Optimal. Leaf size=64 \[ \frac {2 \sqrt {1-(a+b x)^2}}{b \sqrt {\text {ArcCos}(a+b x)}}-\frac {2 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a+b x)}\right )}{b} \]
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Rubi [A]
time = 0.06, antiderivative size = 64, 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 = {4888, 4718,
4810, 3385, 3433} \begin {gather*} \frac {2 \sqrt {1-(a+b x)^2}}{b \sqrt {\text {ArcCos}(a+b x)}}-\frac {2 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a+b x)}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rule 4718
Rule 4810
Rule 4888
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{-1}(a+b x)^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\cos ^{-1}(x)^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {2 \sqrt {1-(a+b x)^2}}{b \sqrt {\cos ^{-1}(a+b x)}}+\frac {2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {\cos ^{-1}(x)}} \, dx,x,a+b x\right )}{b}\\ &=\frac {2 \sqrt {1-(a+b x)^2}}{b \sqrt {\cos ^{-1}(a+b x)}}-\frac {2 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a+b x)\right )}{b}\\ &=\frac {2 \sqrt {1-(a+b x)^2}}{b \sqrt {\cos ^{-1}(a+b x)}}-\frac {4 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a+b x)}\right )}{b}\\ &=\frac {2 \sqrt {1-(a+b x)^2}}{b \sqrt {\cos ^{-1}(a+b x)}}-\frac {2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a+b x)}\right )}{b}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.04, size = 97, normalized size = 1.52 \begin {gather*} -\frac {-2 \sqrt {1-(a+b x)^2}-i \sqrt {-i \text {ArcCos}(a+b x)} \text {Gamma}\left (\frac {1}{2},-i \text {ArcCos}(a+b x)\right )+i \sqrt {i \text {ArcCos}(a+b x)} \text {Gamma}\left (\frac {1}{2},i \text {ArcCos}(a+b x)\right )}{b \sqrt {\text {ArcCos}(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 84, normalized size = 1.31
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (2 \arccos \left (b x +a \right ) \pi \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}}{\sqrt {\pi }}\right )-\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {\pi }\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\right )}{b \sqrt {\pi }\, \arccos \left (b x +a \right )}\) | \(84\) |
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 \frac {1}{\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int \frac {1}{{\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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