Optimal. Leaf size=228 \[ -\frac {b c d^3}{30 x^5}+\frac {b c^3 d^3}{18 x^3}-\frac {b c d^2 e}{4 x^3}-\frac {b c^5 d^3}{6 x}+\frac {3 b c^3 d^2 e}{4 x}-\frac {3 b c d e^2}{2 x}-\frac {1}{6} b c^6 d^3 \text {ArcTan}(c x)+\frac {3}{4} b c^4 d^2 e \text {ArcTan}(c x)-\frac {3}{2} b c^2 d e^2 \text {ArcTan}(c x)-\frac {d^3 (a+b \text {ArcTan}(c x))}{6 x^6}-\frac {3 d^2 e (a+b \text {ArcTan}(c x))}{4 x^4}-\frac {3 d e^2 (a+b \text {ArcTan}(c x))}{2 x^2}+a e^3 \log (x)+\frac {1}{2} i b e^3 \text {PolyLog}(2,-i c x)-\frac {1}{2} i b e^3 \text {PolyLog}(2,i c x) \]
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Rubi [A]
time = 0.15, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5100, 4946,
331, 209, 4940, 2438} \begin {gather*} -\frac {d^3 (a+b \text {ArcTan}(c x))}{6 x^6}-\frac {3 d^2 e (a+b \text {ArcTan}(c x))}{4 x^4}-\frac {3 d e^2 (a+b \text {ArcTan}(c x))}{2 x^2}+a e^3 \log (x)-\frac {1}{6} b c^6 d^3 \text {ArcTan}(c x)+\frac {3}{4} b c^4 d^2 e \text {ArcTan}(c x)-\frac {3}{2} b c^2 d e^2 \text {ArcTan}(c x)-\frac {b c^5 d^3}{6 x}+\frac {b c^3 d^3}{18 x^3}+\frac {3 b c^3 d^2 e}{4 x}-\frac {b c d^3}{30 x^5}-\frac {b c d^2 e}{4 x^3}-\frac {3 b c d e^2}{2 x}+\frac {1}{2} i b e^3 \text {Li}_2(-i c x)-\frac {1}{2} i b e^3 \text {Li}_2(i c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 331
Rule 2438
Rule 4940
Rule 4946
Rule 5100
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{x^7} \, dx &=\int \left (\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^7}+\frac {3 d^2 e \left (a+b \tan ^{-1}(c x)\right )}{x^5}+\frac {3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{x^3}+\frac {e^3 \left (a+b \tan ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^3 \int \frac {a+b \tan ^{-1}(c x)}{x^7} \, dx+\left (3 d^2 e\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^5} \, dx+\left (3 d e^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx+e^3 \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx\\ &=-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+a e^3 \log (x)+\frac {1}{6} \left (b c d^3\right ) \int \frac {1}{x^6 \left (1+c^2 x^2\right )} \, dx+\frac {1}{4} \left (3 b c d^2 e\right ) \int \frac {1}{x^4 \left (1+c^2 x^2\right )} \, dx+\frac {1}{2} \left (3 b c d e^2\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx+\frac {1}{2} \left (i b e^3\right ) \int \frac {\log (1-i c x)}{x} \, dx-\frac {1}{2} \left (i b e^3\right ) \int \frac {\log (1+i c x)}{x} \, dx\\ &=-\frac {b c d^3}{30 x^5}-\frac {b c d^2 e}{4 x^3}-\frac {3 b c d e^2}{2 x}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+a e^3 \log (x)+\frac {1}{2} i b e^3 \text {Li}_2(-i c x)-\frac {1}{2} i b e^3 \text {Li}_2(i c x)-\frac {1}{6} \left (b c^3 d^3\right ) \int \frac {1}{x^4 \left (1+c^2 x^2\right )} \, dx-\frac {1}{4} \left (3 b c^3 d^2 e\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac {1}{2} \left (3 b c^3 d e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {b c d^3}{30 x^5}+\frac {b c^3 d^3}{18 x^3}-\frac {b c d^2 e}{4 x^3}+\frac {3 b c^3 d^2 e}{4 x}-\frac {3 b c d e^2}{2 x}-\frac {3}{2} b c^2 d e^2 \tan ^{-1}(c x)-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+a e^3 \log (x)+\frac {1}{2} i b e^3 \text {Li}_2(-i c x)-\frac {1}{2} i b e^3 \text {Li}_2(i c x)+\frac {1}{6} \left (b c^5 