Optimal. Leaf size=83 \[ -\frac {d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {1}{6} b c \left (c^2 d-3 e\right ) \log \left (c^2 x^2+1\right )-\frac {1}{3} b c \log (x) \left (c^2 d-3 e\right )-\frac {b c d}{6 x^2} \]
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Rubi [A] time = 0.12, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 4976, 12, 446, 77} \[ -\frac {d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {1}{6} b c \left (c^2 d-3 e\right ) \log \left (c^2 x^2+1\right )-\frac {1}{3} b c \log (x) \left (c^2 d-3 e\right )-\frac {b c d}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 77
Rule 446
Rule 4976
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{x}-(b c) \int \frac {-d-3 e x^2}{3 x^3 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac {1}{3} (b c) \int \frac {-d-3 e x^2}{x^3 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {-d-3 e x}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac {1}{6} (b c) \operatorname {Subst}\left (\int \left (-\frac {d}{x^2}+\frac {c^2 d-3 e}{x}+\frac {-c^4 d+3 c^2 e}{1+c^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b c d}{6 x^2}-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac {1}{3} b c \left (c^2 d-3 e\right ) \log (x)+\frac {1}{6} b c \left (c^2 d-3 e\right ) \log \left (1+c^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 98, normalized size = 1.18 \[ -\frac {a d}{3 x^3}-\frac {a e}{x}+\frac {1}{6} b c d \left (c^2 \log \left (c^2 x^2+1\right )-2 c^2 \log (x)-\frac {1}{x^2}\right )-\frac {1}{2} b c e \log \left (c^2 x^2+1\right )-\frac {b d \tan ^{-1}(c x)}{3 x^3}+b c e \log (x)-\frac {b e \tan ^{-1}(c x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 85, normalized size = 1.02 \[ \frac {{\left (b c^{3} d - 3 \, b c e\right )} x^{3} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (b c^{3} d - 3 \, b c e\right )} x^{3} \log \relax (x) - b c d x - 6 \, a e x^{2} - 2 \, a d - 2 \, {\left (3 \, b e x^{2} + b d\right )} \arctan \left (c x\right )}{6 \, x^{3}} \]
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 [A] time = 0.06, size = 97, normalized size = 1.17 \[ -\frac {a d}{3 x^{3}}-\frac {a e}{x}-\frac {b \arctan \left (c x \right ) d}{3 x^{3}}-\frac {b \arctan \left (c x \right ) e}{x}-\frac {c^{3} b d \ln \left (c x \right )}{3}+c b \ln \left (c x \right ) e -\frac {b c d}{6 x^{2}}+\frac {c^{3} b \ln \left (c^{2} x^{2}+1\right ) d}{6}-\frac {c b \ln \left (c^{2} x^{2}+1\right ) e}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 93, normalized size = 1.12 \[ \frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b d - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b e - \frac {a e}{x} - \frac {a d}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 92, normalized size = 1.11 \[ b\,c\,e\,\ln \relax (x)-\frac {a\,e}{x}-\frac {b\,c\,e\,\ln \left (c^2\,x^2+1\right )}{2}-\frac {b\,c\,d}{6\,x^2}-\frac {a\,d}{3\,x^3}-\frac {b\,d\,\mathrm {atan}\left (c\,x\right )}{3\,x^3}-\frac {b\,e\,\mathrm {atan}\left (c\,x\right )}{x}+\frac {b\,c^3\,d\,\ln \left (c^2\,x^2+1\right )}{6}-\frac {b\,c^3\,d\,\ln \relax (x)}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.29, size = 116, normalized size = 1.40 \[ \begin {cases} - \frac {a d}{3 x^{3}} - \frac {a e}{x} - \frac {b c^{3} d \log {\relax (x )}}{3} + \frac {b c^{3} d \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{6} - \frac {b c d}{6 x^{2}} + b c e \log {\relax (x )} - \frac {b c e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2} - \frac {b d \operatorname {atan}{\left (c x \right )}}{3 x^{3}} - \frac {b e \operatorname {atan}{\left (c x \right )}}{x} & \text {for}\: c \neq 0 \\a \left (- \frac {d}{3 x^{3}} - \frac {e}{x}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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