Optimal. Leaf size=190 \[ \frac {1}{3} c^4 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )-2 i c^3 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-6 a c^2 d^4 x+4 i a c d^4 \log (x)-\frac {1}{6} b c^3 d^4 x^2+\frac {8}{3} b c d^4 \log \left (c^2 x^2+1\right )+2 i b c^2 d^4 x-6 b c^2 d^4 x \tan ^{-1}(c x)-2 b c d^4 \text {Li}_2(-i c x)+2 b c d^4 \text {Li}_2(i c x)+b c d^4 \log (x)-2 i b c d^4 \tan ^{-1}(c x) \]
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Rubi [A] time = 0.21, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {4876, 4846, 260, 4852, 266, 36, 29, 31, 4848, 2391, 321, 203, 43} \[ -2 b c d^4 \text {PolyLog}(2,-i c x)+2 b c d^4 \text {PolyLog}(2,i c x)+\frac {1}{3} c^4 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )-2 i c^3 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-6 a c^2 d^4 x+4 i a c d^4 \log (x)-\frac {1}{6} b c^3 d^4 x^2+\frac {8}{3} b c d^4 \log \left (c^2 x^2+1\right )+2 i b c^2 d^4 x-6 b c^2 d^4 x \tan ^{-1}(c x)+b c d^4 \log (x)-2 i b c d^4 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 43
Rule 203
Rule 260
Rule 266
Rule 321
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^2} \, dx &=\int \left (-6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-4 i c^3 d^4 x \left (a+b \tan ^{-1}(c x)\right )+c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d^4 \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx+\left (4 i c d^4\right ) \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx-\left (6 c^2 d^4\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx-\left (4 i c^3 d^4\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (c^4 d^4\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=-6 a c^2 d^4 x-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-2 i c^3 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} c^4 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+4 i a c d^4 \log (x)+\left (b c d^4\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx-\left (2 b c d^4\right ) \int \frac {\log (1-i c x)}{x} \, dx+\left (2 b c d^4\right ) \int \frac {\log (1+i c x)}{x} \, dx-\left (6 b c^2 d^4\right ) \int \tan ^{-1}(c x) \, dx+\left (2 i b c^4 d^4\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {1}{3} \left (b c^5 d^4\right ) \int \frac {x^3}{1+c^2 x^2} \, dx\\ &=-6 a c^2 d^4 x+2 i b c^2 d^4 x-6 b c^2 d^4 x \tan ^{-1}(c x)-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-2 i c^3 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} c^4 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+4 i a c d^4 \log (x)-2 b c d^4 \text {Li}_2(-i c x)+2 b c d^4 \text {Li}_2(i c x)+\frac {1}{2} \left (b c d^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )-\left (2 i b c^2 d^4\right ) \int \frac {1}{1+c^2 x^2} \, dx+\left (6 b c^3 d^4\right ) \int \frac {x}{1+c^2 x^2} \, dx-\frac {1}{6} \left (b c^5 d^4\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )\\ &=-6 a c^2 d^4 x+2 i b c^2 d^4 x-2 i b c d^4 \tan ^{-1}(c x)-6 b c^2 d^4 x \tan ^{-1}(c x)-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-2 i c^3 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} c^4 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+4 i a c d^4 \log (x)+3 b c d^4 \log \left (1+c^2 x^2\right )-2 b c d^4 \text {Li}_2(-i c x)+2 b c d^4 \text {Li}_2(i c x)+\frac {1}{2} \left (b c d^4\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (b c^3 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )-\frac {1}{6} \left (b c^5 d^4\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-6 a c^2 d^4 x+2 i b c^2 d^4 x-\frac {1}{6} b c^3 d^4 x^2-2 i b c d^4 \tan ^{-1}(c x)-6 b c^2 d^4 x \tan ^{-1}(c x)-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-2 i c^3 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} c^4 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+4 i a c d^4 \log (x)+b c d^4 \log (x)+\frac {8}{3} b c d^4 \log \left (1+c^2 x^2\right )-2 b c d^4 \text {Li}_2(-i c x)+2 b c d^4 \text {Li}_2(i c x)\\ \end {align*}
