Optimal. Leaf size=129 \[ -\frac{2 a b^2}{3 \left (a^2+1\right )^2 x}+\frac{\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (a^2+1\right )^3}-\frac{\left (1-3 a^2\right ) b^3 \log \left ((a+b x)^2+1\right )}{6 \left (a^2+1\right )^3}-\frac{a \left (3-a^2\right ) b^3 \tan ^{-1}(a+b x)}{3 \left (a^2+1\right )^3}+\frac{b}{6 \left (a^2+1\right ) x^2}-\frac{\cot ^{-1}(a+b x)}{3 x^3} \]
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Rubi [A] time = 0.111787, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5046, 371, 710, 801, 635, 203, 260} \[ -\frac{2 a b^2}{3 \left (a^2+1\right )^2 x}+\frac{\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (a^2+1\right )^3}-\frac{\left (1-3 a^2\right ) b^3 \log \left ((a+b x)^2+1\right )}{6 \left (a^2+1\right )^3}-\frac{a \left (3-a^2\right ) b^3 \tan ^{-1}(a+b x)}{3 \left (a^2+1\right )^3}+\frac{b}{6 \left (a^2+1\right ) x^2}-\frac{\cot ^{-1}(a+b x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 5046
Rule 371
Rule 710
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a+b x)}{x^4} \, dx &=-\frac{\cot ^{-1}(a+b x)}{3 x^3}-\frac{1}{3} b \int \frac{1}{x^3 \left (1+(a+b x)^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(a+b x)}{3 x^3}-\frac{1}{3} b^3 \operatorname{Subst}\left (\int \frac{1}{(-a+x)^3 \left (1+x^2\right )} \, dx,x,a+b x\right )\\ &=\frac{b}{6 \left (1+a^2\right ) x^2}-\frac{\cot ^{-1}(a+b x)}{3 x^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{-a-x}{(-a+x)^2 \left (1+x^2\right )} \, dx,x,a+b x\right )}{3 \left (1+a^2\right )}\\ &=\frac{b}{6 \left (1+a^2\right ) x^2}-\frac{\cot ^{-1}(a+b x)}{3 x^3}-\frac{b^3 \operatorname{Subst}\left (\int \left (-\frac{2 a}{\left (1+a^2\right ) (a-x)^2}+\frac{1-3 a^2}{\left (1+a^2\right )^2 (a-x)}+\frac{a \left (3-a^2\right )+\left (1-3 a^2\right ) x}{\left (1+a^2\right )^2 \left (1+x^2\right )}\right ) \, dx,x,a+b x\right )}{3 \left (1+a^2\right )}\\ &=\frac{b}{6 \left (1+a^2\right ) x^2}-\frac{2 a b^2}{3 \left (1+a^2\right )^2 x}-\frac{\cot ^{-1}(a+b x)}{3 x^3}+\frac{\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (1+a^2\right )^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{a \left (3-a^2\right )+\left (1-3 a^2\right ) x}{1+x^2} \, dx,x,a+b x\right )}{3 \left (1+a^2\right )^3}\\ &=\frac{b}{6 \left (1+a^2\right ) x^2}-\frac{2 a b^2}{3 \left (1+a^2\right )^2 x}-\frac{\cot ^{-1}(a+b x)}{3 x^3}+\frac{\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (1+a^2\right )^3}-\frac{\left (\left (1-3 a^2\right ) b^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,a+b x\right )}{3 \left (1+a^2\right )^3}-\frac{\left (a \left (3-a^2\right ) b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,a+b x\right )}{3 \left (1+a^2\right )^3}\\ &=\frac{b}{6 \left (1+a^2\right ) x^2}-\frac{2 a b^2}{3 \left (1+a^2\right )^2 x}-\frac{\cot ^{-1}(a+b x)}{3 x^3}-\frac{a \left (3-a^2\right ) b^3 \tan ^{-1}(a+b x)}{3 \left (1+a^2\right )^3}+\frac{\left (1-3 a^2\right ) b^3 \log (x)}{3 \left (1+a^2\right )^3}-\frac{\left (1-3 a^2\right ) b^3 \log \left (1+(a+b x)^2\right )}{6 \left (1+a^2\right )^3}\\ \end{align*}
