Optimal. Leaf size=203 \[ \frac {1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )+4 i a c d^4 x+a d^4 \log (x)-\frac {1}{12} b c^3 d^4 x^3+\frac {2}{3} i b c^2 d^4 x^2-\frac {8}{3} i b d^4 \log \left (c^2 x^2+1\right )+\frac {1}{2} i b d^4 \text {Li}_2(-i c x)-\frac {1}{2} i b d^4 \text {Li}_2(i c x)+\frac {13}{4} b c d^4 x-\frac {13}{4} b d^4 \tan ^{-1}(c x)+4 i b c d^4 x \tan ^{-1}(c x) \]
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Rubi [A] time = 0.21, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {4876, 4846, 260, 4848, 2391, 4852, 321, 203, 266, 43, 302} \[ \frac {1}{2} i b d^4 \text {PolyLog}(2,-i c x)-\frac {1}{2} i b d^4 \text {PolyLog}(2,i c x)+\frac {1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )+4 i a c d^4 x+a d^4 \log (x)-\frac {1}{12} b c^3 d^4 x^3+\frac {2}{3} i b c^2 d^4 x^2-\frac {8}{3} i b d^4 \log \left (c^2 x^2+1\right )+\frac {13}{4} b c d^4 x-\frac {13}{4} b d^4 \tan ^{-1}(c x)+4 i b c d^4 x \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 203
Rule 260
Rule 266
Rule 302
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} \, dx &=\int \left (4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-6 c^2 d^4 x \left (a+b \tan ^{-1}(c x)\right )-4 i c^3 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )+c^4 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d^4 \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx+\left (4 i c d^4\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx-\left (6 c^2 d^4\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx-\left (4 i c^3 d^4\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (c^4 d^4\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=4 i a c d^4 x-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+a d^4 \log (x)+\frac {1}{2} \left (i b d^4\right ) \int \frac {\log (1-i c x)}{x} \, dx-\frac {1}{2} \left (i b d^4\right ) \int \frac {\log (1+i c x)}{x} \, dx+\left (4 i b c d^4\right ) \int \tan ^{-1}(c x) \, dx+\left (3 b c^3 d^4\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {1}{3} \left (4 i b c^4 d^4\right ) \int \frac {x^3}{1+c^2 x^2} \, dx-\frac {1}{4} \left (b c^5 d^4\right ) \int \frac {x^4}{1+c^2 x^2} \, dx\\ &=4 i a c d^4 x+3 b c d^4 x+4 i b c d^4 x \tan ^{-1}(c x)-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+a d^4 \log (x)+\frac {1}{2} i b d^4 \text {Li}_2(-i c x)-\frac {1}{2} i b d^4 \text {Li}_2(i c x)-\left (3 b c d^4\right ) \int \frac {1}{1+c^2 x^2} \, dx-\left (4 i b c^2 d^4\right ) \int \frac {x}{1+c^2 x^2} \, dx+\frac {1}{3} \left (2 i b c^4 d^4\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )-\frac {1}{4} \left (b c^5 d^4\right ) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=4 i a c d^4 x+\frac {13}{4} b c d^4 x-\frac {1}{12} b c^3 d^4 x^3-3 b d^4 \tan ^{-1}(c x)+4 i b c d^4 x \tan ^{-1}(c x)-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+a d^4 \log (x)-2 i b d^4 \log \left (1+c^2 x^2\right )+\frac {1}{2} i b d^4 \text {Li}_2(-i c x)-\frac {1}{2} i b d^4 \text {Li}_2(i c x)-\frac {1}{4} \left (b c d^4\right ) \int \frac {1}{1+c^2 x^2} \, dx+\frac {1}{3} \left (2 i b c^4 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 )\\ &=4 i a c d^4 x+\frac {13}{4} b c d^4 x+\frac {2}{3} i b c^2 d^4 x^2-\frac {1}{12} b c^3 d^4 x^3-\frac {13}{4} b d^4 \tan ^{-1}(c x)+4 i b c d^4 x \tan ^{-1}(c x)-3 c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{3} i c^3 d^4 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} c^4 d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+a d^4 \log (x)-\frac {8}{3} i b d^4 \log \left (1+c^2 x^2\right )+\frac {1}{2} i b d^4 \text {Li}_2(-i c x)-\frac {1}{2} i b d^4 \text {Li}_2(i c x)\\ \end {align*}
