{x^2}-6\,a\,c^2\,d^4\,\ln \relax (x)-\frac {b\,d^4\,\left (c^3\,\mathrm {atan}\left (c\,x\right )+\frac {c^2}{x}\right )}{2\,c}-\frac {b\,c^3\,d^4\,x}{2}-\frac {b\,d^4\,\mathrm {atan}\left (c\,x\right )}{2\,x^2}+b\,c^4\,d^4\,\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )+b\,d^4\,\left (c^2\,\ln \relax (x)-\frac {c^2\,\ln \left (c^2\,x^2+1\right )}{2}\right )\,4{}\mathrm {i}+b\,c^2\,d^4\,\ln \left (c^2\,x^2+1\right )\,2{}\mathrm {i}+b\,c^2\,d^4\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}-b\,c^2\,d^4\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}-a\,c^3\,d^4\,x\,4{}\mathrm {i}-\frac {b\,c\,d^4\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i}}{x}-b\,c^3\,d^4\,x\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
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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