Integrand size = 19, antiderivative size = 161 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (20 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}+\frac {b e x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (20 c^2 d+9 e\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{120 c^4 \sqrt {c^2 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {14, 5347, 12, 470, 327, 223, 212} \[ \int x^2 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (20 c^2 d+9 e\right )}{120 c^4 \sqrt {c^2 x^2}}+\frac {b e x^4 \sqrt {c^2 x^2-1}}{20 c \sqrt {c^2 x^2}}+\frac {b x^2 \sqrt {c^2 x^2-1} \left (20 c^2 d+9 e\right )}{120 c^3 \sqrt {c^2 x^2}} \]
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Rule 12
Rule 14
Rule 212
Rule 223
Rule 327
Rule 470
Rule 5347
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^2 \left (5 d+3 e x^2\right )}{15 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^2 \left (5 d+3 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{15 \sqrt {c^2 x^2}} \\ & = \frac {b e x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (b c \left (-20 d-\frac {9 e}{c^2}\right ) x\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{60 \sqrt {c^2 x^2}} \\ & = \frac {b \left (20 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}+\frac {b e x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (b \left (-20 d-\frac {9 e}{c^2}\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{120 c \sqrt {c^2 x^2}} \\ & = \frac {b \left (20 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}+\frac {b e x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )-\frac {\left (b \left (-20 d-\frac {9 e}{c^2}\right ) x\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{120 c \sqrt {c^2 x^2}} \\ & = \frac {b \left (20 c^2 d+9 e\right ) x^2 \sqrt {-1+c^2 x^2}}{120 c^3 \sqrt {c^2 x^2}}+\frac {b e x^4 \sqrt {-1+c^2 x^2}}{20 c \sqrt {c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (20 c^2 d+9 e\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{120 c^4 \sqrt {c^2 x^2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.75 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {c^2 x^2 \left (8 a c^3 x \left (5 d+3 e x^2\right )+b \sqrt {1-\frac {1}{c^2 x^2}} \left (9 e+c^2 \left (20 d+6 e x^2\right )\right )\right )+8 b c^5 x^3 \left (5 d+3 e x^2\right ) \csc ^{-1}(c x)+b \left (20 c^2 d+9 e\right ) \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{120 c^5} \]
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Time = 0.63 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.58
method | result | size |
parts | \(a \left (\frac {1}{5} e \,x^{5}+\frac {1}{3} d \,x^{3}\right )+\frac {b \,\operatorname {arccsc}\left (c x \right ) e \,x^{5}}{5}+\frac {b \,\operatorname {arccsc}\left (c x \right ) x^{3} d}{3}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} e}{20 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}-1\right ) e}{40 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\) | \(254\) |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d \,c^{3} x^{3}}{3}+\frac {b \,c^{3} \operatorname {arccsc}\left (c x \right ) e \,x^{5}}{5}+\frac {b \left (c^{2} x^{2}-1\right ) d}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} e}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b \left (c^{2} x^{2}-1\right ) e}{40 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{3}}\) | \(267\) |
default | \(\frac {\frac {a \left (\frac {1}{3} d \,c^{5} x^{3}+\frac {1}{5} e \,c^{5} x^{5}\right )}{c^{2}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d \,c^{3} x^{3}}{3}+\frac {b \,c^{3} \operatorname {arccsc}\left (c x \right ) e \,x^{5}}{5}+\frac {b \left (c^{2} x^{2}-1\right ) d}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x^{2} e}{20 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \sqrt {c^{2} x^{2}-1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {3 b \left (c^{2} x^{2}-1\right ) e}{40 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{40 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{3}}\) | \(267\) |
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Time = 0.34 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.06 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {24 \, a c^{5} e x^{5} + 40 \, a c^{5} d x^{3} + 8 \, {\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3} - 5 \, b c^{5} d - 3 \, b c^{5} e\right )} \operatorname {arccsc}\left (c x\right ) - 16 \, {\left (5 \, b c^{5} d + 3 \, b c^{5} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (20 \, b c^{2} d + 9 \, b e\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (6 \, b c^{3} e x^{3} + {\left (20 \, b c^{3} d + 9 \, b c e\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{120 \, c^{5}} \]
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Time = 4.33 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.83 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d x^{3}}{3} + \frac {a e x^{5}}{5} + \frac {b d x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b e x^{5} \operatorname {acsc}{\left (c x \right )}}{5} + \frac {b d \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} + \frac {b e \left (\begin {cases} \frac {c x^{5}}{4 \sqrt {c^{2} x^{2} - 1}} + \frac {x^{3}}{8 c \sqrt {c^{2} x^{2} - 1}} - \frac {3 x}{8 c^{3} \sqrt {c^{2} x^{2} - 1}} + \frac {3 \operatorname {acosh}{\left (c x \right )}}{8 c^{4}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{5}}{4 \sqrt {- c^{2} x^{2} + 1}} - \frac {i x^{3}}{8 c \sqrt {- c^{2} x^{2} + 1}} + \frac {3 i x}{8 c^{3} \sqrt {- c^{2} x^{2} + 1}} - \frac {3 i \operatorname {asin}{\left (c x \right )}}{8 c^{4}} & \text {otherwise} \end {cases}\right )}{5 c} \]
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Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.44 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{5} \, a e x^{5} + \frac {1}{3} \, a d x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arccsc}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b e \]
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Leaf count of result is larger than twice the leaf count of optimal. 822 vs. \(2 (139) = 278\).
Time = 1.16 (sec) , antiderivative size = 822, normalized size of antiderivative = 5.11 \[ \int x^2 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{960} \, {\left (\frac {6 \, b e x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {6 \, a e x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}{c} + \frac {3 \, b e x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c^{2}} + \frac {40 \, b d x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {40 \, a d x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c} + \frac {30 \, b e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {30 \, a e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{3}} + \frac {40 \, b d x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{2}} + \frac {24 \, b e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{4}} + \frac {120 \, b d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {120 \, a d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{3}} + \frac {60 \, b e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {60 \, a e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{5}} + \frac {160 \, b d \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} - \frac {160 \, b d \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{4}} + \frac {72 \, b e \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{6}} - \frac {72 \, b e \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{6}} + \frac {120 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {120 \, a d}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {60 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{7} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {60 \, a e}{c^{7} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {40 \, b d}{c^{6} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {24 \, b e}{c^{8} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {40 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {40 \, a d}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {30 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {30 \, a e}{c^{9} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} - \frac {3 \, b e}{c^{10} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {6 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{11} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}} + \frac {6 \, a e}{c^{11} x^{5} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{5}}\right )} c \]
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Timed out. \[ \int x^2 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^2\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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