d^3\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx+\frac {1}{4} \left (3 b c^5 d^2 e\right ) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {b c d^3}{30 x^5}+\frac {b c^3 d^3}{18 x^3}-\frac {b c d^2 e}{4 x^3}-\frac {b c^5 d^3}{6 x}+\frac {3 b c^3 d^2 e}{4 x}-\frac {3 b c d e^2}{2 x}+\frac {3}{4} b c^4 d^2 e \tan ^{-1}(c x)-\frac {3}{2} b c^2 d e^2 \tan ^{-1}(c x)-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+a e^3 \log (x)+\frac {1}{2} i b e^3 \text {Li}_2(-i c x)-\frac {1}{2} i b e^3 \text {Li}_2(i c x)-\frac {1}{6} \left (b c^7 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {b c d^3}{30 x^5}+\frac {b c^3 d^3}{18 x^3}-\frac {b c d^2 e}{4 x^3}-\frac {b c^5 d^3}{6 x}+\frac {3 b c^3 d^2 e}{4 x}-\frac {3 b c d e^2}{2 x}-\frac {1}{6} b c^6 d^3 \tan ^{-1}(c x)+\frac {3}{4} b c^4 d^2 e \tan ^{-1}(c x)-\frac {3}{2} b c^2 d e^2 \tan ^{-1}(c x)-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {3 d^2 e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {3 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+a e^3 \log (x)+\frac {1}{2} i b e^3 \text {Li}_2(-i c x)-\frac {1}{2} i b e^3 \text {Li}_2(i c x)\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 190, normalized size = 0.83 \begin {gather*} -\frac {a d^3}{6 x^6}-\frac {3 a d^2 e}{4 x^4}-\frac {3 a d e^2}{2 x^2}+\frac {b d^2 e \left (c x \left (-1+3 c^2 x^2\right )+3 \left (-1+c^4 x^4\right ) \text {ArcTan}(c x)\right )}{4 x^4}-\frac {b d^3 \left (c x \left (3-5 c^2 x^2+15 c^4 x^4\right )+15 \left (1+c^6 x^6\right ) \text {ArcTan}(c x)\right )}{90 x^6}-\frac {3 b d e^2 (\text {ArcTan}(c x)+c x (1+c x \text {ArcTan}(c x)))}{2 x^2}+a e^3 \log (x)+\frac {1}{2} i b e^3 (\text {PolyLog}(2,-i c x)-\text {PolyLog}(2,i c x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 315, normalized size = 1.38
method | result | size |
derivativedivides | \(c^{6} \left (-\frac {3 a \,d^{2} e}{4 c^{6} x^{4}}-\frac {a \,d^{3}}{6 c^{6} x^{6}}-\frac {3 a d \,e^{2}}{2 c^{6} x^{2}}+\frac {a \,e^{3} \ln \left (c x \right )}{c^{6}}-\frac {3 b \arctan \left (c x \right ) d^{2} e}{4 c^{6} x^{4}}-\frac {b \arctan \left (c x \right ) d^{3}}{6 c^{6} x^{6}}-\frac {3 b \arctan \left (c x \right ) d \,e^{2}}{2 c^{6} x^{2}}+\frac {b \arctan \left (c x \right ) e^{3} \ln \left (c x \right )}{c^{6}}+\frac {i b \,e^{3} \dilog \left (i c x +1\right )}{2 c^{6}}-\frac {i b \,e^{3} \dilog \left (-i c x +1\right )}{2 c^{6}}+\frac {i b \,e^{3} \ln \left (c x \right ) \ln \left (i c x +1\right )}{2 c^{6}}-\frac {i b \,e^{3} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2 c^{6}}-\frac {b \,d^{3} \arctan \left (c x \right )}{6}+\frac {3 b \,d^{2} e \arctan \left (c x \right )}{4 c^{2}}-\frac {3 b d \,e^{2} \arctan \left (c x \right )}{2 c^{4}}-\frac {b \,d^{3}}{6 c x}+\frac {3 b \,d^{2} e}{4 c^{3} x}-\frac {3 b d \,e^{2}}{2 c^{5} x}-\frac {b \,d^{3}}{30 c^{5} x^{5}}+\frac {b \,d^{3}}{18 c^{3} x^{3}}-\frac {b \,d^{2} e}{4 c^{5} x^{3}}\right )\) | \(315\) |
