Optimal. Leaf size=409 \[ -\frac {e \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d^2}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {b c^2 \tan ^{-1}(c x)}{2 d}-\frac {i b e \text {Li}_2(-i c x)}{2 d^2}+\frac {i b e \text {Li}_2(i c x)}{2 d^2}+\frac {i b e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {i b e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {b c}{2 d x} \]
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Rubi [A] time = 0.48, antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4918, 4852, 325, 203, 4928, 4848, 2391, 4980, 4856, 2402, 2315, 2447} \[ -\frac {i b e \text {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \text {PolyLog}(2,i c x)}{2 d^2}+\frac {i b e \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{4 d^2}-\frac {i b e \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{4 d^2}-\frac {e \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d^2}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {b c^2 \tan ^{-1}(c x)}{2 d}-\frac {b c}{2 d x} \]
Antiderivative was successfully verified.
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Rule 203
Rule 325
Rule 2315
Rule 2391
Rule 2402
Rule 2447
Rule 4848
Rule 4852
Rule 4856
Rule 4918
Rule 4928
Rule 4980
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^3 \left (d+e x^2\right )} \, dx &=\frac {\int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx}{d}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{x \left (d+e x^2\right )} \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d}-\frac {e \int \left (\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{d \left (d+e x^2\right )}\right ) \, dx}{d}\\ &=-\frac {b c}{2 d x}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}-\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx}{d^2}+\frac {e^2 \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx}{d^2}\\ &=-\frac {b c}{2 d x}-\frac {b c^2 \tan ^{-1}(c x)}{2 d}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {(i b e) \int \frac {\log (1-i c x)}{x} \, dx}{2 d^2}+\frac {(i b e) \int \frac {\log (1+i c x)}{x} \, dx}{2 d^2}+\frac {e^2 \int \left (-\frac {a+b \tan ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \tan ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^2}\\ &=-\frac {b c}{2 d x}-\frac {b c^2 \tan ^{-1}(c x)}{2 d}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {i b e \text {Li}_2(-i c x)}{2 d^2}+\frac {i b e \text {Li}_2(i c x)}{2 d^2}-\frac {e^{3/2} \int \frac {a+b \tan ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^2}+\frac {e^{3/2} \int \frac {a+b \tan ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^2}\\ &=-\frac {b c}{2 d x}-\frac {b c^2 \tan ^{-1}(c x)}{2 d}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}-\frac {i b e \text {Li}_2(-i c x)}{2 d^2}+\frac {i b e \text {Li}_2(i c x)}{2 d^2}+2 \frac {(b c e) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 d^2}-\frac {(b c e) \int \frac {\log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 d^2}-\frac {(b c e) \int \frac {\log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 d^2}\\ &=-\frac {b c}{2 d x}-\frac {b c^2 \tan ^{-1}(c x)}{2 d}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}-\frac {i b e \text {Li}_2(-i c x)}{2 d^2}+\frac {i b e \text {Li}_2(i c x)}{2 d^2}-\frac {i b e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {i b e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}+2 \frac {(i b e) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{2 d^2}\\ &=-\frac {b c}{2 d x}-\frac {b c^2 \tan ^{-1}(c x)}{2 d}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}-\frac {i b e \text {Li}_2(-i c x)}{2 d^2}+\frac {i b e \text {Li}_2(i c x)}{2 d^2}+\frac {i b e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {i b e \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}\\ \end {align*}
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Mathematica [C] time = 0.30, size = 507, normalized size = 1.24 \[ \frac {\frac {-a-b \tan ^{-1}(c x)}{2 x^2}-\frac {b c \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-c^2 x^2\right )}{2 x}}{d}-\frac {e \left (-\frac {a \log \left (d+e x^2\right )}{2 d}+\frac {a \log (x)}{d}-\frac {i b \left (\text {Li}_2\left (-\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}-\sqrt {e}}\right )+\log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )\right )}{4 d}-\frac {i b \left (\text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )+\log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )\right )}{4 d}+\frac {i b \left (\text {Li}_2\left (-\frac {\sqrt {e} (i c x+1)}{i c \sqrt {-d}-\sqrt {e}}\right )+\log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )\right )}{4 d}+\frac {i b \left (\text {Li}_2\left (\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )+\log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )\right )}{4 d}+\frac {i b \text {Li}_2(-i c x)}{2 d}-\frac {i b \text {Li}_2(i c x)}{2 d}\right )}{d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x\right ) + a}{e x^{5} + d x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.39, size = 801, normalized size = 1.96 \[ \frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d^{2}}-\frac {a}{2 d \,x^{2}}-\frac {a e \ln \left (c x \right )}{d^{2}}+\frac {b \arctan \left (c x \right ) e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 d^{2}}-\frac {b \arctan \left (c x \right )}{2 d \,x^{2}}-\frac {b \arctan \left (c x \right ) e \ln \left (c x \right )}{d^{2}}-\frac {i b e \dilog \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2}+2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =2\right )-c x +i}{\RootOf \left (e \,\textit {\_Z}^{2}+2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =2\right )}\right )}{4 d^{2}}+\frac {i b e \ln \left (c x +i\right ) \ln \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2}-2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =1\right )-c x -i}{\RootOf \left (e \,\textit {\_Z}^{2}-2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =1\right )}\right )}{4 d^{2}}+\frac {i b e \ln \left (c x +i\right ) \ln \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2}-2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =2\right )-c x -i}{\RootOf \left (e \,\textit {\_Z}^{2}-2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =2\right )}\right )}{4 d^{2}}-\frac {i b e \ln \left (c x \right ) \ln \left (i c x +1\right )}{2 d^{2}}+\frac {i b e \dilog \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2}-2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =1\right )-c x -i}{\RootOf \left (e \,\textit {\_Z}^{2}-2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =1\right )}\right )}{4 d^{2}}-\frac {i b e \ln \left (c x +i\right ) \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{4 d^{2}}+\frac {i b e \ln \left (c x -i\right ) \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{4 d^{2}}-\frac {i b e \dilog \left (i c x +1\right )}{2 d^{2}}-\frac {i b e \ln \left (c x -i\right ) \ln \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2}+2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =1\right )-c x +i}{\RootOf \left (e \,\textit {\_Z}^{2}+2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =1\right )}\right )}{4 d^{2}}+\frac {i b e \dilog \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2}-2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =2\right )-c x -i}{\RootOf \left (e \,\textit {\_Z}^{2}-2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =2\right )}\right )}{4 d^{2}}-\frac {b c}{2 d x}-\frac {b \,c^{2} \arctan \left (c x \right )}{2 d}-\frac {i b e \ln \left (c x -i\right ) \ln \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2}+2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =2\right )-c x +i}{\RootOf \left (e \,\textit {\_Z}^{2}+2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =2\right )}\right )}{4 d^{2}}+\frac {i b e \dilog \left (-i c x +1\right )}{2 d^{2}}+\frac {i b e \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2 d^{2}}-\frac {i b e \dilog \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2}+2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =1\right )-c x +i}{\RootOf \left (e \,\textit {\_Z}^{2}+2 i \textit {\_Z} e +c^{2} d -e , \mathit {index} =1\right )}\right )}{4 d^{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 \[ \frac {1}{2} \, a {\left (\frac {e \log \left (e x^{2} + d\right )}{d^{2}} - \frac {2 \, e \log \relax (x)}{d^{2}} - \frac {1}{d x^{2}}\right )} + 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e x^{5} + d x^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,\left (e\,x^2+d\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x^{3} \left (d + e x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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