Integrand size = 19, antiderivative size = 152 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {2 b c^3 \left (12 c^2 d+25 e\right ) \sqrt {-1+c^2 x^2}}{225 \sqrt {c^2 x^2}}-\frac {b c d \sqrt {-1+c^2 x^2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d+25 e\right ) \sqrt {-1+c^2 x^2}}{225 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3} \]
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Time = 0.07 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 5347, 12, 464, 277, 270} \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {b c \sqrt {c^2 x^2-1} \left (12 c^2 d+25 e\right )}{225 x^2 \sqrt {c^2 x^2}}-\frac {b c d \sqrt {c^2 x^2-1}}{25 x^4 \sqrt {c^2 x^2}}-\frac {2 b c^3 \sqrt {c^2 x^2-1} \left (12 c^2 d+25 e\right )}{225 \sqrt {c^2 x^2}} \]
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Rule 12
Rule 14
Rule 270
Rule 277
Rule 464
Rule 5347
Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}+\frac {(b c x) \int \frac {-3 d-5 e x^2}{15 x^6 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}+\frac {(b c x) \int \frac {-3 d-5 e x^2}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{15 \sqrt {c^2 x^2}} \\ & = -\frac {b c d \sqrt {-1+c^2 x^2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}+\frac {\left (b c \left (-12 c^2 d-25 e\right ) x\right ) \int \frac {1}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{75 \sqrt {c^2 x^2}} \\ & = -\frac {b c d \sqrt {-1+c^2 x^2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d+25 e\right ) \sqrt {-1+c^2 x^2}}{225 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}+\frac {\left (2 b c^3 \left (-12 c^2 d-25 e\right ) x\right ) \int \frac {1}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{225 \sqrt {c^2 x^2}} \\ & = -\frac {2 b c^3 \left (12 c^2 d+25 e\right ) \sqrt {-1+c^2 x^2}}{225 \sqrt {c^2 x^2}}-\frac {b c d \sqrt {-1+c^2 x^2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d+25 e\right ) \sqrt {-1+c^2 x^2}}{225 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.62 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {15 a \left (3 d+5 e x^2\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (25 e x^2 \left (1+2 c^2 x^2\right )+3 d \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )+15 b \left (3 d+5 e x^2\right ) \csc ^{-1}(c x)}{225 x^5} \]
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Time = 0.33 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.84
method | result | size |
parts | \(a \left (-\frac {d}{5 x^{5}}-\frac {e}{3 x^{3}}\right )+b \,c^{5} \left (-\frac {\operatorname {arccsc}\left (c x \right ) d}{5 x^{5} c^{5}}-\frac {\operatorname {arccsc}\left (c x \right ) e}{3 c^{5} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (24 c^{6} d \,x^{4}+50 c^{4} e \,x^{4}+12 c^{4} d \,x^{2}+25 c^{2} e \,x^{2}+9 c^{2} d \right )}{225 c^{8} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{6}}\right )\) | \(127\) |
derivativedivides | \(c^{5} \left (\frac {a \left (-\frac {d}{5 c^{3} x^{5}}-\frac {e}{3 c^{3} x^{3}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsc}\left (c x \right ) d}{5 c^{3} x^{5}}-\frac {\operatorname {arccsc}\left (c x \right ) e}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (24 c^{6} d \,x^{4}+50 c^{4} e \,x^{4}+12 c^{4} d \,x^{2}+25 c^{2} e \,x^{2}+9 c^{2} d \right )}{225 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{2}}\right )\) | \(140\) |
default | \(c^{5} \left (\frac {a \left (-\frac {d}{5 c^{3} x^{5}}-\frac {e}{3 c^{3} x^{3}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsc}\left (c x \right ) d}{5 c^{3} x^{5}}-\frac {\operatorname {arccsc}\left (c x \right ) e}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (24 c^{6} d \,x^{4}+50 c^{4} e \,x^{4}+12 c^{4} d \,x^{2}+25 c^{2} e \,x^{2}+9 c^{2} d \right )}{225 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{2}}\right )\) | \(140\) |
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.58 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {75 \, a e x^{2} + 45 \, a d + 15 \, {\left (5 \, b e x^{2} + 3 \, b d\right )} \operatorname {arccsc}\left (c x\right ) + {\left (2 \, {\left (12 \, b c^{4} d + 25 \, b c^{2} e\right )} x^{4} + {\left (12 \, b c^{2} d + 25 \, b e\right )} x^{2} + 9 \, b d\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, x^{5}} \]
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Time = 4.49 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.84 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=- \frac {a d}{5 x^{5}} - \frac {a e}{3 x^{3}} - \frac {b d \operatorname {acsc}{\left (c x \right )}}{5 x^{5}} - \frac {b e \operatorname {acsc}{\left (c x \right )}}{3 x^{3}} - \frac {b d \left (\begin {cases} \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{15 x} + \frac {4 c^{3} \sqrt {c^{2} x^{2} - 1}}{15 x^{3}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{15 x} + \frac {4 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{15 x^{3}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{5 c} - \frac {b e \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} \]
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Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {1}{75} \, b d {\left (\frac {3 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {15 \, \operatorname {arccsc}\left (c x\right )}{x^{5}}\right )} + \frac {1}{9} \, b e {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arccsc}\left (c x\right )}{x^{3}}\right )} - \frac {a e}{3 \, x^{3}} - \frac {a d}{5 \, x^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.61 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {1}{225} \, {\left (9 \, b c^{4} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 30 \, b c^{4} d {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {45 \, b c^{3} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{x} + 45 \, b c^{4} d \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {90 \, b c^{3} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - 25 \, b c^{2} e {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {45 \, b c^{3} d \arcsin \left (\frac {1}{c x}\right )}{x} + 75 \, b c^{2} e \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {75 \, b c e {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {75 \, b c e \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {75 \, a e}{c x^{3}} + \frac {45 \, a d}{c x^{5}}\right )} c \]
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Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]
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