Mathematica [C] time = 0.131685, size = 126, normalized size = 0.98 \[ \frac{2 \left (1-3 a^2\right ) b^3 x^3 \log (x)+(a-i) b x \left ((a+i) \left (a^2-4 a b x+1\right )+i (a-i)^2 b^2 x^2 \log (a+b x+i)\right )-2 \left (a^2+1\right )^3 \cot ^{-1}(a+b x)+(-1+i a)^3 b^3 x^3 \log (-a-b x+i)}{6 \left (a^2+1\right )^3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 164, normalized size = 1.3 \begin{align*} -{\frac{{\rm arccot} \left (bx+a\right )}{3\,{x}^{3}}}+{\frac{{b}^{3}\ln \left ( 1+ \left ( bx+a \right ) ^{2} \right ){a}^{2}}{2\, \left ({a}^{2}+1 \right ) ^{3}}}-{\frac{{b}^{3}\ln \left ( 1+ \left ( bx+a \right ) ^{2} \right ) }{6\, \left ({a}^{2}+1 \right ) ^{3}}}+{\frac{{b}^{3}\arctan \left ( bx+a \right ){a}^{3}}{3\, \left ({a}^{2}+1 \right ) ^{3}}}-{\frac{{b}^{3}\arctan \left ( bx+a \right ) a}{ \left ({a}^{2}+1 \right ) ^{3}}}+{\frac{b}{ \left ( 6\,{a}^{2}+6 \right ){x}^{2}}}-{\frac{{b}^{3}\ln \left ( bx \right ){a}^{2}}{ \left ({a}^{2}+1 \right ) ^{3}}}+{\frac{{b}^{3}\ln \left ( bx \right ) }{3\, \left ({a}^{2}+1 \right ) ^{3}}}-{\frac{2\,a{b}^{2}}{3\, \left ({a}^{2}+1 \right ) ^{2}x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48873, size = 223, normalized size = 1.73 \begin{align*} \frac{1}{6} \,{\left (\frac{2 \,{\left (a^{3} - 3 \, a\right )} b^{2} \arctan \left (\frac{b^{2} x + a b}{b}\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} + \frac{{\left (3 \, a^{2} - 1\right )} b^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac{2 \,{\left (3 \, a^{2} - 1\right )} b^{2} \log \left (x\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac{4 \, a b x - a^{2} - 1}{{\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}}\right )} b - \frac{\operatorname{arccot}\left (b x + a\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.40332, size = 344, normalized size = 2.67 \begin{align*} \frac{2 \,{\left (a^{3} - 3 \, a\right )} b^{3} x^{3} \arctan \left (b x + a\right ) +{\left (3 \, a^{2} - 1\right )} b^{3} x^{3} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \,{\left (3 \, a^{2} - 1\right )} b^{3} x^{3} \log \left (x\right ) - 4 \,{\left (a^{3} + a\right )} b^{2} x^{2} +{\left (a^{4} + 2 \, a^{2} + 1\right )} b x - 2 \,{\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} \operatorname{arccot}\left (b x + a\right )}{6 \,{\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 33.0395, size = 1125, normalized size = 8.72 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13441, size = 242, normalized size = 1.88 \begin{align*} \frac{1}{6} \, b{\left (\frac{{\left (3 \, a^{2} b^{2} - b^{2}\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac{2 \,{\left (3 \, a^{2} b^{2} - b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} + \frac{2 \,{\left (a^{3} b^{3} - 3 \, a b^{3}\right )} \arctan \left (b x + a\right )}{{\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} b} + \frac{a^{4} + 2 \, a^{2} - 4 \,{\left (a^{3} b + a b\right )} x + 1}{{\left (a^{2} + 1\right )}^{3} x^{2}}\right )} - \frac{\arctan \left (\frac{1}{b x + a}\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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