Optimal. Leaf size=64 \[ \frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{12 c}+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}}}{6 c^3} \]
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Rubi [A] time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5221, 271, 191} \[ \frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{12 c}+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}}}{6 c^3} \]
Antiderivative was successfully verified.
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Rule 191
Rule 271
Rule 5221
Rubi steps
\begin {align*} \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \int \frac {x^2}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{4 c}\\ &=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{6 c^3}\\ &=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 62, normalized size = 0.97 \[ \frac {a x^4}{4}+b \sqrt {\frac {c^2 x^2-1}{c^2 x^2}} \left (\frac {x}{6 c^3}+\frac {x^3}{12 c}\right )+\frac {1}{4} b x^4 \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.51, size = 52, normalized size = 0.81 \[ \frac {3 \, b c^{4} x^{4} \operatorname {arccsc}\left (c x\right ) + 3 \, a c^{4} x^{4} + {\left (b c^{2} x^{2} + 2 \, b\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 352, normalized size = 5.50 \[ \frac {1}{192} \, {\left (\frac {3 \, b x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {3 \, a x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c} + \frac {2 \, b x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{2}} + \frac {12 \, b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {12 \, a x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{3}} + \frac {18 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{4}} + \frac {18 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {18 \, a}{c^{5}} - \frac {18 \, b}{c^{6} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {12 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {12 \, a}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {2 \, b}{c^{8} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {3 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {3 \, a}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 74, normalized size = 1.16 \[ \frac {\frac {a \,c^{4} x^{4}}{4}+b \left (\frac {c^{4} x^{4} \mathrm {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 59, normalized size = 0.92 \[ \frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b \]
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 \[ \int x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.06, size = 107, normalized size = 1.67 \[ \frac {a x^{4}}{4} + \frac {b x^{4} \operatorname {acsc}{\left (c x \right )}}{4} + \frac {b \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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