Optimal. Leaf size=82 \[ -\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{4 x^4}+\frac {3 b \csc ^{-1}\left (\frac {x}{c}\right )}{32 c^4}-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{16 c x^3}-\frac {3 b \sqrt {1-\frac {c^2}{x^2}}}{32 c^3 x} \]
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Rubi [A] time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4842, 12, 335, 321, 216} \[ -\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{4 x^4}-\frac {3 b \sqrt {1-\frac {c^2}{x^2}}}{32 c^3 x}-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{16 c x^3}+\frac {3 b \csc ^{-1}\left (\frac {x}{c}\right )}{32 c^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 321
Rule 335
Rule 4842
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{x^5} \, dx &=-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{4 x^4}-\frac {1}{4} b \int \frac {c}{\sqrt {1-\frac {c^2}{x^2}} x^6} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{4 x^4}-\frac {1}{4} (b c) \int \frac {1}{\sqrt {1-\frac {c^2}{x^2}} x^6} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{4 x^4}+\frac {1}{4} (b c) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{16 c x^3}-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{4 x^4}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx,x,\frac {1}{x}\right )}{16 c}\\ &=-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{16 c x^3}-\frac {3 b \sqrt {1-\frac {c^2}{x^2}}}{32 c^3 x}-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{4 x^4}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x^2}} \, dx,x,\frac {1}{x}\right )}{32 c^3}\\ &=-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{16 c x^3}-\frac {3 b \sqrt {1-\frac {c^2}{x^2}}}{32 c^3 x}+\frac {3 b \csc ^{-1}\left (\frac {x}{c}\right )}{32 c^4}-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 0.94 \[ -\frac {a}{4 x^4}+\frac {3 b \sin ^{-1}\left (\frac {c}{x}\right )}{32 c^4}+b \left (-\frac {3}{32 c^3 x}-\frac {1}{16 c x^3}\right ) \sqrt {\frac {x^2-c^2}{x^2}}-\frac {b \sin ^{-1}\left (\frac {c}{x}\right )}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 67, normalized size = 0.82 \[ -\frac {8 \, a c^{4} + {\left (8 \, b c^{4} - 3 \, b x^{4}\right )} \arcsin \left (\frac {c}{x}\right ) + {\left (2 \, b c^{3} x + 3 \, b c x^{3}\right )} \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}}}{32 \, c^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 111, normalized size = 1.35 \[ -\frac {\frac {8 \, b {\left (\frac {c^{2}}{x^{2}} - 1\right )}^{2} \arcsin \left (\frac {c}{x}\right )}{c^{3}} + \frac {16 \, b {\left (\frac {c^{2}}{x^{2}} - 1\right )} \arcsin \left (\frac {c}{x}\right )}{c^{3}} - \frac {2 \, b {\left (-\frac {c^{2}}{x^{2}} + 1\right )}^{\frac {3}{2}}}{c^{2} x} + \frac {5 \, b \arcsin \left (\frac {c}{x}\right )}{c^{3}} + \frac {5 \, b \sqrt {-\frac {c^{2}}{x^{2}} + 1}}{c^{2} x} + \frac {8 \, a c}{x^{4}}}{32 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 79, normalized size = 0.96 \[ -\frac {\frac {c^{4} a}{4 x^{4}}+b \left (\frac {c^{4} \arcsin \left (\frac {c}{x}\right )}{4 x^{4}}+\frac {c^{3} \sqrt {1-\frac {c^{2}}{x^{2}}}}{16 x^{3}}+\frac {3 c \sqrt {1-\frac {c^{2}}{x^{2}}}}{32 x}-\frac {3 \arcsin \left (\frac {c}{x}\right )}{32}\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 126, normalized size = 1.54 \[ -\frac {1}{32} \, {\left (c {\left (\frac {3 \, x^{3} {\left (-\frac {c^{2}}{x^{2}} + 1\right )}^{\frac {3}{2}} + 5 \, c^{2} x \sqrt {-\frac {c^{2}}{x^{2}} + 1}}{c^{4} x^{4} {\left (\frac {c^{2}}{x^{2}} - 1\right )}^{2} - 2 \, c^{6} x^{2} {\left (\frac {c^{2}}{x^{2}} - 1\right )} + c^{8}} + \frac {3 \, \arctan \left (\frac {x \sqrt {-\frac {c^{2}}{x^{2}} + 1}}{c}\right )}{c^{5}}\right )} + \frac {8 \, \arcsin \left (\frac {c}{x}\right )}{x^{4}}\right )} b - \frac {a}{4 \, x^{4}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {c}{x}\right )}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.06, size = 180, normalized size = 2.20 \[ - \frac {a}{4 x^{4}} - \frac {b c \left (\begin {cases} \frac {i}{4 x^{5} \sqrt {\frac {c^{2}}{x^{2}} - 1}} + \frac {i}{8 c^{2} x^{3} \sqrt {\frac {c^{2}}{x^{2}} - 1}} - \frac {3 i}{8 c^{4} x \sqrt {\frac {c^{2}}{x^{2}} - 1}} + \frac {3 i \operatorname {acosh}{\left (\frac {c}{x} \right )}}{8 c^{5}} & \text {for}\: \left |{\frac {c^{2}}{x^{2}}}\right | > 1 \\- \frac {1}{4 x^{5} \sqrt {- \frac {c^{2}}{x^{2}} + 1}} - \frac {1}{8 c^{2} x^{3} \sqrt {- \frac {c^{2}}{x^{2}} + 1}} + \frac {3}{8 c^{4} x \sqrt {- \frac {c^{2}}{x^{2}} + 1}} - \frac {3 \operatorname {asin}{\left (\frac {c}{x} \right )}}{8 c^{5}} & \text {otherwise} \end {cases}\right )}{4} - \frac {b \operatorname {asin}{\left (\frac {c}{x} \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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