Integrand size = 19, antiderivative size = 153 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (3 c^2 d+2 e\right ) x \sqrt {-1+c^2 x^2}}{12 c^5 \sqrt {c^2 x^2}}+\frac {b \left (3 c^2 d+4 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{36 c^5 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right ) \]
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Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 5347, 12, 457, 78} \[ \int x^3 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (3 c^2 d+4 e\right )}{36 c^5 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (3 c^2 d+2 e\right )}{12 c^5 \sqrt {c^2 x^2}}+\frac {b e x \left (c^2 x^2-1\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}} \]
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Rule 12
Rule 14
Rule 78
Rule 457
Rule 5347
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^3 \left (3 d+2 e x^2\right )}{12 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^3 \left (3 d+2 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{12 \sqrt {c^2 x^2}} \\ & = \frac {1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \text {Subst}\left (\int \frac {x (3 d+2 e x)}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{24 \sqrt {c^2 x^2}} \\ & = \frac {1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \text {Subst}\left (\int \left (\frac {3 c^2 d+2 e}{c^4 \sqrt {-1+c^2 x}}+\frac {\left (3 c^2 d+4 e\right ) \sqrt {-1+c^2 x}}{c^4}+\frac {2 e \left (-1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{24 \sqrt {c^2 x^2}} \\ & = \frac {b \left (3 c^2 d+2 e\right ) x \sqrt {-1+c^2 x^2}}{12 c^5 \sqrt {c^2 x^2}}+\frac {b \left (3 c^2 d+4 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{36 c^5 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{5/2}}{30 c^5 \sqrt {c^2 x^2}}+\frac {1}{4} d x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \csc ^{-1}(c x)\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.63 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{180} x \left (15 a x^3 \left (3 d+2 e x^2\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (16 e+c^2 \left (30 d+8 e x^2\right )+3 c^4 \left (5 d x^2+2 e x^4\right )\right )}{c^5}+15 b x^3 \left (3 d+2 e x^2\right ) \csc ^{-1}(c x)\right ) \]
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Time = 0.72 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79
method | result | size |
parts | \(a \left (\frac {1}{6} e \,x^{6}+\frac {1}{4} d \,x^{4}\right )+\frac {b \left (\frac {c^{4} \operatorname {arccsc}\left (c x \right ) e \,x^{6}}{6}+\frac {\operatorname {arccsc}\left (c x \right ) c^{4} d \,x^{4}}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (6 c^{4} e \,x^{4}+15 c^{4} d \,x^{2}+8 c^{2} e \,x^{2}+30 c^{2} d +16 e \right )}{180 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )}{c^{4}}\) | \(121\) |
derivativedivides | \(\frac {-\frac {a \left (\frac {c^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}{2}-\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{3}\right )}{2 c^{2} e^{2}}-\frac {b \,c^{4} \operatorname {arccsc}\left (c x \right ) d^{3}}{12 e^{2}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d \,c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\operatorname {arccsc}\left (c x \right ) x^{6}}{6}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{12 e^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) c x d}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b c e \left (c^{2} x^{2}-1\right ) x^{3}}{30 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {2 b e \left (c^{2} x^{2}-1\right ) x}{45 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {4 b e \left (c^{2} x^{2}-1\right )}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{4}}\) | \(333\) |
default | \(\frac {-\frac {a \left (\frac {c^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}{2}-\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{3}\right )}{2 c^{2} e^{2}}-\frac {b \,c^{4} \operatorname {arccsc}\left (c x \right ) d^{3}}{12 e^{2}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d \,c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\operatorname {arccsc}\left (c x \right ) x^{6}}{6}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d^{3} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{12 e^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) c x d}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b c e \left (c^{2} x^{2}-1\right ) x^{3}}{30 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {2 b e \left (c^{2} x^{2}-1\right ) x}{45 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {4 b e \left (c^{2} x^{2}-1\right )}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{4}}\) | \(333\) |
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Time = 0.31 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.69 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {30 \, a c^{6} e x^{6} + 45 \, a c^{6} d x^{4} + 15 \, {\left (2 \, b c^{6} e x^{6} + 3 \, b c^{6} d x^{4}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (6 \, b c^{4} e x^{4} + 30 \, b c^{2} d + {\left (15 \, b c^{4} d + 8 \, b c^{2} e\right )} x^{2} + 16 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{180 \, c^{6}} \]
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Time = 2.41 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.78 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d x^{4}}{4} + \frac {a e x^{6}}{6} + \frac {b d x^{4} \operatorname {acsc}{\left (c x \right )}}{4} + \frac {b e x^{6} \operatorname {acsc}{\left (c x \right )}}{6} + \frac {b d \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} + \frac {b e \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} \]
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Time = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.93 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{6} \, a e x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b e \]
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Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (131) = 262\).
Time = 0.34 (sec) , antiderivative size = 900, normalized size of antiderivative = 5.88 \[ \int x^3 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
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Timed out. \[ \int x^3 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^3\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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