Optimal. Leaf size=63 \[ \frac{b \tanh ^{-1}\left (\frac{1-a (a+b x)}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right )}{\sqrt{1-a^2}}-\frac{\cos ^{-1}(a+b x)}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.075788, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4806, 4744, 725, 206} \[ \frac{b \tanh ^{-1}\left (\frac{1-a (a+b x)}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right )}{\sqrt{1-a^2}}-\frac{\cos ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4806
Rule 4744
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}(a+b x)}{x^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos ^{-1}(x)}{\left (-\frac{a}{b}+\frac{x}{b}\right )^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\cos ^{-1}(a+b x)}{x}-\operatorname{Subst}\left (\int \frac{1}{\left (-\frac{a}{b}+\frac{x}{b}\right ) \sqrt{1-x^2}} \, dx,x,a+b x\right )\\ &=-\frac{\cos ^{-1}(a+b x)}{x}+\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 )\\ &=-\frac{\cos ^{-1}(a+b x)}{x}+\frac{b \tanh ^{-1}\left (\frac{b \left (\frac{1}{b}-\frac{a (a+b x)}{b}\right )}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right )}{\sqrt{1-a^2}}\\ \end{align*}
Mathematica [A] time = 0.0550105, size = 79, normalized size = 1.25 \[ \frac{b \left (\log \left (\sqrt{1-a^2} \sqrt{-a^2-2 a b x-b^2 x^2+1}-a^2-a b x+1\right )-\log (x)\right )}{\sqrt{1-a^2}}-\frac{\cos ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 77, normalized size = 1.2 \begin{align*} -{\frac{\arccos \left ( bx+a \right ) }{x}}+{b\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 ){\frac{1}{\sqrt{-{a}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 3.01077, size = 857, normalized size = 13.6 \begin{align*} \left [-\frac{\sqrt{-a^{2} + 1} b x \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 ) + 2 \,{\left (a^{2} - 1\right )} x \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^{2} -{\left (a^{2} - 1\right )} x - 1\right )} \arccos \left (b x + a\right )}{2 \,{\left (a^{2} - 1\right )} x}, -\frac{\sqrt{a^{2} - 1} b x \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 ) +{\left (a^{2} - 1\right )} x \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 ) +{\left (a^{2} -{\left (a^{2} - 1\right )} x - 1\right )} \arccos \left (b x + a\right )}{{\left (a^{2} - 1\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acos}{\left (a + b x \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.31559, size = 107, normalized size = 1.7 \begin{align*} -\frac{2 \, b^{2} \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 )}{\sqrt{a^{2} - 1}{\left | b \right |}} - \frac{\arccos \left (b x + a\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]