Optimal. Leaf size=60 \[ -\frac {1}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {1}{9} b c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {a+b \csc ^{-1}(c x)}{3 x^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5329, 272, 45}
\begin {gather*} -\frac {a+b \csc ^{-1}(c x)}{3 x^3}+\frac {1}{9} b c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {1}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 5329
Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x^4} \, dx &=-\frac {a+b \csc ^{-1}(c x)}{3 x^3}-\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^5} \, dx}{3 c}\\ &=-\frac {a+b \csc ^{-1}(c x)}{3 x^3}+\frac {b \text {Subst}\left (\int \frac {x}{\sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{6 c}\\ &=-\frac {a+b \csc ^{-1}(c x)}{3 x^3}+\frac {b \text {Subst}\left (\int \left (\frac {c^2}{\sqrt {1-\frac {x}{c^2}}}-c^2 \sqrt {1-\frac {x}{c^2}}\right ) \, dx,x,\frac {1}{x^2}\right )}{6 c}\\ &=-\frac {1}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {1}{9} b c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {a+b \csc ^{-1}(c x)}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 59, normalized size = 0.98 \begin {gather*} -\frac {a}{3 x^3}+b \left (-\frac {2 c^3}{9}-\frac {c}{9 x^2}\right ) \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \csc ^{-1}(c x)}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 75, normalized size = 1.25
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {a}{3 c^{3} x^{3}}+b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\) | \(75\) |
default | \(c^{3} \left (-\frac {a}{3 c^{3} x^{3}}+b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 58, normalized size = 0.97 \begin {gather*} \frac {1}{9} \, b {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arccsc}\left (c x\right )}{x^{3}}\right )} - \frac {a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 39, normalized size = 0.65 \begin {gather*} -\frac {3 \, b \operatorname {arccsc}\left (c x\right ) + {\left (2 \, b c^{2} x^{2} + b\right )} \sqrt {c^{2} x^{2} - 1} + 3 \, a}{9 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.15, size = 112, normalized size = 1.87 \begin {gather*} - \frac {a}{3 x^{3}} - \frac {b \operatorname {acsc}{\left (c x \right )}}{3 x^{3}} - \frac {b \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 87, normalized size = 1.45 \begin {gather*} \frac {1}{9} \, {\left (b c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {3 \, b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - \frac {3 \, b c \arcsin \left (\frac {1}{c x}\right )}{x} - \frac {3 \, a}{c x^{3}}\right )} 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 {1}{c\,x}\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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