Optimal. Leaf size=137 \[ \frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \text {ArcCos}(c x)}}-\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 \sqrt {2 \pi } S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c} \]
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Rubi [A]
time = 0.17, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4718, 4810,
3387, 3386, 3432, 3385, 3433} \begin {gather*} -\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 \sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \text {ArcCos}(c x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4718
Rule 4810
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cos ^{-1}(c x)\right )^{3/2}} \, dx &=\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \cos ^{-1}(c x)}}+\frac {(2 c) \int \frac {x}{\sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}} \, dx}{b}\\ &=\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \cos ^{-1}(c x)}}-\frac {2 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{b c}\\ &=\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \cos ^{-1}(c x)}}-\frac {\left (2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{b c}-\frac {\left (2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{b c}\\ &=\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \cos ^{-1}(c x)}}-\frac {\left (4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{b^2 c}-\frac {\left (4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{b^2 c}\\ &=\frac {2 \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \cos ^{-1}(c x)}}-\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 \sqrt {2 \pi } S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} c}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.12, size = 150, normalized size = 1.09 \begin {gather*} -\frac {i e^{-\frac {i a}{b}} \left (2 i e^{\frac {i a}{b}} \sqrt {1-c^2 x^2}-\sqrt {-\frac {i (a+b \text {ArcCos}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {i (a+b \text {ArcCos}(c x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \text {ArcCos}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {i (a+b \text {ArcCos}(c x))}{b}\right )\right )}{b c \sqrt {a+b \text {ArcCos}(c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 157, normalized size = 1.15
method | result | size |
default | \(-\frac {2 \left (\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )-\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right )+\sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right )\right )}{c b \sqrt {a +b \arccos \left (c x \right )}}\) | \(157\) |
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 [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}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {3}{2}}}\, 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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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