Integrand size = 17, antiderivative size = 138 \[ \int x \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (2 c^2 d+e\right ) x \sqrt {-1+c^2 x^2}}{4 c^3 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{3/2}}{12 c^3 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b c d^2 x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{4 e \sqrt {c^2 x^2}} \]
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Time = 0.07 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5345, 457, 90, 65, 211} \[ \int x \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b c d^2 x \arctan \left (\sqrt {c^2 x^2-1}\right )}{4 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right )}{4 c^3 \sqrt {c^2 x^2}}+\frac {b e x \left (c^2 x^2-1\right )^{3/2}}{12 c^3 \sqrt {c^2 x^2}} \]
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Rule 65
Rule 90
Rule 211
Rule 457
Rule 5345
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^2}{x \sqrt {-1+c^2 x^2}} \, dx}{4 e \sqrt {c^2 x^2}} \\ & = \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {(b c x) \text {Subst}\left (\int \frac {(d+e x)^2}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{8 e \sqrt {c^2 x^2}} \\ & = \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {(b c x) \text {Subst}\left (\int \left (\frac {e \left (2 c^2 d+e\right )}{c^2 \sqrt {-1+c^2 x}}+\frac {d^2}{x \sqrt {-1+c^2 x}}+\frac {e^2 \sqrt {-1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{8 e \sqrt {c^2 x^2}} \\ & = \frac {b \left (2 c^2 d+e\right ) x \sqrt {-1+c^2 x^2}}{4 c^3 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{3/2}}{12 c^3 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {\left (b c d^2 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{8 e \sqrt {c^2 x^2}} \\ & = \frac {b \left (2 c^2 d+e\right ) x \sqrt {-1+c^2 x^2}}{4 c^3 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{3/2}}{12 c^3 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {\left (b d^2 x\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{4 c e \sqrt {c^2 x^2}} \\ & = \frac {b \left (2 c^2 d+e\right ) x \sqrt {-1+c^2 x^2}}{4 c^3 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{3/2}}{12 c^3 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{4 e}+\frac {b c d^2 x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{4 e \sqrt {c^2 x^2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57 \[ \int x \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {x \left (3 a c^3 x \left (2 d+e x^2\right )+b \sqrt {1-\frac {1}{c^2 x^2}} \left (2 e+c^2 \left (6 d+e x^2\right )\right )+3 b c^3 x \left (2 d+e x^2\right ) \csc ^{-1}(c x)\right )}{12 c^3} \]
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Time = 0.74 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.57
method | result | size |
parts | \(\frac {a \left (e \,x^{2}+d \right )^{2}}{4 e}+\frac {b \,\operatorname {arccsc}\left (c x \right ) e \,x^{4}}{4}+\frac {b \,\operatorname {arccsc}\left (c x \right ) x^{2} d}{2}+\frac {b \,d^{2} \operatorname {arccsc}\left (c x \right )}{4 e}+\frac {b \left (c^{2} x^{2}-1\right ) x e}{12 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) d}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b e \left (c^{2} x^{2}-1\right )}{6 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\) | \(217\) |
derivativedivides | \(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}{4 c^{2} e}+\frac {b \,c^{2} \operatorname {arccsc}\left (c x \right ) d^{2}}{4 e}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d \,c^{2} x^{2}}{2}+\frac {b \,c^{2} e \,\operatorname {arccsc}\left (c x \right ) x^{4}}{4}-\frac {b c \sqrt {c^{2} x^{2}-1}\, d^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) d}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b e \left (c^{2} x^{2}-1\right ) x}{12 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b e \left (c^{2} x^{2}-1\right )}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{2}}\) | \(238\) |
default | \(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}{4 c^{2} e}+\frac {b \,c^{2} \operatorname {arccsc}\left (c x \right ) d^{2}}{4 e}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d \,c^{2} x^{2}}{2}+\frac {b \,c^{2} e \,\operatorname {arccsc}\left (c x \right ) x^{4}}{4}-\frac {b c \sqrt {c^{2} x^{2}-1}\, d^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) d}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b e \left (c^{2} x^{2}-1\right ) x}{12 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b e \left (c^{2} x^{2}-1\right )}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{2}}\) | \(238\) |
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Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.62 \[ \int x \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {3 \, a c^{4} e x^{4} + 6 \, a c^{4} d x^{2} + 3 \, {\left (b c^{4} e x^{4} + 2 \, b c^{4} d x^{2}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (b c^{2} e x^{2} + 6 \, b c^{2} d + 2 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, c^{4}} \]
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Time = 1.58 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.28 \[ \int x \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d x^{2}}{2} + \frac {a e x^{4}}{4} + \frac {b d x^{2} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b e x^{4} \operatorname {acsc}{\left (c x \right )}}{4} + \frac {b d \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} + \frac {b e \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} \]
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Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.71 \[ \int x \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{4} \, a e x^{4} + \frac {1}{2} \, a d x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b e \]
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Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (118) = 236\).
Time = 0.33 (sec) , antiderivative size = 556, normalized size of antiderivative = 4.03 \[ \int x \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{192} \, {\left (\frac {3 \, b e 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 e x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c} + \frac {2 \, b e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{2}} + \frac {24 \, b d x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {24 \, a d x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c} + \frac {12 \, b e 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 e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{3}} + \frac {48 \, b d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{2}} + \frac {18 \, b e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{4}} + \frac {48 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {48 \, a d}{c^{3}} + \frac {18 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {18 \, a e}{c^{5}} - \frac {48 \, b d}{c^{4} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {18 \, b e}{c^{6} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {24 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {24 \, a d}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {12 \, b e \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 e}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {2 \, b e}{c^{8} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {3 \, b e \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 e}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}\right )} c \]
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Timed out. \[ \int x \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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