Integrand size = 21, antiderivative size = 281 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{49 x^7}+\frac {2 b d \left (15 c^2 d+49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1225 x^5}+\frac {b \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{11025 x^3}+\frac {2 b c^2 \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{11025 x}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3} \]
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Time = 0.14 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 6436, 12, 1279, 464, 277, 270} \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}+\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{49 x^7}+\frac {2 b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (15 c^2 d+49 e\right )}{1225 x^5}+\frac {2 b c^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right )}{11025 x}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right )}{11025 x^3} \]
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Rule 12
Rule 270
Rule 276
Rule 277
Rule 464
Rule 1279
Rule 6436
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-15 d^2-42 d e x^2-35 e^2 x^4}{105 x^8 \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}+\frac {1}{105} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-15 d^2-42 d e x^2-35 e^2 x^4}{x^8 \sqrt {1-c^2 x^2}} \, dx \\ & = \frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{49 x^7}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {1}{735} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {6 d \left (15 c^2 d+49 e\right )+245 e^2 x^2}{x^6 \sqrt {1-c^2 x^2}} \, dx \\ & = \frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{49 x^7}+\frac {2 b d \left (15 c^2 d+49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1225 x^5}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {\left (b \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^4 \sqrt {1-c^2 x^2}} \, dx}{3675} \\ & = \frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{49 x^7}+\frac {2 b d \left (15 c^2 d+49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1225 x^5}+\frac {b \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{11025 x^3}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {\left (2 b c^2 \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{11025} \\ & = \frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{49 x^7}+\frac {2 b d \left (15 c^2 d+49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1225 x^5}+\frac {b \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{11025 x^3}+\frac {2 b c^2 \left (360 c^4 d^2+1176 c^2 d e+1225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{11025 x}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.57 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\frac {-105 a \left (15 d^2+42 d e x^2+35 e^2 x^4\right )+b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (1225 e^2 x^4 \left (1+2 c^2 x^2\right )+294 d e x^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )+45 d^2 \left (5+6 c^2 x^2+8 c^4 x^4+16 c^6 x^6\right )\right )-105 b \left (15 d^2+42 d e x^2+35 e^2 x^4\right ) \text {sech}^{-1}(c x)}{11025 x^7} \]
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Time = 0.63 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.74
method | result | size |
parts | \(a \left (-\frac {e^{2}}{3 x^{3}}-\frac {2 d e}{5 x^{5}}-\frac {d^{2}}{7 x^{7}}\right )+b \,c^{7} \left (-\frac {\operatorname {arcsech}\left (c x \right ) e^{2}}{3 c^{7} x^{3}}-\frac {2 \,\operatorname {arcsech}\left (c x \right ) d e}{5 c^{7} x^{5}}-\frac {\operatorname {arcsech}\left (c x \right ) d^{2}}{7 x^{7} c^{7}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (720 c^{10} d^{2} x^{6}+2352 c^{8} d e \,x^{6}+360 c^{8} d^{2} x^{4}+2450 c^{6} e^{2} x^{6}+1176 c^{6} d e \,x^{4}+270 c^{6} d^{2} x^{2}+1225 c^{4} e^{2} x^{4}+882 c^{4} d e \,x^{2}+225 c^{4} d^{2}\right )}{11025 c^{10} x^{6}}\right )\) | \(209\) |
derivativedivides | \(c^{7} \left (\frac {a \left (-\frac {2 d e}{5 c^{3} x^{5}}-\frac {d^{2}}{7 c^{3} x^{7}}-\frac {e^{2}}{3 c^{3} x^{3}}\right )}{c^{4}}+\frac {b \left (-\frac {2 \,\operatorname {arcsech}\left (c x \right ) d e}{5 c^{3} x^{5}}-\frac {\operatorname {arcsech}\left (c x \right ) d^{2}}{7 c^{3} x^{7}}-\frac {\operatorname {arcsech}\left (c x \right ) e^{2}}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (720 c^{10} d^{2} x^{6}+2352 c^{8} d e \,x^{6}+360 c^{8} d^{2} x^{4}+2450 c^{6} e^{2} x^{6}+1176 c^{6} d e \,x^{4}+270 c^{6} d^{2} x^{2}+1225 c^{4} e^{2} x^{4}+882 c^{4} d e \,x^{2}+225 c^{4} d^{2}\right )}{11025 c^{6} x^{6}}\right )}{c^{4}}\right )\) | \(225\) |
default | \(c^{7} \left (\frac {a \left (-\frac {2 d e}{5 c^{3} x^{5}}-\frac {d^{2}}{7 c^{3} x^{7}}-\frac {e^{2}}{3 c^{3} x^{3}}\right )}{c^{4}}+\frac {b \left (-\frac {2 \,\operatorname {arcsech}\left (c x \right ) d e}{5 c^{3} x^{5}}-\frac {\operatorname {arcsech}\left (c x \right ) d^{2}}{7 c^{3} x^{7}}-\frac {\operatorname {arcsech}\left (c x \right ) e^{2}}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (720 c^{10} d^{2} x^{6}+2352 c^{8} d e \,x^{6}+360 c^{8} d^{2} x^{4}+2450 c^{6} e^{2} x^{6}+1176 c^{6} d e \,x^{4}+270 c^{6} d^{2} x^{2}+1225 c^{4} e^{2} x^{4}+882 c^{4} d e \,x^{2}+225 c^{4} d^{2}\right )}{11025 c^{6} x^{6}}\right )}{c^{4}}\right )\) | \(225\) |
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Time = 0.27 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.71 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=-\frac {3675 \, a e^{2} x^{4} + 4410 \, a d e x^{2} + 1575 \, a d^{2} + 105 \, {\left (35 \, b e^{2} x^{4} + 42 \, b d e x^{2} + 15 \, b d^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, {\left (360 \, b c^{7} d^{2} + 1176 \, b c^{5} d e + 1225 \, b c^{3} e^{2}\right )} x^{7} + {\left (360 \, b c^{5} d^{2} + 1176 \, b c^{3} d e + 1225 \, b c e^{2}\right )} x^{5} + 225 \, b c d^{2} x + 18 \, {\left (15 \, b c^{3} d^{2} + 49 \, b c d e\right )} x^{3}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{11025 \, x^{7}} \]
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\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{8}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.83 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\frac {1}{245} \, b d^{2} {\left (\frac {5 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {7}{2}} + 21 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 35 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 35 \, c^{8} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {35 \, \operatorname {arsech}\left (c x\right )}{x^{7}}\right )} + \frac {2}{75} \, b d e {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {15 \, \operatorname {arsech}\left (c x\right )}{x^{5}}\right )} + \frac {1}{9} \, b e^{2} {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {3 \, \operatorname {arsech}\left (c x\right )}{x^{3}}\right )} - \frac {a e^{2}}{3 \, x^{3}} - \frac {2 \, a d e}{5 \, x^{5}} - \frac {a d^{2}}{7 \, x^{7}} \]
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\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{8}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]
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