Optimal. Leaf size=49 \[ a x+b x \sin ^{-1}\left (c x^2\right )+\frac {2 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}}-\frac {2 b E\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}} \]
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Rubi [A] time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4840, 12, 307, 221, 1199, 424} \[ a x+b x \sin ^{-1}\left (c x^2\right )+\frac {2 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}}-\frac {2 b E\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 221
Rule 307
Rule 424
Rule 1199
Rule 4840
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=a x+b \int \sin ^{-1}\left (c x^2\right ) \, dx\\ &=a x+b x \sin ^{-1}\left (c x^2\right )-b \int \frac {2 c x^2}{\sqrt {1-c^2 x^4}} \, dx\\ &=a x+b x \sin ^{-1}\left (c x^2\right )-(2 b c) \int \frac {x^2}{\sqrt {1-c^2 x^4}} \, dx\\ &=a x+b x \sin ^{-1}\left (c x^2\right )+(2 b) \int \frac {1}{\sqrt {1-c^2 x^4}} \, dx-(2 b) \int \frac {1+c x^2}{\sqrt {1-c^2 x^4}} \, dx\\ &=a x+b x \sin ^{-1}\left (c x^2\right )+\frac {2 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}}-(2 b) \int \frac {\sqrt {1+c x^2}}{\sqrt {1-c x^2}} \, dx\\ &=a x+b x \sin ^{-1}\left (c x^2\right )-\frac {2 b E\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}}+\frac {2 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.80 \[ a x-\frac {2}{3} b c x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^2 x^4\right )+b x \sin ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b \arcsin \left (c x^{2}\right ) + a, x\right ) \]
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 \[ \int b \arcsin \left (c x^{2}\right ) + a\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 1.45 \[ a x +b \left (x \arcsin \left (c \,x^{2}\right )+\frac {2 \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \left (\EllipticF \left (x \sqrt {c}, i\right )-\EllipticE \left (x \sqrt {c}, i\right )\right )}{\sqrt {c}\, \sqrt {-c^{2} x^{4}+1}}\right ) \]
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 \[ {\left (x \arctan \left (c x^{2}, \sqrt {c x^{2} + 1} \sqrt {-c x^{2} + 1}\right ) + 2 \, c \int \frac {x^{2} e^{\left (\frac {1}{2} \, \log \left (c x^{2} + 1\right ) + \frac {1}{2} \, \log \left (-c x^{2} + 1\right )\right )}}{c^{4} x^{8} - c^{2} x^{4} + {\left (c^{2} x^{4} - 1\right )} e^{\left (\log \left (c x^{2} + 1\right ) + \log \left (-c x^{2} + 1\right )\right )}}\,{d x}\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int a+b\,\mathrm {asin}\left (c\,x^2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.00, size = 49, normalized size = 1.00 \[ a x + b \left (- \frac {c x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + x \operatorname {asin}{\left (c x^{2} \right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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