Optimal. Leaf size=102 \[ -\frac {6 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}+\frac {6 b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{x} \]
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Rubi [A] time = 0.10, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6285, 3296, 2637} \[ -\frac {6 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}+\frac {6 b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{x} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 6285
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^2} \, dx &=-\left (c \operatorname {Subst}\left (\int (a+b x)^3 \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}+(3 b c) \operatorname {Subst}\left (\int (a+b x)^2 \cosh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}-\left (6 b^2 c\right ) \operatorname {Subst}\left (\int (a+b x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {6 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}+\left (6 b^3 c\right ) \operatorname {Subst}\left (\int \cosh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {6 b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{x}-\frac {6 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 165, normalized size = 1.62 \[ -\frac {a^3+3 b \text {sech}^{-1}(c x) \left (a^2-2 a b \sqrt {\frac {1-c x}{c x+1}} (c x+1)+2 b^2\right )-3 a^2 b \sqrt {\frac {1-c x}{c x+1}} (c x+1)-3 b^2 \text {sech}^{-1}(c x)^2 \left (b \sqrt {\frac {1-c x}{c x+1}} (c x+1)-a\right )+6 a b^2-6 b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)+b^3 \text {sech}^{-1}(c x)^3}{x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 228, normalized size = 2.24 \[ -\frac {b^{3} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + a^{3} + 6 \, a b^{2} - 3 \, {\left (b^{3} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 3 \, {\left (2 \, a b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a^{2} b - 2 \, b^{3}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{x} \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 227, normalized size = 2.23 \[ c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{3}}{c x}+3 \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {6 \,\mathrm {arcsech}\left (c x \right )}{c x}+6 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{c x}+2 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {2}{c x}\right )+3 a^{2} b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 144, normalized size = 1.41 \[ -\frac {b^{3} \operatorname {arsech}\left (c x\right )^{3}}{x} + 3 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} a^{2} b + 6 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right ) - \frac {1}{x}\right )} a b^{2} + 3 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right )^{2} + 2 \, c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {2 \, \operatorname {arsech}\left (c x\right )}{x}\right )} b^{3} - \frac {3 \, a b^{2} \operatorname {arsech}\left (c x\right )^{2}}{x} - \frac {a^{3}}{x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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