Optimal. Leaf size=82 \[ \frac {6 \sqrt {1-(a+b x)^2}}{b}-\frac {6 (a+b x) \text {ArcCos}(a+b x)}{b}-\frac {3 \sqrt {1-(a+b x)^2} \text {ArcCos}(a+b x)^2}{b}+\frac {(a+b x) \text {ArcCos}(a+b x)^3}{b} \]
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Rubi [A]
time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4888, 4716,
4768, 267} \begin {gather*} \frac {(a+b x) \text {ArcCos}(a+b x)^3}{b}-\frac {3 \sqrt {1-(a+b x)^2} \text {ArcCos}(a+b x)^2}{b}-\frac {6 (a+b x) \text {ArcCos}(a+b x)}{b}+\frac {6 \sqrt {1-(a+b x)^2}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 4716
Rule 4768
Rule 4888
Rubi steps
\begin {align*} \int \cos ^{-1}(a+b x)^3 \, dx &=\frac {\text {Subst}\left (\int \cos ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \cos ^{-1}(a+b x)^3}{b}+\frac {3 \text {Subst}\left (\int \frac {x \cos ^{-1}(x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {3 \sqrt {1-(a+b x)^2} \cos ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \cos ^{-1}(a+b x)^3}{b}-\frac {6 \text {Subst}\left (\int \cos ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=-\frac {6 (a+b x) \cos ^{-1}(a+b x)}{b}-\frac {3 \sqrt {1-(a+b x)^2} \cos ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \cos ^{-1}(a+b x)^3}{b}-\frac {6 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {6 \sqrt {1-(a+b x)^2}}{b}-\frac {6 (a+b x) \cos ^{-1}(a+b x)}{b}-\frac {3 \sqrt {1-(a+b x)^2} \cos ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \cos ^{-1}(a+b x)^3}{b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 74, normalized size = 0.90 \begin {gather*} \frac {6 \sqrt {1-(a+b x)^2}-6 (a+b x) \text {ArcCos}(a+b x)-3 \sqrt {1-(a+b x)^2} \text {ArcCos}(a+b x)^2+(a+b x) \text {ArcCos}(a+b x)^3}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 71, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {\arccos \left (b x +a \right )^{3} \left (b x +a \right )-3 \arccos \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}+6 \sqrt {1-\left (b x +a \right )^{2}}-6 \left (b x +a \right ) \arccos \left (b x +a \right )}{b}\) | \(71\) |
default | \(\frac {\arccos \left (b x +a \right )^{3} \left (b x +a \right )-3 \arccos \left (b x +a \right )^{2} \sqrt {1-\left (b x +a \right )^{2}}+6 \sqrt {1-\left (b x +a \right )^{2}}-6 \left (b x +a \right ) \arccos \left (b x +a \right )}{b}\) | \(71\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.73, size = 66, normalized size = 0.80 \begin {gather*} \frac {{\left (b x + a\right )} \arccos \left (b x + a\right )^{3} - 6 \, {\left (b x + a\right )} \arccos \left (b x + a\right ) - 3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (\arccos \left (b x + a\right )^{2} - 2\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 109, normalized size = 1.33 \begin {gather*} \begin {cases} \frac {a \operatorname {acos}^{3}{\left (a + b x \right )}}{b} - \frac {6 a \operatorname {acos}{\left (a + b x \right )}}{b} + x \operatorname {acos}^{3}{\left (a + b x \right )} - 6 x \operatorname {acos}{\left (a + b x \right )} - \frac {3 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname {acos}^{2}{\left (a + b x \right )}}{b} + \frac {6 \sqrt {- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{b} & \text {for}\: b \neq 0 \\x \operatorname {acos}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 78, normalized size = 0.95 \begin {gather*} \frac {{\left (b x + a\right )} \arccos \left (b x + a\right )^{3}}{b} - \frac {3 \, \sqrt {-{\left (b x + a\right )}^{2} + 1} \arccos \left (b x + a\right )^{2}}{b} - \frac {6 \, {\left (b x + a\right )} \arccos \left (b x + a\right )}{b} + \frac {6 \, \sqrt {-{\left (b x + a\right )}^{2} + 1}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 60, normalized size = 0.73 \begin {gather*} -\frac {\left (3\,{\mathrm {acos}\left (a+b\,x\right )}^2-6\right )\,\sqrt {1-{\left (a+b\,x\right )}^2}}{b}-\frac {\left (6\,\mathrm {acos}\left (a+b\,x\right )-{\mathrm {acos}\left (a+b\,x\right )}^3\right )\,\left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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