Optimal. Leaf size=238 \[ -\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (30 c^2 d+49 e\right )}{1225 x^5}+\frac {4 b c^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (30 c^2 d+49 e\right )}{3675 x^3}+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{49 x^7}+\frac {8 b c^4 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (30 c^2 d+49 e\right )}{3675 x} \]
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Rubi [A] time = 0.12, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 6301, 12, 453, 271, 264} \[ -\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}+\frac {8 b c^4 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (30 c^2 d+49 e\right )}{3675 x}+\frac {4 b c^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (30 c^2 d+49 e\right )}{3675 x^3}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (30 c^2 d+49 e\right )}{1225 x^5}+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{49 x^7} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 264
Rule 271
Rule 453
Rule 6301
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-5 d-7 e x^2}{35 x^8 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}+\frac {1}{35} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-5 d-7 e x^2}{x^8 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{49 x^7}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}+\frac {1}{245} \left (b \left (-30 c^2 d-49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^6 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{49 x^7}+\frac {b \left (30 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 \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}+\frac {\left (4 b c^2 \left (-30 c^2 d-49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^4 \sqrt {1-c^2 x^2}} \, dx}{1225}\\ &=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{49 x^7}+\frac {b \left (30 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 {4 b c^2 \left (30 c^2 d+49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3675 x^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}+\frac {\left (8 b c^4 \left (-30 c^2 d-49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{3675}\\ &=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{49 x^7}+\frac {b \left (30 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 {4 b c^2 \left (30 c^2 d+49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3675 x^3}+\frac {8 b c^4 \left (30 c^2 d+49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3675 x}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 117, normalized size = 0.49 \[ \frac {-105 a \left (5 d+7 e x^2\right )+b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (49 e x^2 \left (8 c^4 x^4+4 c^2 x^2+3\right )+15 d \left (16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right )\right )-105 b \text {sech}^{-1}(c x) \left (5 d+7 e x^2\right )}{3675 x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 149, normalized size = 0.63 \[ -\frac {735 \, a e x^{2} + 525 \, a d + 105 \, {\left (7 \, b e x^{2} + 5 \, b d\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (8 \, {\left (30 \, b c^{7} d + 49 \, b c^{5} e\right )} x^{7} + 4 \, {\left (30 \, b c^{5} d + 49 \, b c^{3} e\right )} x^{5} + 75 \, b c d x + 3 \, {\left (30 \, b c^{3} d + 49 \, b c e\right )} x^{3}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{3675 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 160, normalized size = 0.67 \[ c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\mathrm {arcsech}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\mathrm {arcsech}\left (c x \right ) e}{5 c^{5} x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (240 c^{8} d \,x^{6}+392 c^{6} e \,x^{6}+120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 c^{2} x^{2} e +75 c^{2} d \right )}{3675 c^{6} x^{6}}\right )}{c^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 165, normalized size = 0.69 \[ \frac {1}{245} \, b d {\left (\frac {5 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {7}{2}} + 21 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 35 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 35 \, c^{8} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {35 \, \operatorname {arsech}\left (c x\right )}{x^{7}}\right )} + \frac {1}{75} \, b e {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {15 \, \operatorname {arsech}\left (c x\right )}{x^{5}}\right )} - \frac {a e}{5 \, x^{5}} - \frac {a d}{7 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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