Optimal. Leaf size=227 \[ \frac {4 i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {3 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+a c^4 d^4 \log (x)+\frac {1}{2} i b c^4 d^4 \text {Li}_2(-i c x)-\frac {1}{2} i b c^4 d^4 \text {Li}_2(i c x)-\frac {16}{3} i b c^4 d^4 \log (x)+\frac {13}{4} b c^4 d^4 \tan ^{-1}(c x)+\frac {13 b c^3 d^4}{4 x}-\frac {2 i b c^2 d^4}{3 x^2}+\frac {8}{3} i b c^4 d^4 \log \left (c^2 x^2+1\right )-\frac {b c d^4}{12 x^3} \]
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Rubi [A] time = 0.23, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {4876, 4852, 325, 203, 266, 44, 36, 29, 31, 4848, 2391} \[ \frac {1}{2} i b c^4 d^4 \text {PolyLog}(2,-i c x)-\frac {1}{2} i b c^4 d^4 \text {PolyLog}(2,i c x)+\frac {3 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+\frac {4 i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+a c^4 d^4 \log (x)-\frac {2 i b c^2 d^4}{3 x^2}+\frac {8}{3} i b c^4 d^4 \log \left (c^2 x^2+1\right )+\frac {13 b c^3 d^4}{4 x}-\frac {16}{3} i b c^4 d^4 \log (x)+\frac {13}{4} b c^4 d^4 \tan ^{-1}(c x)-\frac {b c d^4}{12 x^3} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 44
Rule 203
Rule 266
Rule 325
Rule 2391
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^5} \, dx &=\int \left (\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^5}+\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^4}-\frac {6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^3}-\frac {4 i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+\frac {c^4 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^4 \int \frac {a+b \tan ^{-1}(c x)}{x^5} \, dx+\left (4 i c d^4\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^4} \, dx-\left (6 c^2 d^4\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx-\left (4 i c^3 d^4\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx+\left (c^4 d^4\right ) \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx\\ &=-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+\frac {4 i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+a c^4 d^4 \log (x)+\frac {1}{4} \left (b c d^4\right ) \int \frac {1}{x^4 \left (1+c^2 x^2\right )} \, dx+\frac {1}{3} \left (4 i b c^2 d^4\right ) \int \frac {1}{x^3 \left (1+c^2 x^2\right )} \, dx-\left (3 b c^3 d^4\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx+\frac {1}{2} \left (i b c^4 d^4\right ) \int \frac {\log (1-i c x)}{x} \, dx-\frac {1}{2} \left (i b c^4 d^4\right ) \int \frac {\log (1+i c x)}{x} \, dx-\left (4 i b c^4 d^4\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c d^4}{12 x^3}+\frac {3 b c^3 d^4}{x}-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+\frac {4 i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+a c^4 d^4 \log (x)+\frac {1}{2} i b c^4 d^4 \text {Li}_2(-i c x)-\frac {1}{2} i b c^4 d^4 \text {Li}_2(i c x)+\frac {1}{3} \left (2 i b c^2 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{4} \left (b c^3 d^4\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\left (2 i b c^4 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\left (3 b c^5 d^4\right ) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {b c d^4}{12 x^3}+\frac {13 b c^3 d^4}{4 x}+3 b c^4 d^4 \tan ^{-1}(c x)-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+\frac {4 i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+a c^4 d^4 \log (x)+\frac {1}{2} i b c^4 d^4 \text {Li}_2(-i c x)-\frac {1}{2} i b c^4 d^4 \text {Li}_2(i c x)+\frac {1}{3} \left (2 i b c^2 d^4\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^2}-\frac {c^2}{x}+\frac {c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )-\left (2 i b c^4 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{4} \left (b c^5 d^4\right ) \int \frac {1}{1+c^2 x^2} \, dx+\left (2 i b c^6 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b c d^4}{12 x^3}-\frac {2 i b c^2 d^4}{3 x^2}+\frac {13 b c^3 d^4}{4 x}+\frac {13}{4} b c^4 d^4 \tan ^{-1}(c x)-\frac {d^4 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+\frac {4 i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+a c^4 d^4 \log (x)-\frac {16}{3} i b c^4 d^4 \log (x)+\frac {8}{3} i b c^4 d^4 \log \left (1+c^2 x^2\right )+\frac {1}{2} i b c^4 d^4 \text {Li}_2(-i c x)-\frac {1}{2} i b c^4 d^4 \text {Li}_2(i c x)\\ \end {align*}
