Optimal. Leaf size=89 \[ -\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\text {ArcCos}(a+b x)}}{2 b}+\frac {(a+b x) \text {ArcCos}(a+b x)^{3/2}}{b}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a+b x)}\right )}{2 b} \]
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Rubi [A]
time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4888, 4716,
4768, 4720, 3386, 3432} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a+b x)}\right )}{2 b}+\frac {(a+b x) \text {ArcCos}(a+b x)^{3/2}}{b}-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\text {ArcCos}(a+b x)}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3386
Rule 3432
Rule 4716
Rule 4720
Rule 4768
Rule 4888
Rubi steps
\begin {align*} \int \cos ^{-1}(a+b x)^{3/2} \, dx &=\frac {\text {Subst}\left (\int \cos ^{-1}(x)^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}+\frac {3 \text {Subst}\left (\int \frac {x \sqrt {\cos ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\cos ^{-1}(a+b x)}}{2 b}+\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {\cos ^{-1}(x)}} \, dx,x,a+b x\right )}{4 b}\\ &=-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\cos ^{-1}(a+b x)}}{2 b}+\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}+\frac {3 \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a+b x)\right )}{4 b}\\ &=-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\cos ^{-1}(a+b x)}}{2 b}+\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}+\frac {3 \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a+b x)}\right )}{2 b}\\ &=-\frac {3 \sqrt {1-(a+b x)^2} \sqrt {\cos ^{-1}(a+b x)}}{2 b}+\frac {(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}+\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a+b x)}\right )}{2 b}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.03, size = 76, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {-i \text {ArcCos}(a+b x)} \text {Gamma}\left (\frac {5}{2},-i \text {ArcCos}(a+b x)\right )+\sqrt {i \text {ArcCos}(a+b x)} \text {Gamma}\left (\frac {5}{2},i \text {ArcCos}(a+b x)\right )}{2 b \sqrt {\text {ArcCos}(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 105, normalized size = 1.18
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (-2 \arccos \left (b x +a \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, b x -2 \arccos \left (b x +a \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a +3 \sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {\pi }\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}-3 \pi \,\mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}}{\sqrt {\pi }}\right )\right )}{4 b \sqrt {\pi }}\) | \(105\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \operatorname {acos}^{\frac {3}{2}}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.52, size = 139, normalized size = 1.56 \begin {gather*} \frac {\arccos \left (b x + a\right )^{\frac {3}{2}} e^{\left (i \, \arccos \left (b x + a\right )\right )}}{2 \, b} + \frac {\arccos \left (b x + a\right )^{\frac {3}{2}} e^{\left (-i \, \arccos \left (b x + a\right )\right )}}{2 \, b} + \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (b x + a\right )}\right )}{16 \, b} - \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arccos \left (b x + a\right )}\right )}{16 \, b} + \frac {3 i \, \sqrt {\arccos \left (b x + a\right )} e^{\left (i \, \arccos \left (b x + a\right )\right )}}{4 \, b} - \frac {3 i \, \sqrt {\arccos \left (b x + a\right )} e^{\left (-i \, \arccos \left (b x + a\right )\right )}}{4 \, b} \end {gather*}
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 \begin {gather*} \int {\mathrm {acos}\left (a+b\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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