Optimal. Leaf size=171 \[ -\frac {2 d \text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}},x\right )}{b c}-\frac {2 \sqrt {\pi } d \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {2 \sqrt {\pi } d \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}} \]
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Rubi [A] time = 0.78, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {d-c^2 d x^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {d-c^2 d x^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(2 d) \int \frac {\sqrt {1-c^2 x^2}}{x^2 \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac {(4 c d) \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(4 d) \operatorname {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac {(2 d) \int \left (-\frac {c^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}+\frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}\right ) \, dx}{b c}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(4 d) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}+\frac {\cos (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}+\frac {(2 c d) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(2 d) \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac {\left (2 d \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac {\left (2 d \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac {\left (4 d \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b^2}-\frac {\left (4 d \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b^2}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {2 d \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {2 d \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2}}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}\\ \end {align*}
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Mathematica [A] time = 2.02, size = 0, normalized size = 0.00 \[ \int \frac {d-c^2 d x^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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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(-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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maple [A] time = 0.86, size = 0, normalized size = 0.00 \[ \int \frac {-c^{2} d \,x^{2}+d}{x \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {c^{2} d x^{2} - d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {d-c^2\,d\,x^2}{x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ - d \left (\int \frac {c^{2} x^{2}}{a x \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a x \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b x \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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