Optimal. Leaf size=62 \[ -\frac {b \sqrt {1-\frac {c^2}{x^2}}}{3 c^3}+\frac {b \left (1-\frac {c^2}{x^2}\right )^{3/2}}{9 c^3}-\frac {a+b \text {ArcSin}\left (\frac {c}{x}\right )}{3 x^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4926, 12, 272,
45} \begin {gather*} -\frac {a+b \text {ArcSin}\left (\frac {c}{x}\right )}{3 x^3}+\frac {b \left (1-\frac {c^2}{x^2}\right )^{3/2}}{9 c^3}-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{3 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 272
Rule 4926
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{x^4} \, dx &=-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{3 x^3}-\frac {1}{3} b \int \frac {c}{\sqrt {1-\frac {c^2}{x^2}} x^5} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{3 x^3}-\frac {1}{3} (b c) \int \frac {1}{\sqrt {1-\frac {c^2}{x^2}} x^5} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{3 x^3}+\frac {1}{6} (b c) \text {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{3 x^3}+\frac {1}{6} (b c) \text {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {b \sqrt {1-\frac {c^2}{x^2}}}{3 c^3}+\frac {b \left (1-\frac {c^2}{x^2}\right )^{3/2}}{9 c^3}-\frac {a+b \sin ^{-1}\left (\frac {c}{x}\right )}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 60, normalized size = 0.97 \begin {gather*} -\frac {a}{3 x^3}+b \left (-\frac {2}{9 c^3}-\frac {1}{9 c x^2}\right ) \sqrt {\frac {-c^2+x^2}{x^2}}-\frac {b \text {ArcSin}\left (\frac {c}{x}\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 67, normalized size = 1.08
method | result | size |
derivativedivides | \(-\frac {\frac {a \,c^{3}}{3 x^{3}}+b \left (\frac {c^{3} \arcsin \left (\frac {c}{x}\right )}{3 x^{3}}+\frac {c^{2} \sqrt {1-\frac {c^{2}}{x^{2}}}}{9 x^{2}}+\frac {2 \sqrt {1-\frac {c^{2}}{x^{2}}}}{9}\right )}{c^{3}}\) | \(67\) |
default | \(-\frac {\frac {a \,c^{3}}{3 x^{3}}+b \left (\frac {c^{3} \arcsin \left (\frac {c}{x}\right )}{3 x^{3}}+\frac {c^{2} \sqrt {1-\frac {c^{2}}{x^{2}}}}{9 x^{2}}+\frac {2 \sqrt {1-\frac {c^{2}}{x^{2}}}}{9}\right )}{c^{3}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 58, normalized size = 0.94 \begin {gather*} \frac {1}{9} \, {\left (c {\left (\frac {{\left (-\frac {c^{2}}{x^{2}} + 1\right )}^{\frac {3}{2}}}{c^{4}} - \frac {3 \, \sqrt {-\frac {c^{2}}{x^{2}} + 1}}{c^{4}}\right )} - \frac {3 \, \arcsin \left (\frac {c}{x}\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.37, size = 57, normalized size = 0.92 \begin {gather*} -\frac {3 \, b c^{3} \arcsin \left (\frac {c}{x}\right ) + 3 \, a c^{3} + {\left (b c^{2} x + 2 \, b x^{3}\right )} \sqrt {-\frac {c^{2} - x^{2}}{x^{2}}}}{9 \, c^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.79, size = 112, normalized size = 1.81 \begin {gather*} - \frac {a}{3 x^{3}} - \frac {b c \left (\begin {cases} \frac {\sqrt {-1 + \frac {x^{2}}{c^{2}}}}{3 c x^{3}} + \frac {2 \sqrt {-1 + \frac {x^{2}}{c^{2}}}}{3 c^{3} x} & \text {for}\: \left |{\frac {x^{2}}{c^{2}}}\right | > 1 \\\frac {i \sqrt {1 - \frac {x^{2}}{c^{2}}}}{3 c x^{3}} + \frac {2 i \sqrt {1 - \frac {x^{2}}{c^{2}}}}{3 c^{3} x} & \text {otherwise} \end {cases}\right )}{3} - \frac {b \operatorname {asin}{\left (\frac {c}{x} \right )}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 88, normalized size = 1.42 \begin {gather*} -\frac {\frac {3 \, b {\left (\frac {c^{2}}{x^{2}} - 1\right )} \arcsin \left (\frac {c}{x}\right )}{c x} - \frac {b {\left (-\frac {c^{2}}{x^{2}} + 1\right )}^{\frac {3}{2}}}{c^{2}} + \frac {3 \, b \arcsin \left (\frac {c}{x}\right )}{c x} + \frac {3 \, b \sqrt {-\frac {c^{2}}{x^{2}} + 1}}{c^{2}} + \frac {3 \, a c}{x^{3}}}{9 \, c} \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 \frac {a+b\,\mathrm {asin}\left (\frac {c}{x}\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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