Optimal. Leaf size=81 \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{3 x^3}+\frac {2}{3} b c^{3/2} F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )-\frac {2}{3} b c^{3/2} E\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )-\frac {2 b c \sqrt {1-c^2 x^4}}{3 x} \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4842, 12, 325, 307, 221, 1199, 424} \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{3 x^3}-\frac {2 b c \sqrt {1-c^2 x^4}}{3 x}+\frac {2}{3} b c^{3/2} F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )-\frac {2}{3} b c^{3/2} E\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 221
Rule 307
Rule 325
Rule 424
Rule 1199
Rule 4842
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c x^2\right )}{x^4} \, dx &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{3 x^3}+\frac {1}{3} b \int \frac {2 c}{x^2 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{3 x^3}+\frac {1}{3} (2 b c) \int \frac {1}{x^2 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {2 b c \sqrt {1-c^2 x^4}}{3 x}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{3 x^3}-\frac {1}{3} \left (2 b c^3\right ) \int \frac {x^2}{\sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {2 b c \sqrt {1-c^2 x^4}}{3 x}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{3 x^3}+\frac {1}{3} \left (2 b c^2\right ) \int \frac {1}{\sqrt {1-c^2 x^4}} \, dx-\frac {1}{3} \left (2 b c^2\right ) \int \frac {1+c x^2}{\sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {2 b c \sqrt {1-c^2 x^4}}{3 x}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{3 x^3}+\frac {2}{3} b c^{3/2} F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )-\frac {1}{3} \left (2 b c^2\right ) \int \frac {\sqrt {1+c x^2}}{\sqrt {1-c x^2}} \, dx\\ &=-\frac {2 b c \sqrt {1-c^2 x^4}}{3 x}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{3 x^3}-\frac {2}{3} b c^{3/2} E\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )+\frac {2}{3} b c^{3/2} F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.21, size = 89, normalized size = 1.10 \[ -\frac {a+2 b c x^2 \sqrt {1-c^2 x^4}+2 i b \sqrt {-c} c x^3 \left (E\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )\right )+b \sin ^{-1}\left (c x^2\right )}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (c x^{2}\right ) + a}{x^{4}}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (c x^{2}\right ) + a}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 97, normalized size = 1.20 \[ -\frac {a}{3 x^{3}}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{3 x^{3}}+\frac {2 c \left (-\frac {\sqrt {-c^{2} x^{4}+1}}{x}+\frac {\sqrt {c}\, \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \left (\EllipticF \left (x \sqrt {c}, i\right )-\EllipticE \left (x \sqrt {c}, i\right )\right )}{\sqrt {-c^{2} x^{4}+1}}\right )}{3}\right ) \]
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 \[ -\frac {{\left (2 \, c x^{3} \int \frac {e^{\left (\frac {1}{2} \, \log \left (c x^{2} + 1\right ) + \frac {1}{2} \, \log \left (-c x^{2} + 1\right )\right )}}{c^{4} x^{10} - c^{2} x^{6} - {\left (c^{2} x^{6} - x^{2}\right )} {\left (c x^{2} + 1\right )} {\left (c x^{2} - 1\right )}}\,{d x} + \arctan \left (c x^{2}, \sqrt {c x^{2} + 1} \sqrt {-c x^{2} + 1}\right )\right )} b}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.66, size = 60, normalized size = 0.74 \[ - \frac {a}{3 x^{3}} + \frac {b c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{6 x \Gamma \left (\frac {3}{4}\right )} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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