Optimal. Leaf size=65 \[ \frac {\text {Si}\left (\cos ^{-1}(a+b x)\right )}{2 b}+\frac {a+b x}{2 b \cos ^{-1}(a+b x)}+\frac {\sqrt {1-(a+b x)^2}}{2 b \cos ^{-1}(a+b x)^2} \]
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Rubi [A] time = 0.08, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4804, 4622, 4720, 4624, 3299} \[ \frac {\text {Si}\left (\cos ^{-1}(a+b x)\right )}{2 b}+\frac {a+b x}{2 b \cos ^{-1}(a+b x)}+\frac {\sqrt {1-(a+b x)^2}}{2 b \cos ^{-1}(a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4622
Rule 4624
Rule 4720
Rule 4804
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{-1}(a+b x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\cos ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ &=\frac {\sqrt {1-(a+b x)^2}}{2 b \cos ^{-1}(a+b x)^2}+\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \cos ^{-1}(x)^2} \, dx,x,a+b x\right )}{2 b}\\ &=\frac {\sqrt {1-(a+b x)^2}}{2 b \cos ^{-1}(a+b x)^2}+\frac {a+b x}{2 b \cos ^{-1}(a+b x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{\cos ^{-1}(x)} \, dx,x,a+b x\right )}{2 b}\\ &=\frac {\sqrt {1-(a+b x)^2}}{2 b \cos ^{-1}(a+b x)^2}+\frac {a+b x}{2 b \cos ^{-1}(a+b x)}+\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\cos ^{-1}(a+b x)\right )}{2 b}\\ &=\frac {\sqrt {1-(a+b x)^2}}{2 b \cos ^{-1}(a+b x)^2}+\frac {a+b x}{2 b \cos ^{-1}(a+b x)}+\frac {\text {Si}\left (\cos ^{-1}(a+b x)\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 65, normalized size = 1.00 \[ \frac {\text {Si}\left (\cos ^{-1}(a+b x)\right )}{2 b}+\frac {a+b x}{2 b \cos ^{-1}(a+b x)}+\frac {\sqrt {1-(a+b x)^2}}{2 b \cos ^{-1}(a+b x)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\arccos \left (b x + a\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 57, normalized size = 0.88 \[ \frac {\operatorname {Si}\left (\arccos \left (b x + a\right )\right )}{2 \, b} + \frac {b x + a}{2 \, b \arccos \left (b x + a\right )} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{2 \, b \arccos \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 53, normalized size = 0.82 \[ \frac {\frac {\sqrt {1-\left (b x +a \right )^{2}}}{2 \arccos \left (b x +a \right )^{2}}+\frac {b x +a}{2 \arccos \left (b x +a \right )}+\frac {\Si \left (\arccos \left (b x +a \right )\right )}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b \arctan \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1}, b x + a\right )^{2} \int \frac {1}{\arctan \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1}, b x + a\right )}\,{d x} - {\left (b x + a\right )} \arctan \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1}, b x + a\right ) - \sqrt {b x + a + 1} \sqrt {-b x - a + 1}}{2 \, b \arctan \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1}, b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {acos}\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \frac {1}{\operatorname {acos}^{3}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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