Integrand size = 12, antiderivative size = 89 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{40 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{20 c}+\frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {3 b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{40 c^5} \]
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Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5329, 272, 44, 65, 214} \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {3 b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{40 c^5}+\frac {b x^4 \sqrt {1-\frac {1}{c^2 x^2}}}{20 c}+\frac {3 b x^2 \sqrt {1-\frac {1}{c^2 x^2}}}{40 c^3} \]
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Rule 44
Rule 65
Rule 214
Rule 272
Rule 5329
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \int \frac {x^3}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{5 c} \\ & = \frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {b \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{10 c} \\ & = \frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{20 c}+\frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {(3 b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{40 c^3} \\ & = \frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{40 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{20 c}+\frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {(3 b) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{80 c^5} \\ & = \frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{40 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{20 c}+\frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {(3 b) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )}{40 c^3} \\ & = \frac {3 b \sqrt {1-\frac {1}{c^2 x^2}} x^2}{40 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^4}{20 c}+\frac {1}{5} x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {3 b \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{40 c^5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a x^5}{5}+b \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}} \left (\frac {3 x^2}{40 c^3}+\frac {x^4}{20 c}\right )+\frac {1}{5} b x^5 \csc ^{-1}(c x)+\frac {3 b \log \left (x \left (1+\sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}\right )\right )}{40 c^5} \]
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Time = 0.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.58
method | result | size |
parts | \(\frac {a \,x^{5}}{5}+\frac {x^{5} b \,\operatorname {arccsc}\left (c x \right )}{5}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2}}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right )}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\) | \(141\) |
derivativedivides | \(\frac {\frac {a \,c^{5} x^{5}}{5}+\frac {\operatorname {arccsc}\left (c x \right ) b \,c^{5} x^{5}}{5}+\frac {b \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{5}}\) | \(148\) |
default | \(\frac {\frac {a \,c^{5} x^{5}}{5}+\frac {\operatorname {arccsc}\left (c x \right ) b \,c^{5} x^{5}}{5}+\frac {b \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{5}}\) | \(148\) |
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Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.19 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {8 \, a c^{5} x^{5} - 16 \, b c^{5} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 8 \, {\left (b c^{5} x^{5} - b c^{5}\right )} \operatorname {arccsc}\left (c x\right ) - 3 \, b \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (2 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} - 1}}{40 \, c^{5}} \]
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Time = 3.38 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.97 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a x^{5}}{5} + \frac {b x^{5} \operatorname {acsc}{\left (c x \right )}}{5} + \frac {b \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} \]
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Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.48 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{5} \, a x^{5} + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arccsc}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b \]
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Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (75) = 150\).
Time = 0.63 (sec) , antiderivative size = 480, normalized size of antiderivative = 5.39 \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{320} \, {\left (\frac {2 \, b x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {2 \, a x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}{c} + \frac {b x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c^{2}} + \frac {10 \, b x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {10 \, a x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{3}} + \frac {8 \, b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{4}} + \frac {20 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {20 \, a x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{5}} + \frac {24 \, b \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {24 \, b \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{6}} + \frac {20 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{7} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {20 \, a}{c^{7} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {8 \, b}{c^{8} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {10 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {10 \, a}{c^{9} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} - \frac {b}{c^{10} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {2 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{11} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} + \frac {2 \, a}{c^{11} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}\right )} c \]
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Timed out. \[ \int x^4 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^4\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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