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Mathematica [A] time = 0.15, size = 181, normalized size = 0.95 \[ \frac {d^4 \left (2 a c^4 x^4-12 i a c^3 x^3-36 a c^2 x^2+24 i a c x \log (x)-6 a+2 b c^4 x^4 \tan ^{-1}(c x)-b c^3 x^3-12 i b c^3 x^3 \tan ^{-1}(c x)+12 i b c^2 x^2+16 b c x \log \left (c^2 x^2+1\right )-36 b c^2 x^2 \tan ^{-1}(c x)-12 b c x \text {Li}_2(-i c x)+12 b c x \text {Li}_2(i c x)+6 b c x \log (c x)-12 i b c x \tan ^{-1}(c x)-6 b \tan ^{-1}(c x)\right )}{6 x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, a c^{4} d^{4} x^{4} - 8 i \, a c^{3} d^{4} x^{3} - 12 \, a c^{2} d^{4} x^{2} + 8 i \, a c d^{4} x + 2 \, a d^{4} + {\left (i \, b c^{4} d^{4} x^{4} + 4 \, b c^{3} d^{4} x^{3} - 6 i \, b c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + i \, b d^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{2 \, x^{2}}, x\right ) \]
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 = 264, normalized size = 1.39 \[ -6 a \,c^{2} d^{4} x +\frac {d^{4} a \,c^{4} x^{3}}{3}+2 i b \,c^{2} d^{4} x -2 i d^{4} a \,c^{3} x^{2}-\frac {d^{4} a}{x}-6 b \,c^{2} d^{4} x \arctan \left (c x \right )+\frac {d^{4} b \arctan \left (c x \right ) c^{4} x^{3}}{3}-2 i b c \,d^{4} \arctan \left (c x \right )+4 i c \,d^{4} b \arctan \left (c x \right ) \ln \left (c x \right )-\frac {d^{4} b \arctan \left (c x \right )}{x}-2 c \,d^{4} b \ln \left (c x \right ) \ln \left (i c x +1\right )+2 c \,d^{4} b \ln \left (c x \right ) \ln \left (-i c x +1\right )-2 c \,d^{4} b \dilog \left (i c x +1\right )+2 c \,d^{4} b \dilog \left (-i c x +1\right )+4 i c \,d^{4} a \ln \left (c x \right )-\frac {b \,c^{3} d^{4} x^{2}}{6}+c \,d^{4} b \ln \left (c x \right )+\frac {8 b c \,d^{4} \ln \left (c^{2} x^{2}+1\right )}{3}-2 i d^{4} b \arctan \left (c x \right ) c^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 241, normalized size = 1.27 \[ \frac {1}{3} \, a c^{4} d^{4} x^{3} - 2 i \, a c^{3} d^{4} x^{2} - \frac {1}{6} \, b c^{3} d^{4} x^{2} - 6 \, a c^{2} d^{4} x + 2 i \, b c^{2} d^{4} x - \frac {1}{6} \, {\left (6 i \, \pi - 1\right )} b c d^{4} \log \left (c^{2} x^{2} + 1\right ) + 4 i \, b c d^{4} \arctan \left (c x\right ) \log \left (c x\right ) - 3 \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b c d^{4} + 2 \, b c d^{4} {\rm Li}_2\left (i \, c x + 1\right ) - 2 \, b c d^{4} {\rm Li}_2\left (-i \, c x + 1\right ) + 4 i \, a c d^{4} \log \relax (x) - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b d^{4} - \frac {a d^{4}}{x} + \frac {1}{6} \, {\left (2 \, b c^{4} d^{4} x^{3} - 12 i \, b c^{3} d^{4} x^{2} - 12 i \, b c d^{4}\right )} \arctan \left (c x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 253, normalized size = 1.33 \[ \left \{\begin {array}{cl} -\frac {a\,d^4}{x} & \text {\ if\ \ }c=0\\ \frac {a\,c^4\,d^4\,x^3}{3}-\frac {a\,d^4}{x}+\frac {b\,d^4\,\left (c^2\,\ln \relax (x)-\frac {c^2\,\ln \left (c^2\,x^2+1\right )}{2}\right )}{c}+2\,b\,c\,d^4\,\left ({\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )-{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\right )+3\,b\,c\,d^4\,\ln \left (c^2\,x^2+1\right )-6\,a\,c^2\,d^4\,x-\frac {b\,c^3\,d^4\,\left (\frac {x^2}{2}-\frac {\ln \left (c^2\,x^2+1\right )}{2\,c^2}\right )}{3}-\frac {b\,d^4\,\mathrm {atan}\left (c\,x\right )}{x}-6\,b\,c^2\,d^4\,x\,\mathrm {atan}\left (c\,x\right )+\frac {b\,c^4\,d^4\,x^3\,\mathrm {atan}\left (c\,x\right )}{3}-a\,c^3\,d^4\,x^2\,2{}\mathrm {i}+b\,c^2\,d^4\,x\,2{}\mathrm {i}+a\,c\,d^4\,\ln \relax (x)\,4{}\mathrm {i}-b\,c^3\,d^4\,\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )\,4{}\mathrm {i} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
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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