Optimal. Leaf size=99 \[ \frac {\sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} c^2}-\frac {\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} c^2} \]
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Rubi [A] time = 0.17, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4636, 4406, 12, 3306, 3305, 3351, 3304, 3352} \[ \frac {\sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {\pi } \sqrt {b}}\right )}{2 \sqrt {b} c^2}-\frac {\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rule 4636
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+b \cos ^{-1}(c x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{c^2}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{c^2}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{2 c^2}\\ &=-\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{2 c^2}+\frac {\sin \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{2 c^2}\\ &=-\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{b c^2}+\frac {\sin \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{b c^2}\\ &=-\frac {\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} c^2}+\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{2 \sqrt {b} c^2}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 91, normalized size = 0.92 \[ -\frac {\sqrt {\pi } \sqrt {\frac {1}{b}} \left (\cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {\pi }}\right )-\sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {\pi }}\right )\right )}{2 c^2} \]
Antiderivative was successfully verified.
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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 [A] time = 1.28, size = 139, normalized size = 1.40 \[ \frac {\sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {b \arccos \left (c x\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}}\right ) e^{\left (-\frac {2 \, a i}{b}\right )}}{4 \, {\left (\frac {b^{\frac {3}{2}} i}{{\left | b \right |}} - \sqrt {b}\right )} c^{2}} + \frac {\sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}}\right ) e^{\left (\frac {2 \, a i}{b}\right )}}{4 \, \sqrt {b} c^{2} {\left (\frac {b i}{{\left | b \right |}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 80, normalized size = 0.81 \[ -\frac {\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \left (\cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )-\sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )\right )}{2 c^{2}} \]
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 \[ \int \frac {x}{\sqrt {b \arccos \left (c x\right ) + a}}\,{d x} \]
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 \[ \int \frac {x}{\sqrt {a+b\,\mathrm {acos}\left (c\,x\right )}} \,d x \]
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 \[ \int \frac {x}{\sqrt {a + b \operatorname {acos}{\left (c x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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