Optimal. Leaf size=49 \[ a x+b x \text {ArcSin}\left (c x^2\right )-\frac {2 b E\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}}+\frac {2 b F\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}} \]
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Rubi [A]
time = 0.03, 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 = {4924, 12, 313,
227, 1213, 435} \begin {gather*} a x+b x \text {ArcSin}\left (c x^2\right )+\frac {2 b F\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}}-\frac {2 b E\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 227
Rule 313
Rule 435
Rule 1213
Rule 4924
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 39, normalized size = 0.80 \begin {gather*} a x+b x \text {ArcSin}\left (c x^2\right )-\frac {2}{3} b c x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^2 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 71, normalized size = 1.45
method | result | size |
default | \(a x +b \left (x \arcsin \left (c \,x^{2}\right )+\frac {2 \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \left (\EllipticF \left (x \sqrt {c}, i\right )-\EllipticE \left (x \sqrt {c}, i\right )\right )}{\sqrt {c}\, \sqrt {-c^{2} x^{4}+1}}\right )\) | \(71\) |
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 [A]
time = 0.40, size = 41, normalized size = 0.84 \begin {gather*} \frac {b c x^{2} \arcsin \left (c x^{2}\right ) + a c x^{2} + 2 \, \sqrt {-c^{2} x^{4} + 1} b}{c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.61, size = 49, normalized size = 1.00 \begin {gather*} 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 ) \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 = -0.02 \begin {gather*} \int a+b\,\mathrm {asin}\left (c\,x^2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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