Integrand size = 19, antiderivative size = 195 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt {-1+c^2 x^2}}{6 c^5 \sqrt {c^2 x^2}}+\frac {b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {b c d^3 x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{6 e \sqrt {c^2 x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5345, 457, 90, 65, 211} \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {b c d^3 x \arctan \left (\sqrt {c^2 x^2-1}\right )}{6 e \sqrt {c^2 x^2}}+\frac {b e x \left (c^2 x^2-1\right )^{3/2} \left (3 c^2 d+2 e\right )}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{6 c^5 \sqrt {c^2 x^2}} \]
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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 )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^3}{x \sqrt {-1+c^2 x^2}} \, dx}{6 e \sqrt {c^2 x^2}} \\ & = \frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {(b c x) \text {Subst}\left (\int \frac {(d+e x)^3}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{12 e \sqrt {c^2 x^2}} \\ & = \frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {(b c x) \text {Subst}\left (\int \left (\frac {e \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{c^4 \sqrt {-1+c^2 x}}+\frac {d^3}{x \sqrt {-1+c^2 x}}+\frac {e^2 \left (3 c^2 d+2 e\right ) \sqrt {-1+c^2 x}}{c^4}+\frac {e^3 \left (-1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{12 e \sqrt {c^2 x^2}} \\ & = \frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt {-1+c^2 x^2}}{6 c^5 \sqrt {c^2 x^2}}+\frac {b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {\left (b c d^3 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{12 e \sqrt {c^2 x^2}} \\ & = \frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt {-1+c^2 x^2}}{6 c^5 \sqrt {c^2 x^2}}+\frac {b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {\left (b d^3 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 )}{6 c e \sqrt {c^2 x^2}} \\ & = \frac {b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) x \sqrt {-1+c^2 x^2}}{6 c^5 \sqrt {c^2 x^2}}+\frac {b e \left (3 c^2 d+2 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{18 c^5 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^3 \left (a+b \csc ^{-1}(c x)\right )}{6 e}+\frac {b c d^3 x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{6 e \sqrt {c^2 x^2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.64 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{90} x \left (15 a x \left (3 d^2+3 d e x^2+e^2 x^4\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (8 e^2+2 c^2 e \left (15 d+2 e x^2\right )+3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )\right )}{c^5}+15 b x \left (3 d^2+3 d e x^2+e^2 x^4\right ) \csc ^{-1}(c x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(351\) vs. \(2(169)=338\).
Time = 0.97 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.81
method | result | size |
parts | \(\frac {a \left (e \,x^{2}+d \right )^{3}}{6 e}+\frac {b \,\operatorname {arccsc}\left (c x \right ) e^{2} x^{6}}{6}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d e \,x^{4}}{2}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} x^{2}}{2}+\frac {b \,d^{3} \operatorname {arccsc}\left (c x \right )}{6 e}+\frac {b \left (c^{2} x^{2}-1\right ) x^{3} e^{2}}{30 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) x d e}{6 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {2 b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{45 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{2 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{3 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {4 b \,e^{2} \left (c^{2} x^{2}-1\right )}{45 c^{7} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\) | \(352\) |
derivativedivides | \(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \,c^{2} \operatorname {arccsc}\left (c x \right ) d^{3}}{6 e}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} c^{2} x^{2}}{2}+\frac {b \,c^{2} e \,\operatorname {arccsc}\left (c x \right ) d \,x^{4}}{2}+\frac {b \,c^{2} e^{2} \operatorname {arccsc}\left (c x \right ) x^{6}}{6}-\frac {b c \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b e \left (c^{2} x^{2}-1\right ) x d}{6 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right ) x^{3}}{30 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {2 b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {4 b \,e^{2} \left (c^{2} x^{2}-1\right )}{45 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{2}}\) | \(376\) |
default | \(\frac {\frac {a \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{6 c^{4} e}+\frac {b \,c^{2} \operatorname {arccsc}\left (c x \right ) d^{3}}{6 e}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} c^{2} x^{2}}{2}+\frac {b \,c^{2} e \,\operatorname {arccsc}\left (c x \right ) d \,x^{4}}{2}+\frac {b \,c^{2} e^{2} \operatorname {arccsc}\left (c x \right ) x^{6}}{6}-\frac {b c \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{6 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {b e \left (c^{2} x^{2}-1\right ) x d}{6 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \,e^{2} \left (c^{2} x^{2}-1\right ) x^{3}}{30 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b e \left (c^{2} x^{2}-1\right ) d}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {2 b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {4 b \,e^{2} \left (c^{2} x^{2}-1\right )}{45 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{2}}\) | \(376\) |
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Time = 0.31 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.78 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {15 \, a c^{6} e^{2} x^{6} + 45 \, a c^{6} d e x^{4} + 45 \, a c^{6} d^{2} x^{2} + 15 \, {\left (b c^{6} e^{2} x^{6} + 3 \, b c^{6} d e x^{4} + 3 \, b c^{6} d^{2} x^{2}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (3 \, b c^{4} e^{2} x^{4} + 45 \, b c^{4} d^{2} + 30 \, b c^{2} d e + 8 \, b e^{2} + {\left (15 \, b c^{4} d e + 4 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{90 \, c^{6}} \]
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Time = 2.78 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.81 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d^{2} x^{2}}{2} + \frac {a d e x^{4}}{2} + \frac {a e^{2} x^{6}}{6} + \frac {b d^{2} x^{2} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b d e x^{4} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b e^{2} x^{6} \operatorname {acsc}{\left (c x \right )}}{6} + \frac {b d^{2} \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} + \frac {b d e \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{2 c} + \frac {b e^{2} \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} \]
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Time = 0.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.97 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{6} \, a e^{2} x^{6} + \frac {1}{2} \, a d e x^{4} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d^{2} + \frac {1}{6} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d e + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b e^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1160 vs. \(2 (169) = 338\).
Time = 0.39 (sec) , antiderivative size = 1160, normalized size of antiderivative = 5.95 \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
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Timed out. \[ \int x \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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