Optimal. Leaf size=183 \[ -\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}+\frac {2 b c^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (12 c^2 d+25 e\right )}{225 x}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (12 c^2 d+25 e\right )}{225 x^3}+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{25 x^5} \]
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Rubi [A] time = 0.10, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ -\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}+\frac {2 b c^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (12 c^2 d+25 e\right )}{225 x}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (12 c^2 d+25 e\right )}{225 x^3}+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{25 x^5} \]
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^6} \, dx &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e \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 {-3 d-5 e x^2}{15 x^6 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}+\frac {1}{15} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-3 d-5 e x^2}{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}}{25 x^5}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}+\frac {1}{75} \left (b \left (-12 c^2 d-25 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\\ &=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{25 x^5}+\frac {b \left (12 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}+\frac {1}{225} \left (2 b c^2 \left (-12 c^2 d-25 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\\ &=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{25 x^5}+\frac {b \left (12 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x^3}+\frac {2 b c^2 \left (12 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 101, normalized size = 0.55 \[ \frac {-15 a \left (3 d+5 e x^2\right )+b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (25 e x^2 \left (2 c^2 x^2+1\right )+3 d \left (8 c^4 x^4+4 c^2 x^2+3\right )\right )-15 b \text {sech}^{-1}(c x) \left (3 d+5 e x^2\right )}{225 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 128, normalized size = 0.70 \[ -\frac {75 \, a e x^{2} + 45 \, a d + 15 \, {\left (5 \, b e x^{2} + 3 \, b d\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, {\left (12 \, b c^{5} d + 25 \, b c^{3} e\right )} x^{5} + 9 \, b c d x + {\left (12 \, b c^{3} d + 25 \, b c e\right )} x^{3}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{225 \, x^{5}} \]
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 \[ \int \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 142, normalized size = 0.78 \[ c^{5} \left (\frac {a \left (-\frac {e}{3 c^{3} x^{3}}-\frac {d}{5 c^{3} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\mathrm {arcsech}\left (c x \right ) e}{3 c^{3} x^{3}}-\frac {\mathrm {arcsech}\left (c x \right ) d}{5 c^{3} x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (24 c^{6} d \,x^{4}+50 c^{4} e \,x^{4}+12 c^{4} d \,x^{2}+25 c^{2} x^{2} e +9 c^{2} d \right )}{225 c^{4} x^{4}}\right )}{c^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 132, normalized size = 0.72 \[ \frac {1}{75} \, b d {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {15 \, \operatorname {arsech}\left (c x\right )}{x^{5}}\right )} + \frac {1}{9} \, b e {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {3 \, \operatorname {arsech}\left (c x\right )}{x^{3}}\right )} - \frac {a e}{3 \, x^{3}} - \frac {a d}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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