Integrand size = 21, antiderivative size = 183 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {b c \left (24 c^4 d^2+100 c^2 d e+225 e^2\right ) \sqrt {-1+c^2 x^2}}{225 \sqrt {c^2 x^2}}-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {2 b c d \left (6 c^2 d+25 e\right ) \sqrt {-1+c^2 x^2}}{225 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x} \]
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Time = 0.12 (sec) , antiderivative size = 183, 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 = {276, 5347, 12, 1279, 464, 270} \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x}-\frac {b c d^2 \sqrt {c^2 x^2-1}}{25 x^4 \sqrt {c^2 x^2}}-\frac {2 b c d \sqrt {c^2 x^2-1} \left (6 c^2 d+25 e\right )}{225 x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {c^2 x^2-1} \left (24 c^4 d^2+100 c^2 d e+225 e^2\right )}{225 \sqrt {c^2 x^2}} \]
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Rule 12
Rule 270
Rule 276
Rule 464
Rule 1279
Rule 5347
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+\frac {(b c x) \int \frac {-3 d^2-10 d e x^2-15 e^2 x^4}{15 x^6 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+\frac {(b c x) \int \frac {-3 d^2-10 d e x^2-15 e^2 x^4}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{15 \sqrt {c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {-1+c^2 x^2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+\frac {(b c x) \int \frac {-2 d \left (6 c^2 d+25 e\right )-75 e^2 x^2}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{75 \sqrt {c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {-1+c^2 x^2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {2 b c d \left (6 c^2 d+25 e\right ) \sqrt {-1+c^2 x^2}}{225 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x}+\frac {\left (b c \left (-225 e^2-4 c^2 d \left (6 c^2 d+25 e\right )\right ) x\right ) \int \frac {1}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{225 \sqrt {c^2 x^2}} \\ & = -\frac {b c \left (225 e^2+4 c^2 d \left (6 c^2 d+25 e\right )\right ) \sqrt {-1+c^2 x^2}}{225 \sqrt {c^2 x^2}}-\frac {b c d^2 \sqrt {-1+c^2 x^2}}{25 x^4 \sqrt {c^2 x^2}}-\frac {2 b c d \left (6 c^2 d+25 e\right ) \sqrt {-1+c^2 x^2}}{225 x^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{x} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {15 a \left (3 d^2+10 d e x^2+15 e^2 x^4\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (225 e^2 x^4+50 d e x^2 \left (1+2 c^2 x^2\right )+3 d^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )+15 b \left (3 d^2+10 d e x^2+15 e^2 x^4\right ) \csc ^{-1}(c x)}{225 x^5} \]
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Time = 0.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.96
method | result | size |
parts | \(a \left (-\frac {e^{2}}{x}-\frac {d^{2}}{5 x^{5}}-\frac {2 d e}{3 x^{3}}\right )+b \,c^{5} \left (-\frac {\operatorname {arccsc}\left (c x \right ) e^{2}}{c^{5} x}-\frac {\operatorname {arccsc}\left (c x \right ) d^{2}}{5 x^{5} c^{5}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) d e}{3 c^{5} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (24 c^{8} d^{2} x^{4}+100 c^{6} d e \,x^{4}+12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 c^{10} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{6}}\right )\) | \(175\) |
derivativedivides | \(c^{5} \left (\frac {a \left (-\frac {e^{2}}{c x}-\frac {d^{2}}{5 c \,x^{5}}-\frac {2 d e}{3 c \,x^{3}}\right )}{c^{4}}+\frac {b \left (-\frac {\operatorname {arccsc}\left (c x \right ) e^{2}}{c x}-\frac {\operatorname {arccsc}\left (c x \right ) d^{2}}{5 c \,x^{5}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) d e}{3 c \,x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (24 c^{8} d^{2} x^{4}+100 c^{6} d e \,x^{4}+12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{4}}\right )\) | \(191\) |
default | \(c^{5} \left (\frac {a \left (-\frac {e^{2}}{c x}-\frac {d^{2}}{5 c \,x^{5}}-\frac {2 d e}{3 c \,x^{3}}\right )}{c^{4}}+\frac {b \left (-\frac {\operatorname {arccsc}\left (c x \right ) e^{2}}{c x}-\frac {\operatorname {arccsc}\left (c x \right ) d^{2}}{5 c \,x^{5}}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) d e}{3 c \,x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (24 c^{8} d^{2} x^{4}+100 c^{6} d e \,x^{4}+12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )}{c^{4}}\right )\) | \(191\) |
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Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {225 \, a e^{2} x^{4} + 150 \, a d e x^{2} + 45 \, a d^{2} + 15 \, {\left (15 \, b e^{2} x^{4} + 10 \, b d e x^{2} + 3 \, b d^{2}\right )} \operatorname {arccsc}\left (c x\right ) + {\left ({\left (24 \, b c^{4} d^{2} + 100 \, b c^{2} d e + 225 \, b e^{2}\right )} x^{4} + 9 \, b d^{2} + 2 \, {\left (6 \, b c^{2} d^{2} + 25 \, b d e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, x^{5}} \]
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Time = 5.19 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.83 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=- \frac {a d^{2}}{5 x^{5}} - \frac {2 a d e}{3 x^{3}} - \frac {a e^{2}}{x} - b c e^{2} \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d^{2} \operatorname {acsc}{\left (c x \right )}}{5 x^{5}} - \frac {2 b d e \operatorname {acsc}{\left (c x \right )}}{3 x^{3}} - \frac {b e^{2} \operatorname {acsc}{\left (c x \right )}}{x} - \frac {b d^{2} \left (\begin {cases} \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{15 x} + \frac {4 c^{3} \sqrt {c^{2} x^{2} - 1}}{15 x^{3}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{15 x} + \frac {4 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{15 x^{3}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{5 c} - \frac {2 b d e \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} \]
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Time = 0.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.99 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-{\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} b e^{2} - \frac {1}{75} \, b d^{2} {\left (\frac {3 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {15 \, \operatorname {arccsc}\left (c x\right )}{x^{5}}\right )} + \frac {2}{9} \, b d e {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arccsc}\left (c x\right )}{x^{3}}\right )} - \frac {a e^{2}}{x} - \frac {2 \, a d e}{3 \, x^{3}} - \frac {a d^{2}}{5 \, x^{5}} \]
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Time = 0.30 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.72 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {1}{225} \, {\left (9 \, b c^{4} d^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 30 \, b c^{4} d^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {45 \, b c^{3} d^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{x} + 45 \, b c^{4} d^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {90 \, b c^{3} d^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - 50 \, b c^{2} d e {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {45 \, b c^{3} d^{2} \arcsin \left (\frac {1}{c x}\right )}{x} + 150 \, b c^{2} d e \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {150 \, b c d e {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {150 \, b c d e \arcsin \left (\frac {1}{c x}\right )}{x} + 225 \, b e^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {225 \, b e^{2} \arcsin \left (\frac {1}{c x}\right )}{c x} + \frac {225 \, a e^{2}}{c x} + \frac {150 \, a d e}{c x^{3}} + \frac {45 \, a d^{2}}{c x^{5}}\right )} c \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]
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