Optimal. Leaf size=122 \[ \frac {4 b c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x}+\frac {2 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}-\frac {4 b^2 c^2}{9 x}-\frac {2 b^2}{27 x^3} \]
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Rubi [A] time = 0.10, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6285, 5447, 3310, 3296, 2638} \[ \frac {4 b c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x}+\frac {2 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}-\frac {4 b^2 c^2}{9 x}-\frac {2 b^2}{27 x^3} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3310
Rule 5447
Rule 6285
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x^4} \, dx &=-\left (c^3 \operatorname {Subst}\left (\int (a+b x)^2 \cosh ^2(x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (2 b c^3\right ) \operatorname {Subst}\left (\int (a+b x) \cosh ^3(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {2 b^2}{27 x^3}+\frac {2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{9} \left (4 b c^3\right ) \operatorname {Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {2 b^2}{27 x^3}+\frac {2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}+\frac {4 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}-\frac {1}{9} \left (4 b^2 c^3\right ) \operatorname {Subst}\left (\int \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {2 b^2}{27 x^3}-\frac {4 b^2 c^2}{9 x}+\frac {2 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}+\frac {4 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{9 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 134, normalized size = 1.10 \[ \frac {-9 a^2+6 a b \sqrt {\frac {1-c x}{c x+1}} \left (2 c^3 x^3+2 c^2 x^2+c x+1\right )+6 b \text {sech}^{-1}(c x) \left (b \sqrt {\frac {1-c x}{c x+1}} \left (2 c^3 x^3+2 c^2 x^2+c x+1\right )-3 a\right )-2 b^2 \left (6 c^2 x^2+1\right )-9 b^2 \text {sech}^{-1}(c x)^2}{27 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 181, normalized size = 1.48 \[ -\frac {12 \, b^{2} c^{2} x^{2} + 9 \, b^{2} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 9 \, a^{2} + 2 \, b^{2} + 6 \, {\left (3 \, a b - {\left (2 \, b^{2} c^{3} x^{3} + b^{2} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 6 \, {\left (2 \, a b c^{3} x^{3} + a b c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{27 \, x^{3}} \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 192, normalized size = 1.57 \[ c^{3} \left (-\frac {a^{2}}{3 c^{3} x^{3}}+b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {4 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9 c^{2} x^{2}}-\frac {4}{9 c x}-\frac {2}{27 c^{3} x^{3}}\right )+2 a b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2}{9} \, a b {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {3 \, \operatorname {arsech}\left (c x\right )}{x^{3}}\right )} + b^{2} \int \frac {\log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{2}}{x^{4}}\,{d x} - \frac {a^{2}}{3 \, x^{3}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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