Optimal. Leaf size=144 \[ \frac{a b^2 \sqrt{1-(a+b x)^2}}{2 \left (1-a^2\right )^2 x}+\frac{\left (2 a^2+1\right ) b^3 \tanh ^{-1}\left (\frac{1-a (a+b x)}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right )}{6 \left (1-a^2\right )^{5/2}}+\frac{b \sqrt{1-(a+b x)^2}}{6 \left (1-a^2\right ) x^2}-\frac{\cos ^{-1}(a+b x)}{3 x^3} \]
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Rubi [A] time = 0.179291, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4806, 4744, 745, 807, 725, 206} \[ \frac{a b^2 \sqrt{1-(a+b x)^2}}{2 \left (1-a^2\right )^2 x}+\frac{\left (2 a^2+1\right ) b^3 \tanh ^{-1}\left (\frac{1-a (a+b x)}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right )}{6 \left (1-a^2\right )^{5/2}}+\frac{b \sqrt{1-(a+b x)^2}}{6 \left (1-a^2\right ) x^2}-\frac{\cos ^{-1}(a+b x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 4806
Rule 4744
Rule 745
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}(a+b x)}{x^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos ^{-1}(x)}{\left (-\frac{a}{b}+\frac{x}{b}\right )^4} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\cos ^{-1}(a+b x)}{3 x^3}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{a}{b}+\frac{x}{b}\right )^3 \sqrt{1-x^2}} \, dx,x,a+b x\right )\\ &=\frac{b \sqrt{1-(a+b x)^2}}{6 \left (1-a^2\right ) x^2}-\frac{\cos ^{-1}(a+b x)}{3 x^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\frac{2 a}{b}+\frac{x}{b}}{\left (-\frac{a}{b}+\frac{x}{b}\right )^2 \sqrt{1-x^2}} \, dx,x,a+b x\right )}{6 \left (1-a^2\right )}\\ &=\frac{b \sqrt{1-(a+b x)^2}}{6 \left (1-a^2\right ) x^2}+\frac{a b^2 \sqrt{1-(a+b x)^2}}{2 \left (1-a^2\right )^2 x}-\frac{\cos ^{-1}(a+b x)}{3 x^3}-\frac{\left (\left (1+2 a^2\right ) b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{a}{b}+\frac{x}{b}\right ) \sqrt{1-x^2}} \, dx,x,a+b x\right )}{6 \left (1-a^2\right )^2}\\ &=\frac{b \sqrt{1-(a+b x)^2}}{6 \left (1-a^2\right ) x^2}+\frac{a b^2 \sqrt{1-(a+b x)^2}}{2 \left (1-a^2\right )^2 x}-\frac{\cos ^{-1}(a+b x)}{3 x^3}+\frac{\left (\left (1+2 a^2\right ) b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{b^2}-\frac{a^2}{b^2}-x^2} \, dx,x,\frac{\frac{1}{b}-\frac{a (a+b x)}{b}}{\sqrt{1-(a+b x)^2}}\right )}{6 \left (1-a^2\right )^2}\\ &=\frac{b \sqrt{1-(a+b x)^2}}{6 \left (1-a^2\right ) x^2}+\frac{a b^2 \sqrt{1-(a+b x)^2}}{2 \left (1-a^2\right )^2 x}-\frac{\cos ^{-1}(a+b x)}{3 x^3}+\frac{\left (1+2 a^2\right ) b^3 \tanh ^{-1}\left (\frac{1-a (a+b x)}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right )}{6 \left (1-a^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.184182, size = 168, normalized size = 1.17 \[ \frac{\sqrt{1-a^2} b x \left (-a^2+3 a b x+1\right ) \sqrt{-a^2-2 a b x-b^2 x^2+1}-\left (2 a^2+1\right ) b^3 x^3 \log (x)+\left (2 a^2+1\right ) b^3 x^3 \log \left (\sqrt{1-a^2} \sqrt{-a^2-2 a b x-b^2 x^2+1}-a^2-a b x+1\right )-2 \left (1-a^2\right )^{5/2} \cos ^{-1}(a+b x)}{6 \left (1-a^2\right )^{5/2} x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 227, normalized size = 1.6 \begin{align*} -{\frac{\arccos \left ( bx+a \right ) }{3\,{x}^{3}}}+{\frac{b}{ \left ( -6\,{a}^{2}+6 \right ){x}^{2}}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}+{\frac{{b}^{2}a}{2\, \left ( -{a}^{2}+1 \right ) ^{2}x}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}+{\frac{{b}^{3}{a}^{2}}{2}\ln \left ({\frac{1}{bx} \left ( -2\,{a}^{2}+2-2\,xab+2\,\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \right ) } \right ) \left ( -{a}^{2}+1 \right ) ^{-{\frac{5}{2}}}}+{\frac{{b}^{3}}{6}\ln \left ({\frac{1}{bx} \left ( -2\,{a}^{2}+2-2\,xab+2\,\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \right ) } \right ) \left ( -{a}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.54357, size = 1328, normalized size = 9.22 \begin{align*} \left [-\frac{{\left (2 \, a^{2} + 1\right )} \sqrt{-a^{2} + 1} b^{3} x^{3} \log \left (\frac{{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \,{\left (a^{3} - a\right )} b x - 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a b x + a^{2} - 1\right )} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 4 \,{\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 4 \,{\left (a^{6} - 3 \, a^{4} -{\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} + 3 \, a^{2} - 1\right )} \arccos \left (b x + a\right ) - 2 \,{\left (3 \,{\left (a^{3} - a\right )} b^{2} x^{2} -{\left (a^{4} - 2 \, a^{2} + 1\right )} b x\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{12 \,{\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}, -\frac{{\left (2 \, a^{2} + 1\right )} \sqrt{a^{2} - 1} b^{3} x^{3} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a b x + a^{2} - 1\right )} \sqrt{a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \,{\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + 2 \,{\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 2 \,{\left (a^{6} - 3 \, a^{4} -{\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3} + 3 \, a^{2} - 1\right )} \arccos \left (b x + a\right ) -{\left (3 \,{\left (a^{3} - a\right )} b^{2} x^{2} -{\left (a^{4} - 2 \, a^{2} + 1\right )} b x\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \,{\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acos}{\left (a + b x \right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.35558, size = 752, normalized size = 5.22 \begin{align*} -\frac{1}{3} \, b{\left (\frac{{\left (2 \, a^{2} b^{3} + b^{3}\right )} \arctan \left (\frac{\frac{{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt{a^{2} - 1}}\right )}{{\left (a^{4}{\left | b \right |} - 2 \, a^{2}{\left | b \right |} +{\left | b \right |}\right )} \sqrt{a^{2} - 1}} - \frac{\frac{4 \,{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )}^{2} a^{4} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} + 4 \, a^{4} b^{3} - \frac{11 \,{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )} a^{3} b^{3}}{b^{2} x + a b} - \frac{5 \,{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )}^{3} a^{3} b^{3}}{{\left (b^{2} x + a b\right )}^{3}} + \frac{7 \,{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )}^{2} a^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{2}} - a^{2} b^{3} + \frac{2 \,{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )} a b^{3}}{b^{2} x + a b} + \frac{2 \,{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )}^{3} a b^{3}}{{\left (b^{2} x + a b\right )}^{3}} - \frac{2 \,{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )}^{2} b^{3}}{{\left (b^{2} x + a b\right )}^{2}}}{{\left (a^{6}{\left | b \right |} - 2 \, a^{4}{\left | b \right |} + a^{2}{\left | b \right |}\right )}{\left (\frac{{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac{2 \,{\left (\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}^{2}}\right )} - \frac{\arccos \left (b x + a\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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