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Mathematica [A] time = 0.15, size = 174, normalized size = 0.86 \[ \frac {1}{12} d^4 \left (3 a c^4 x^4-16 i a c^3 x^3-36 a c^2 x^2+48 i a c x+12 a \log (x)+3 b c^4 x^4 \tan ^{-1}(c x)-b c^3 x^3-16 i b c^3 x^3 \tan ^{-1}(c x)+8 i b c^2 x^2-32 i b \log \left (c^2 x^2+1\right )-36 b c^2 x^2 \tan ^{-1}(c x)+6 i b \text {Li}_2(-i c x)-6 i b \text {Li}_2(i c x)+39 b c x+48 i b c x \tan ^{-1}(c x)-39 b \tan ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, a c^{4} d^{4} x^{4} - 8 i \, a c^{3} d^{4} x^{3} - 12 \, a c^{2} d^{4} x^{2} + 8 i \, a c d^{4} x + 2 \, a d^{4} + {\left (i \, b c^{4} d^{4} x^{4} + 4 \, b c^{3} d^{4} x^{3} - 6 i \, b c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + i \, b d^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{2 \, x}, x\right ) \]
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 = 260, normalized size = 1.28 \[ \frac {i d^{4} b \dilog \left (i c x +1\right )}{2}+\frac {d^{4} a \,c^{4} x^{4}}{4}+\frac {i d^{4} b \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-3 d^{4} a \,c^{2} x^{2}+d^{4} a \ln \left (c x \right )-\frac {4 i d^{4} b \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {d^{4} b \arctan \left (c x \right ) c^{4} x^{4}}{4}-\frac {i d^{4} b \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-3 d^{4} b \arctan \left (c x \right ) c^{2} x^{2}+d^{4} b \ln \left (c x \right ) \arctan \left (c x \right )+\frac {2 i b \,c^{2} d^{4} x^{2}}{3}-\frac {i d^{4} b \dilog \left (-i c x +1\right )}{2}-\frac {4 i d^{4} a \,c^{3} x^{3}}{3}+4 i a c \,d^{4} x +\frac {13 b c \,d^{4} x}{4}-\frac {b \,c^{3} d^{4} x^{3}}{12}+4 i b c \,d^{4} x \arctan \left (c x \right )-\frac {8 i b \,d^{4} \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {13 b \,d^{4} \arctan \left (c x \right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 220, normalized size = 1.08 \[ \frac {1}{4} \, a c^{4} d^{4} x^{4} - \frac {4}{3} i \, a c^{3} d^{4} x^{3} - \frac {1}{12} \, b c^{3} d^{4} x^{3} - 3 \, a c^{2} d^{4} x^{2} + \frac {2}{3} i \, b c^{2} d^{4} x^{2} + 4 i \, a c d^{4} x + \frac {13}{4} \, b c d^{4} x - \frac {1}{12} \, {\left (3 \, \pi + 8 i\right )} b d^{4} \log \left (c^{2} x^{2} + 1\right ) + b d^{4} \arctan \left (c x\right ) \log \left (c x\right ) + 2 i \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{4} - \frac {1}{2} i \, b d^{4} {\rm Li}_2\left (i \, c x + 1\right ) + \frac {1}{2} i \, b d^{4} {\rm Li}_2\left (-i \, c x + 1\right ) + a d^{4} \log \relax (x) + \frac {1}{12} \, {\left (3 \, b c^{4} d^{4} x^{4} - 16 i \, b c^{3} d^{4} x^{3} - 36 \, b c^{2} d^{4} x^{2} - 39 \, b d^{4}\right )} \arctan \left (c x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 248, normalized size = 1.22 \[ \left \{\begin {array}{cl} a\,d^4\,\ln \relax (x) & \text {\ if\ \ }c=0\\ a\,d^4\,\ln \relax (x)-b\,d^4\,\ln \left (c^2\,x^2+1\right )\,2{}\mathrm {i}-\frac {b\,d^4\,\left (3\,\mathrm {atan}\left (c\,x\right )-3\,c\,x+c^3\,x^3\right )}{12}-\frac {b\,d^4\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {b\,d^4\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-3\,a\,c^2\,d^4\,x^2-\frac {a\,c^3\,d^4\,x^3\,4{}\mathrm {i}}{3}+\frac {a\,c^4\,d^4\,x^4}{4}+a\,c\,d^4\,x\,4{}\mathrm {i}+3\,b\,c\,d^4\,x+\frac {b\,c^2\,d^4\,\left (\frac {x^2}{2}-\frac {\ln \left (c^2\,x^2+1\right )}{2\,c^2}\right )\,4{}\mathrm {i}}{3}-6\,b\,c^2\,d^4\,\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )-\frac {b\,c^3\,d^4\,x^3\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i}}{3}+\frac {b\,c^4\,d^4\,x^4\,\mathrm {atan}\left (c\,x\right )}{4}+b\,c\,d^4\,x\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
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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