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Mathematica [C] time = 0.13, size = 227, normalized size = 1.00 \[ \frac {d^4 \left (12 a c^4 x^4 \log (x)+48 i a c^3 x^3+36 a c^2 x^2-16 i a c x-3 a+6 i b c^4 x^4 \text {Li}_2(-i c x)-6 i b c^4 x^4 \text {Li}_2(i c x)-64 i b c^4 x^4 \log (x)+48 i b c^3 x^3 \tan ^{-1}(c x)-b c x \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-c^2 x^2\right )-8 i b c^2 x^2+36 b c^2 x^2 \tan ^{-1}(c x)+32 i b c^4 x^4 \log \left (c^2 x^2+1\right )+36 b c^3 x^3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-c^2 x^2\right )-16 i b c x \tan ^{-1}(c x)-3 b \tan ^{-1}(c x)\right )}{12 x^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, a c^{4} d^{4} x^{4} - 8 i \, a c^{3} d^{4} x^{3} - 12 \, a c^{2} d^{4} x^{2} + 8 i \, a c d^{4} x + 2 \, a d^{4} + {\left (i \, b c^{4} d^{4} x^{4} + 4 \, b c^{3} d^{4} x^{3} - 6 i \, b c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + i \, b d^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{2 \, x^{5}}, x\right ) \]
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 = 298, normalized size = 1.31 \[ \frac {4 i c^{3} d^{4} b \arctan \left (c x \right )}{x}+c^{4} d^{4} a \ln \left (c x \right )-\frac {4 i c \,d^{4} b \arctan \left (c x \right )}{3 x^{3}}-\frac {d^{4} a}{4 x^{4}}+\frac {3 c^{2} d^{4} a}{x^{2}}-\frac {4 i c \,d^{4} a}{3 x^{3}}+c^{4} d^{4} b \ln \left (c x \right ) \arctan \left (c x \right )+\frac {i c^{4} d^{4} b \dilog \left (i c x +1\right )}{2}-\frac {d^{4} b \arctan \left (c x \right )}{4 x^{4}}+\frac {3 c^{2} d^{4} b \arctan \left (c x \right )}{x^{2}}+\frac {4 i c^{3} d^{4} a}{x}-\frac {2 i b \,c^{2} d^{4}}{3 x^{2}}-\frac {b c \,d^{4}}{12 x^{3}}+\frac {13 b \,c^{3} d^{4}}{4 x}-\frac {16 i c^{4} d^{4} b \ln \left (c x \right )}{3}+\frac {13 b \,c^{4} d^{4} \arctan \left (c x \right )}{4}+\frac {8 i b \,c^{4} d^{4} \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {i c^{4} d^{4} b \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i c^{4} d^{4} b \dilog \left (-i c x +1\right )}{2}-\frac {i c^{4} d^{4} b \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2} \]
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 \[ b c^{4} d^{4} \int \frac {\arctan \left (c x\right )}{x}\,{d x} + a c^{4} d^{4} \log \relax (x) + 2 i \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b c^{3} d^{4} + 3 \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b c^{2} d^{4} + \frac {2}{3} i \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c d^{4} + \frac {4 i \, a c^{3} d^{4}}{x} + \frac {1}{12} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b d^{4} + \frac {3 \, a c^{2} d^{4}}{x^{2}} - \frac {4 i \, a c d^{4}}{3 \, x^{3}} - \frac {a d^{4}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 298, normalized size = 1.31 \[ \left \{\begin {array}{cl} -\frac {a\,d^4}{4\,x^4} & \text {\ if\ \ }c=0\\ 3\,b\,c\,d^4\,\left (c^3\,\mathrm {atan}\left (c\,x\right )+\frac {c^2}{x}\right )-\frac {b\,d^4\,\left (\frac {\frac {c^2}{3}-c^4\,x^2}{x^3}-c^5\,\mathrm {atan}\left (c\,x\right )\right )}{4\,c}-\frac {b\,c^4\,d^4\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {b\,c^4\,d^4\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-b\,c^2\,d^4\,\left (c^2\,\ln \relax (x)-\frac {c^2\,\ln \left (c^2\,x^2+1\right )}{2}\right )\,4{}\mathrm {i}+\frac {a\,d^4\,\left (36\,c^2\,x^2+12\,c^4\,x^4\,\ln \relax (x)-3-c\,x\,16{}\mathrm {i}+c^3\,x^3\,48{}\mathrm {i}\right )}{12\,x^4}-\frac {b\,d^4\,\mathrm {atan}\left (c\,x\right )}{4\,x^4}-\frac {b\,d^4\,\left (c^4\,\ln \relax (x)-\frac {c^4\,\ln \left (-\frac {c^4\,\left (3\,c^2\,x^2+1\right )}{2}-c^4\right )}{2}+\frac {c^2}{2\,x^2}\right )\,4{}\mathrm {i}}{3}-\frac {b\,c\,d^4\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i}}{3\,x^3}+\frac {3\,b\,c^2\,d^4\,\mathrm {atan}\left (c\,x\right )}{x^2}+\frac {b\,c^3\,d^4\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i}}{x} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
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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