Integrand size = 19, antiderivative size = 196 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (4 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b \left (4 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{5/2}}{120 c^7 \sqrt {c^2 x^2}}+\frac {b e x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right ) \]
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Time = 0.11 (sec) , antiderivative size = 196, 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} \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \left (c^2 x^2-1\right )^{5/2} \left (4 c^2 d+9 e\right )}{120 c^7 \sqrt {c^2 x^2}}+\frac {b x \left (c^2 x^2-1\right )^{3/2} \left (8 c^2 d+9 e\right )}{72 c^7 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (4 c^2 d+3 e\right )}{24 c^7 \sqrt {c^2 x^2}}+\frac {b e x \left (c^2 x^2-1\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}} \]
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Rule 12
Rule 14
Rule 78
Rule 457
Rule 5347
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{24 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \int \frac {x^5 \left (4 d+3 e x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{24 \sqrt {c^2 x^2}} \\ & = \frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \text {Subst}\left (\int \frac {x^2 (4 d+3 e x)}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{48 \sqrt {c^2 x^2}} \\ & = \frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right )+\frac {(b c x) \text {Subst}\left (\int \left (\frac {4 c^2 d+3 e}{c^6 \sqrt {-1+c^2 x}}+\frac {\left (8 c^2 d+9 e\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {\left (4 c^2 d+9 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {3 e \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 (4 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2}}{24 c^7 \sqrt {c^2 x^2}}+\frac {b \left (8 c^2 d+9 e\right ) x \left (-1+c^2 x^2\right )^{3/2}}{72 c^7 \sqrt {c^2 x^2}}+\frac {b \left (4 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 x \left (-1+c^2 x^2\right )^{7/2}}{56 c^7 \sqrt {c^2 x^2}}+\frac {1}{6} d x^6 \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{8} e x^8 \left (a+b \csc ^{-1}(c x)\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.59 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {x \left (105 a x^5 \left (4 d+3 e x^2\right )+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} \left (144 e+8 c^2 \left (28 d+9 e x^2\right )+2 c^4 \left (56 d x^2+27 e x^4\right )+c^6 \left (84 d x^4+45 e x^6\right )\right )}{c^7}+105 b x^5 \left (4 d+3 e x^2\right ) \csc ^{-1}(c x)\right )}{2520} \]
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Time = 0.64 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.71
method | result | size |
parts | \(a \left (\frac {1}{8} e \,x^{8}+\frac {1}{6} x^{6} d \right )+\frac {b \left (\frac {c^{6} \operatorname {arccsc}\left (c x \right ) e \,x^{8}}{8}+\frac {\operatorname {arccsc}\left (c x \right ) d \,x^{6} c^{6}}{6}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )}{c^{6}}\) | \(139\) |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccsc}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\operatorname {arccsc}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}}\) | \(152\) |
default | \(\frac {\frac {a \left (\frac {1}{6} c^{8} d \,x^{6}+\frac {1}{8} e \,c^{8} x^{8}\right )}{c^{2}}+\frac {b \left (\frac {\operatorname {arccsc}\left (c x \right ) d \,c^{8} x^{6}}{6}+\frac {\operatorname {arccsc}\left (c x \right ) e \,c^{8} x^{8}}{8}+\frac {\left (c^{2} x^{2}-1\right ) \left (45 c^{6} e \,x^{6}+84 c^{6} d \,x^{4}+54 c^{4} e \,x^{4}+112 c^{4} d \,x^{2}+72 c^{2} e \,x^{2}+224 c^{2} d +144 e \right )}{2520 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{6}}\) | \(152\) |
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Time = 0.31 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.65 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {315 \, a c^{8} e x^{8} + 420 \, a c^{8} d x^{6} + 105 \, {\left (3 \, b c^{8} e x^{8} + 4 \, b c^{8} d x^{6}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (45 \, b c^{6} e x^{6} + 6 \, {\left (14 \, b c^{6} d + 9 \, b c^{4} e\right )} x^{4} + 224 \, b c^{2} d + 8 \, {\left (14 \, b c^{4} d + 9 \, b c^{2} e\right )} x^{2} + 144 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{2520 \, c^{8}} \]
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Time = 3.86 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.86 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a d x^{6}}{6} + \frac {a e x^{8}}{8} + \frac {b d x^{6} \operatorname {acsc}{\left (c x \right )}}{6} + \frac {b e x^{8} \operatorname {acsc}{\left (c x \right )}}{8} + \frac {b d \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} - 1}}{5 c} + \frac {4 x^{2} \sqrt {c^{2} x^{2} - 1}}{15 c^{3}} + \frac {8 \sqrt {c^{2} x^{2} - 1}}{15 c^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c} + \frac {4 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{3}} + \frac {8 i \sqrt {- c^{2} x^{2} + 1}}{15 c^{5}} & \text {otherwise} \end {cases}\right )}{6 c} + \frac {b e \left (\begin {cases} \frac {x^{6} \sqrt {c^{2} x^{2} - 1}}{7 c} + \frac {6 x^{4} \sqrt {c^{2} x^{2} - 1}}{35 c^{3}} + \frac {8 x^{2} \sqrt {c^{2} x^{2} - 1}}{35 c^{5}} + \frac {16 \sqrt {c^{2} x^{2} - 1}}{35 c^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{6} \sqrt {- c^{2} x^{2} + 1}}{7 c} + \frac {6 i x^{4} \sqrt {- c^{2} x^{2} + 1}}{35 c^{3}} + \frac {8 i x^{2} \sqrt {- c^{2} x^{2} + 1}}{35 c^{5}} + \frac {16 i \sqrt {- c^{2} x^{2} + 1}}{35 c^{7}} & \text {otherwise} \end {cases}\right )}{8 c} \]
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Time = 0.19 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.93 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{8} \, a e x^{8} + \frac {1}{6} \, a d x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \operatorname {arccsc}\left (c x\right ) + \frac {3 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 10 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{5}}\right )} b d + \frac {1}{280} \, {\left (35 \, x^{8} \operatorname {arccsc}\left (c x\right ) + \frac {5 \, c^{6} x^{7} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} + 21 \, c^{4} x^{5} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 35 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{7}}\right )} b e \]
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Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (168) = 336\).
Time = 0.43 (sec) , antiderivative size = 1244, normalized size of antiderivative = 6.35 \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
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Timed out. \[ \int x^5 \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^5\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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