Optimal. Leaf size=64 \[ \frac {2 \sqrt {1-(a+b x)^2}}{b \sqrt {\cos ^{-1}(a+b x)}}-\frac {2 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a+b x)}\right )}{b} \]
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Rubi [A] time = 0.09, 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 = {4804, 4622, 4724, 3304, 3352} \[ \frac {2 \sqrt {1-(a+b x)^2}}{b \sqrt {\cos ^{-1}(a+b x)}}-\frac {2 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4622
Rule 4724
Rule 4804
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{-1}(a+b x)^{3/2}} \, dx &=\frac {\operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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] time = 0.05, size = 97, normalized size = 1.52 \[ -\frac {-2 \sqrt {1-(a+b x)^2}-i \sqrt {-i \cos ^{-1}(a+b x)} \Gamma \left (\frac {1}{2},-i \cos ^{-1}(a+b x)\right )+i \sqrt {i \cos ^{-1}(a+b x)} \Gamma \left (\frac {1}{2},i \cos ^{-1}(a+b x)\right )}{b \sqrt {\cos ^{-1}(a+b x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{\arccos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 84, normalized size = 1.31 \[ -\frac {\sqrt {2}\, \left (2 \pi \arccos \left (b x +a \right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (b x +a \right )}}{\sqrt {\pi }}\right )-\sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (b x +a \right )}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\right )}{b \sqrt {\pi }\, \arccos \left (b x +a \right )} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {1}{{\mathrm {acos}\left (a+b\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\operatorname {acos}^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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