Optimal. Leaf size=162 \[ -\frac {b^2-3 a c}{a^2 \left (b^2-4 a c\right ) x^2}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )}-\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {2 b \log (x)}{a^3}+\frac {b \log \left (a+b x^2+c x^4\right )}{2 a^3} \]
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Rubi [A]
time = 0.16, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1128, 754, 814,
648, 632, 212, 642} \begin {gather*} \frac {b \log \left (a+b x^2+c x^4\right )}{2 a^3}-\frac {2 b \log (x)}{a^3}-\frac {b^2-3 a c}{a^2 x^2 \left (b^2-4 a c\right )}-\frac {\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}+\frac {-2 a c+b^2+b c x^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 754
Rule 814
Rule 1128
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \frac {-2 \left (b^2-3 a c\right )-2 b c x}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )}-\frac {\text {Subst}\left (\int \left (\frac {2 \left (-b^2+3 a c\right )}{a x^2}-\frac {2 b \left (-b^2+4 a c\right )}{a^2 x}+\frac {2 \left (-b^4+5 a b^2 c-3 a^2 c^2-b c \left (b^2-4 a c\right ) x\right )}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}\\ &=-\frac {b^2-3 a c}{a^2 \left (b^2-4 a c\right ) x^2}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )}-\frac {2 b \log (x)}{a^3}-\frac {\text {Subst}\left (\int \frac {-b^4+5 a b^2 c-3 a^2 c^2-b c \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{a^3 \left (b^2-4 a c\right )}\\ &=-\frac {b^2-3 a c}{a^2 \left (b^2-4 a c\right ) x^2}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )}-\frac {2 b \log (x)}{a^3}+\frac {b \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^3}+\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^3 \left (b^2-4 a c\right )}\\ &=-\frac {b^2-3 a c}{a^2 \left (b^2-4 a c\right ) x^2}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )}-\frac {2 b \log (x)}{a^3}+\frac {b \log \left (a+b x^2+c x^4\right )}{2 a^3}-\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{a^3 \left (b^2-4 a c\right )}\\ &=-\frac {b^2-3 a c}{a^2 \left (b^2-4 a c\right ) x^2}+\frac {b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )}-\frac {\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {2 b \log (x)}{a^3}+\frac {b \log \left (a+b x^2+c x^4\right )}{2 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 248, normalized size = 1.53 \begin {gather*} \frac {-\frac {a}{x^2}-\frac {a \left (b^3-3 a b c+b^2 c x^2-2 a c^2 x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-4 b \log (x)+\frac {\left (b^4-6 a b^2 c+6 a^2 c^2+b^3 \sqrt {b^2-4 a c}-4 a b c \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (-b^4+6 a b^2 c-6 a^2 c^2+b^3 \sqrt {b^2-4 a c}-4 a b c \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 213, normalized size = 1.31
method | result | size |
default | \(-\frac {\frac {\frac {a c \left (2 a c -b^{2}\right ) x^{2}}{4 a c -b^{2}}+\frac {a b \left (3 a c -b^{2}\right )}{4 a c -b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (-4 a b \,c^{2}+b^{3} c \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{c}+\frac {4 \left (3 a^{2} c^{2}-5 a \,b^{2} c +b^{4}-\frac {\left (-4 a b \,c^{2}+b^{3} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{2 a^{3}}-\frac {1}{2 a^{2} x^{2}}-\frac {2 b \ln \left (x \right )}{a^{3}}\) | \(213\) |
risch | \(\frac {-\frac {c \left (3 a c -b^{2}\right ) x^{4}}{a^{2} \left (4 a c -b^{2}\right )}-\frac {b \left (7 a c -2 b^{2}\right ) x^{2}}{2 \left (4 a c -b^{2}\right ) a^{2}}-\frac {1}{2 a}}{x^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {2 b \ln \left (x \right )}{a^{3}}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (64 a^{6} c^{3}-48 a^{5} b^{2} c^{2}+12 b^{4} a^{4} c -b^{6} a^{3}\right ) \textit {\_Z}^{2}+\left (-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) \textit {\_Z} +9 a \,c^{4}-2 b^{2} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (160 a^{7} c^{3}-128 a^{6} b^{2} c^{2}+34 a^{5} b^{4} c -3 b^{6} a^{4}\right ) \textit {\_R}^{2}+\left (-68 a^{4} b \,c^{3}+33 a^{3} b^{3} c^{2}-4 a^{2} b^{5} c \right ) \textit {\_R} +18 a^{2} c^{4}-12 a \,b^{2} c^{3}+2 b^{4} c^{2}\right ) x^{2}+\left (-16 a^{7} b \,c^{2}+8 a^{6} b^{3} c -a^{5} b^{5}\right ) \textit {\_R}^{2}+\left (12 a^{5} c^{3}-39 a^{4} b^{2} c^{2}+17 a^{3} b^{4} c -2 b^{6} a^{2}\right ) \textit {\_R} +24 a^{2} b \,c^{3}-14 a \,b^{3} c^{2}+2 b^{5} c \right )\right )\) | \(390\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 492 vs.
\(2 (154) = 308\).
time = 0.48, size = 1007, normalized size = 6.22 \begin {gather*} \left [-\frac {a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + 2 \, {\left (a b^{4} c - 7 \, a^{2} b^{2} c^{2} + 12 \, a^{3} c^{3}\right )} x^{4} + {\left (2 \, a b^{5} - 15 \, a^{2} b^{3} c + 28 \, a^{3} b c^{2}\right )} x^{2} + {\left ({\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{6} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x^{4} + {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2}\right )} x^{2}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - {\left ({\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x^{4} + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left ({\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x^{4} + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (a^{3} b^{4} c - 8 \, a^{4} b^{2} c^{2} + 16 \, a^{5} c^{3}\right )} x^{6} + {\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{4} + {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} x^{2}\right )}}, -\frac {a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + 2 \, {\left (a b^{4} c - 7 \, a^{2} b^{2} c^{2} + 12 \, a^{3} c^{3}\right )} x^{4} + {\left (2 \, a b^{5} - 15 \, a^{2} b^{3} c + 28 \, a^{3} b c^{2}\right )} x^{2} + 2 \, {\left ({\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{6} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x^{4} + {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2}\right )} x^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left ({\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x^{4} + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left ({\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x^{4} + {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left ({\left (a^{3} b^{4} c - 8 \, a^{4} b^{2} c^{2} + 16 \, a^{5} c^{3}\right )} x^{6} + {\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{4} + {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.72, size = 182, normalized size = 1.12 \begin {gather*} \frac {{\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{2} c x^{4} - 6 \, a c^{2} x^{4} + 2 \, b^{3} x^{2} - 7 \, a b c x^{2} + a b^{2} - 4 \, a^{2} c}{2 \, {\left (c x^{6} + b x^{4} + a x^{2}\right )} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} + \frac {b \log \left (c x^{4} + b x^{2} + a\right )}{2 \, a^{3}} - \frac {b \log \left (x^{2}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.81, size = 2500, normalized size = 15.43 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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