Integrand size = 21, antiderivative size = 242 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (6 c^4 d^2+8 c^2 d e+3 e^2\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b e \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right ) \]
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Time = 0.16 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {272, 45, 5347, 12, 1265, 785} \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e x \left (c^2 x^2-1\right )^{5/2} \left (8 c^2 d+9 e\right )}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (6 c^4 d^2+16 c^2 d e+9 e^2\right )}{72 c^7 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (6 c^4 d^2+8 c^2 d e+3 e^2\right )}{24 c^7 \sqrt {c^2 x^2}} \]
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Rule 12
Rule 45
Rule 272
Rule 785
Rule 1265
Rule 5347
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{24 \sqrt {c^2 x^2}} \\ & = \frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \text {Subst}\left (\int \frac {x \left (6 d^2+8 d e x+3 e^2 x^2\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}} \\ & = \frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \text {Subst}\left (\int \left (\frac {6 c^4 d^2+8 c^2 d e+3 e^2}{c^6 \sqrt {-1+c^2 x}}+\frac {\left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {e \left (8 c^2 d+9 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {3 e^2 \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}} \\ & = \frac {b \left (6 c^4 d^2+8 c^2 d e+3 e^2\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (6 c^4 d^2+16 c^2 d e+9 e^2\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b e \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e^2 x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{4} d^2 x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \csc ^{-1}(c x)\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.66 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {x \left (105 a x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (144 e^2+8 c^2 e \left (56 d+9 e x^2\right )+c^4 \left (420 d^2+224 d e x^2+54 e^2 x^4\right )+3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )\right )}{c^7}+105 b x^3 \left (6 d^2+8 d e x^2+3 e^2 x^4\right ) \csc ^{-1}(c x)\right )}{2520} \]
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Time = 1.05 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.82
method | result | size |
parts | \(a \left (\frac {1}{8} e^{2} x^{8}+\frac {1}{3} d e \,x^{6}+\frac {1}{4} x^{4} d^{2}\right )+\frac {b \left (\frac {c^{4} \operatorname {arccsc}\left (c x \right ) e^{2} x^{8}}{8}+\frac {c^{4} \operatorname {arccsc}\left (c x \right ) d e \,x^{6}}{3}+\frac {\operatorname {arccsc}\left (c x \right ) d^{2} x^{4} c^{4}}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e^{2} x^{6}+168 c^{6} d e \,x^{4}+210 c^{6} d^{2} x^{2}+54 c^{4} e^{2} x^{4}+224 c^{4} d e \,x^{2}+420 c^{4} d^{2}+72 c^{2} e^{2} x^{2}+448 c^{2} d e +144 e^{2}\right )}{2520 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )}{c^{4}}\) | \(198\) |
derivativedivides | \(\frac {-\frac {a \left (\frac {c^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}-\frac {b \,c^{4} \operatorname {arccsc}\left (c x \right ) d^{4}}{24 e^{2}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\operatorname {arccsc}\left (c x \right ) d \,x^{6}}{3}+\frac {b \,c^{4} e^{2} \operatorname {arccsc}\left (c x \right ) x^{8}}{8}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{24 e^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) c x \,d^{2}}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b c e \left (c^{2} x^{2}-1\right ) x^{3} d}{15 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b c \,e^{2} \left (c^{2} x^{2}-1\right ) x^{5}}{56 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {4 b e \left (c^{2} x^{2}-1\right ) x d}{45 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \,e^{2} \left (c^{2} x^{2}-1\right ) x^{3}}{140 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {8 b e \left (c^{2} x^{2}-1\right ) d}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{35 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 b \,e^{2} \left (c^{2} x^{2}-1\right )}{35 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{4}}\) | \(511\) |
default | \(\frac {-\frac {a \left (\frac {c^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )^{3}}{3}-\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4}}{4}\right )}{2 c^{4} e^{2}}-\frac {b \,c^{4} \operatorname {arccsc}\left (c x \right ) d^{4}}{24 e^{2}}+\frac {b \,\operatorname {arccsc}\left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {b \,c^{4} e \,\operatorname {arccsc}\left (c x \right ) d \,x^{6}}{3}+\frac {b \,c^{4} e^{2} \operatorname {arccsc}\left (c x \right ) x^{8}}{8}+\frac {b \,c^{3} \sqrt {c^{2} x^{2}-1}\, d^{4} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{24 e^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) c x \,d^{2}}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b c e \left (c^{2} x^{2}-1\right ) x^{3} d}{15 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b c \,e^{2} \left (c^{2} x^{2}-1\right ) x^{5}}{56 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {b \left (c^{2} x^{2}-1\right ) d^{2}}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}+\frac {4 b e \left (c^{2} x^{2}-1\right ) x d}{45 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {3 b \,e^{2} \left (c^{2} x^{2}-1\right ) x^{3}}{140 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {8 b e \left (c^{2} x^{2}-1\right ) d}{45 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}+\frac {b \left (c^{2} x^{2}-1\right ) x \,e^{2}}{35 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {2 b \,e^{2} \left (c^{2} x^{2}-1\right )}{35 c^{5} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}}{c^{4}}\) | \(511\) |
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Time = 0.34 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.77 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {315 \, a c^{8} e^{2} x^{8} + 840 \, a c^{8} d e x^{6} + 630 \, a c^{8} d^{2} x^{4} + 105 \, {\left (3 \, b c^{8} e^{2} x^{8} + 8 \, b c^{8} d e x^{6} + 6 \, b c^{8} d^{2} x^{4}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (45 \, b c^{6} e^{2} x^{6} + 420 \, b c^{4} d^{2} + 448 \, b c^{2} d e + 6 \, {\left (28 \, b c^{6} d e + 9 \, b c^{4} e^{2}\right )} x^{4} + 144 \, b e^{2} + 2 \, {\left (105 \, b c^{6} d^{2} + 112 \, b c^{4} d e + 36 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{2520 \, c^{8}} \]
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Time = 4.39 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.04 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {acsc}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {acsc}{\left (c x \right )}}{8} + \frac {b d^{2} \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} + \frac {b d e \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{3 c} + \frac {b e^{2} \left (\begin {cases} \frac {x^{6} \sqrt {c^{2} x^{2} - 1}}{7 c} + \frac {6 x^{4} \sqrt {c^{2} x^{2} - 1}}{35 c^{3}} + \frac {8 x^{2} \sqrt {c^{2} x^{2} - 1}}{35 c^{5}} + \frac {16 \sqrt {c^{2} x^{2} - 1}}{35 c^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{6} \sqrt {- c^{2} x^{2} + 1}}{7 c} + \frac {6 i x^{4} \sqrt {- c^{2} x^{2} + 1}}{35 c^{3}} + \frac {8 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{5}} + \frac {16 i \sqrt {- c^{2} x^{2} + 1}}{35 c^{7}} & \text {otherwise} \end {cases}\right )}{8 c} \]
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Time = 0.21 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.05 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{8} \, a e^{2} x^{8} + \frac {1}{3} \, a d e x^{6} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d e + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arccsc}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} + 21 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 35 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1706 vs. \(2 (212) = 424\).
Time = 0.45 (sec) , antiderivative size = 1706, normalized size of antiderivative = 7.05 \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
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Timed out. \[ \int x^3 \left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^3\,{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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