Optimal. Leaf size=162 \[ -\frac{\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac{b^2-3 a c}{a^2 x^2 \left (b^2-4 a c\right )}+\frac{b \log \left (a+b x^2+c x^4\right )}{2 a^3}-\frac{2 b \log (x)}{a^3}+\frac{-2 a c+b^2+b c x^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.250251, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1585, 1114, 740, 800, 634, 618, 206, 628} \[ -\frac{\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac{b^2-3 a c}{a^2 x^2 \left (b^2-4 a c\right )}+\frac{b \log \left (a+b x^2+c x^4\right )}{2 a^3}-\frac{2 b \log (x)}{a^3}+\frac{-2 a c+b^2+b c x^2}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1585
Rule 1114
Rule 740
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x \left (a x+b x^3+c x^5\right )^2} \, dx &=\int \frac{1}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.274994, size = 248, normalized size = 1.53 \[ \frac{\frac{\left (6 a^2 c^2+b^3 \sqrt{b^2-4 a c}-6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^4\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\left (-6 a^2 c^2+b^3 \sqrt{b^2-4 a c}+6 a b^2 c-4 a b c \sqrt{b^2-4 a c}-b^4\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{a \left (-3 a b c-2 a c^2 x^2+b^2 c x^2+b^3\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{a}{x^2}-4 b \log (x)}{2 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.017, size = 352, normalized size = 2.2 \begin{align*} -{\frac{{c}^{2}{x}^{2}}{a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{c{x}^{2}{b}^{2}}{2\,{a}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{3\,bc}{2\,a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{{b}^{3}}{2\,{a}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+2\,{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) b}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}}{2\,{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }}-6\,{\frac{{c}^{2}}{a \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+6\,{\frac{{b}^{2}c}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{4}}{{a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{2\,{a}^{2}{x}^{2}}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \,{\left (b^{2} c - 3 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (2 \, b^{3} - 7 \, a b c\right )} x^{2}}{2 \,{\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{6} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{4} +{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}\right )}} - \frac{-2 \, \int \frac{{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 5 \, a b^{2} c + 3 \, a^{2} c^{2}\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{a^{3} b^{2} - 4 \, a^{4} c} - \frac{2 \, b \log \left (x\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.19399, size = 2103, normalized size = 12.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 72.9524, size = 906, normalized size = 5.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 25.5326, size = 246, normalized size = 1.52 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]