default | \(c^{6} \left (-\frac {3 a \,d^{2} e}{4 c^{6} x^{4}}-\frac {a \,d^{3}}{6 c^{6} x^{6}}-\frac {3 a d \,e^{2}}{2 c^{6} x^{2}}+\frac {a \,e^{3} \ln \left (c x \right )}{c^{6}}-\frac {3 b \arctan \left (c x \right ) d^{2} e}{4 c^{6} x^{4}}-\frac {b \arctan \left (c x \right ) d^{3}}{6 c^{6} x^{6}}-\frac {3 b \arctan \left (c x \right ) d \,e^{2}}{2 c^{6} x^{2}}+\frac {b \arctan \left (c x \right ) e^{3} \ln \left (c x \right )}{c^{6}}+\frac {i b \,e^{3} \dilog \left (i c x +1\right )}{2 c^{6}}-\frac {i b \,e^{3} \dilog \left (-i c x +1\right )}{2 c^{6}}+\frac {i b \,e^{3} \ln \left (c x \right ) \ln \left (i c x +1\right )}{2 c^{6}}-\frac {i b \,e^{3} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2 c^{6}}-\frac {b \,d^{3} \arctan \left (c x \right )}{6}+\frac {3 b \,d^{2} e \arctan \left (c x \right )}{4 c^{2}}-\frac {3 b d \,e^{2} \arctan \left (c x \right )}{2 c^{4}}-\frac {b \,d^{3}}{6 c x}+\frac {3 b \,d^{2} e}{4 c^{3} x}-\frac {3 b d \,e^{2}}{2 c^{5} x}-\frac {b \,d^{3}}{30 c^{5} x^{5}}+\frac {b \,d^{3}}{18 c^{3} x^{3}}-\frac {b \,d^{2} e}{4 c^{5} x^{3}}\right )\) | \(315\) |
risch | \(a \,e^{3} \ln \left (-i c x \right )+\frac {3 i b \,e^{2} d \ln \left (i c x +1\right )}{4 x^{2}}-\frac {3 i b \,c^{2} e^{2} d \ln \left (i c x \right )}{4}-\frac {a \,d^{3}}{6 x^{6}}-\frac {i b \,d^{3} \ln \left (-i c x +1\right )}{12 x^{6}}+\frac {i c^{6} b \,d^{3} \ln \left (-i c x \right )}{12}-\frac {i b \,c^{6} d^{3} \ln \left (i c x \right )}{12}+\frac {i b \,d^{3} \ln \left (i c x +1\right )}{12 x^{6}}-\frac {3 i b \,d^{2} e \ln \left (-i c x +1\right )}{8 x^{4}}-\frac {b c \,d^{2} e}{4 x^{3}}+\frac {3 b \,c^{3} d^{2} e}{4 x}-\frac {3 b c d \,e^{2}}{2 x}+\frac {3 b \,c^{4} d^{2} e \arctan \left (c x \right )}{4}-\frac {3 b \,c^{2} d \,e^{2} \arctan \left (c x \right )}{2}-\frac {b \,c^{6} d^{3} \arctan \left (c x \right )}{6}+\frac {i b \,e^{3} \dilog \left (i c x +1\right )}{2}-\frac {i b \,e^{3} \dilog \left (-i c x +1\right )}{2}-\frac {3 a \,d^{2} e}{4 x^{4}}-\frac {3 a d \,e^{2}}{2 x^{2}}-\frac {b c \,d^{3}}{30 x^{5}}+\frac {b \,c^{3} d^{3}}{18 x^{3}}-\frac {b \,c^{5} d^{3}}{6 x}-\frac {3 i c^{4} b \,d^{2} e \ln \left (-i c x \right )}{8}-\frac {3 i b d \,e^{2} \ln \left (-i c x +1\right )}{4 x^{2}}+\frac {3 i c^{2} b d \,e^{2} \ln \left (-i c x \right )}{4}+\frac {3 i b \,d^{2} e \ln \left (i c x +1\right )}{8 x^{4}}+\frac {3 i b \,c^{4} d^{2} e \ln \left (i c x \right )}{8}\) | \(394\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}}{x^{7}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.84, size = 261, normalized size = 1.14 \begin {gather*} \left \{\begin {array}{cl} a\,e^3\,\ln \left (x\right )-\frac {\frac {a\,d^3}{6}+\frac {3\,a\,d^2\,e\,x^2}{4}+\frac {3\,a\,d\,e^2\,x^4}{2}}{x^6} & \text {\ if\ \ }c=0\\ a\,e^3\,\ln \left (x\right )-\frac {\frac {a\,d^3}{6}+\frac {3\,a\,d^2\,e\,x^2}{4}+\frac {3\,a\,d\,e^2\,x^4}{2}}{x^6}-3\,b\,d^2\,e\,\left (\frac {\mathrm {atan}\left (c\,x\right )}{4\,x^4}+\frac {\frac {\frac {c^2}{3}-c^4\,x^2}{x^3}-c^5\,\mathrm {atan}\left (c\,x\right )}{4\,c}\right )-\frac {b\,d^3\,\left (\frac {c^6\,x^4-\frac {c^4\,x^2}{3}+\frac {c^2}{5}}{x^5}+c^7\,\mathrm {atan}\left (c\,x\right )\right )}{6\,c}-3\,b\,d\,e^2\,\left (\frac {c^3\,\mathrm {atan}\left (c\,x\right )+\frac {c^2}{x}}{2\,c}+\frac {\mathrm {atan}\left (c\,x\right )}{2\,x^2}\right )-\frac {b\,d^3\,\mathrm {atan}\left (c\,x\right )}{6\,x^6}-\frac {b\,e^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {b\,e^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} & \text {\ if\ \ }c\neq 0 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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