Optimal. Leaf size=103 \[ \frac {a b^2 \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{2 \left (1-a^2\right )^{3/2}}+\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\cos ^{-1}(a+b x)}{2 x^2} \]
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Rubi [A] time = 0.11, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4806, 4744, 731, 725, 206} \[ \frac {a b^2 \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{2 \left (1-a^2\right )^{3/2}}+\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\cos ^{-1}(a+b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 731
Rule 4744
Rule 4806
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a+b x)}{x^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\cos ^{-1}(a+b x)}{2 x^2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\cos ^{-1}(a+b x)}{2 x^2}-\frac {(a b) \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 \left (1-a^2\right )}\\ &=\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\cos ^{-1}(a+b x)}{2 x^2}+\frac {(a b) \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 )}{2 \left (1-a^2\right )}\\ &=\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\cos ^{-1}(a+b x)}{2 x^2}+\frac {a b^2 \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{2 \left (1-a^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 126, normalized size = 1.22 \[ -\frac {\cos ^{-1}(a+b x)-\frac {b x \left (\sqrt {1-a^2} \sqrt {-a^2-2 a b x-b^2 x^2+1}+a b x \log \left (\sqrt {1-a^2} \sqrt {-a^2-2 a b x-b^2 x^2+1}-a^2-a b x+1\right )-a b x \log (x)\right )}{\left (1-a^2\right )^{3/2}}}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 482, normalized size = 4.68 \[ \left [-\frac {\sqrt {-a^{2} + 1} a b^{2} x^{2} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} b x + 2 \, {\left (a^{4} - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} - 2 \, a^{2} + 1\right )} \arccos \left (b x + a\right )}{4 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}, \frac {\sqrt {a^{2} - 1} a b^{2} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} b x - {\left (a^{4} - {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2} - 2 \, a^{2} + 1\right )} \arccos \left (b x + a\right )}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 242, normalized size = 2.35 \[ {\left (\frac {a b^{2} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{{\left (a^{2} {\left | b \right |} - {\left | b \right |}\right )} \sqrt {a^{2} - 1}} - \frac {a b^{2} - \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} b^{2}}{b^{2} x + a b}}{{\left (a^{3} {\left | b \right |} - a {\left | b \right |}\right )} {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}}\right )} b - \frac {\arccos \left (b x + a\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 118, normalized size = 1.15 \[ -\frac {\arccos \left (b x +a \right )}{2 x^{2}}+\frac {b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) x}+\frac {b^{2} a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acos}\left (a+b\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acos}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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