Optimal. Leaf size=163 \[ -\frac {3 b^2 (1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{4 x^2}-\frac {1}{4} c^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 x^2}-\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 x^2}-\frac {3}{8} b^3 c^2 \text {sech}^{-1}(c x)+\frac {3 b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{8 x^2} \]
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Rubi [A] time = 0.12, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6285, 5446, 3311, 32, 2635, 8} \[ -\frac {3 b^2 (1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{4 x^2}-\frac {1}{4} c^2 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 x^2}-\frac {(1-c x) (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 x^2}-\frac {3}{8} b^3 c^2 \text {sech}^{-1}(c x)+\frac {3 b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2635
Rule 3311
Rule 5446
Rule 6285
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^3} \, dx &=-\left (c^2 \operatorname {Subst}\left (\int (a+b x)^3 \cosh (x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 x^2}+\frac {1}{2} \left (3 b c^2\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sinh ^2(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {3 b^2 (1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{4 x^2}+\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 x^2}-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 x^2}-\frac {1}{4} \left (3 b c^2\right ) \operatorname {Subst}\left (\int (a+b x)^2 \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{4} \left (3 b^3 c^2\right ) \operatorname {Subst}\left (\int \sinh ^2(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {3 b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 x^2}-\frac {3 b^2 (1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{4 x^2}+\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 x^2}-\frac {1}{4} c^2 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 x^2}-\frac {1}{8} \left (3 b^3 c^2\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {3 b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{8 x^2}-\frac {3}{8} b^3 c^2 \text {sech}^{-1}(c x)-\frac {3 b^2 (1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{4 x^2}+\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{4 x^2}-\frac {1}{4} c^2 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {(1-c x) (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^3}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 245, normalized size = 1.50 \[ \frac {-4 a^3-3 b c^2 x^2 \left (2 a^2+b^2\right ) \log (x)+3 b c^2 x^2 \left (2 a^2+b^2\right ) \log \left (c x \sqrt {\frac {1-c x}{c x+1}}+\sqrt {\frac {1-c x}{c x+1}}+1\right )+3 b \left (2 a^2+b^2\right ) \sqrt {\frac {1-c x}{c x+1}} (c x+1)-6 b \text {sech}^{-1}(c x) \left (2 a^2-2 a b \sqrt {\frac {1-c x}{c x+1}} (c x+1)+b^2\right )+6 b^2 \text {sech}^{-1}(c x)^2 \left (a \left (c^2 x^2-2\right )+b \sqrt {\frac {1-c x}{c x+1}} (c x+1)\right )-6 a b^2+2 b^3 \left (c^2 x^2-2\right ) \text {sech}^{-1}(c x)^3}{8 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 271, normalized size = 1.66 \[ \frac {2 \, {\left (b^{3} c^{2} x^{2} - 2 \, b^{3}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} + 3 \, {\left (2 \, a^{2} b + b^{3}\right )} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 4 \, a^{3} - 6 \, a b^{2} + 6 \, {\left (a b^{2} c^{2} x^{2} + b^{3} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, a b^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, {\left (4 \, a b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + {\left (2 \, a^{2} b + b^{3}\right )} c^{2} x^{2} - 4 \, a^{2} b - 2 \, b^{3}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{8 \, x^{2}} \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 321, normalized size = 1.97 \[ c^{2} \left (-\frac {a^{3}}{2 c^{2} x^{2}}+b^{3} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{3}}{2 c^{2} x^{2}}+\frac {3 \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}+\frac {\mathrm {arcsech}\left (c x \right )^{3}}{4}-\frac {3 \,\mathrm {arcsech}\left (c x \right )}{4 c^{2} x^{2}}+\frac {3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{8 c x}+\frac {3 \,\mathrm {arcsech}\left (c x \right )}{8}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{2 c^{2} x^{2}}+\frac {\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{2 c x}+\frac {\mathrm {arcsech}\left (c x \right )^{2}}{4}-\frac {1}{4 c^{2} x^{2}}\right )+3 a^{2} b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{2 c^{2} x^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\right )}{4 c x \sqrt {-c^{2} x^{2}+1}}\right )\right ) \]
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 \[ -\frac {3}{8} \, a^{2} b {\left (\frac {\frac {2 \, c^{4} x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} - 1} - c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} + 1\right ) + c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} - 1\right )}{c} + \frac {4 \, \operatorname {arsech}\left (c x\right )}{x^{2}}\right )} - \frac {a^{3}}{2 \, x^{2}} + \int \frac {b^{3} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{3}}{x^{3}} + \frac {3 \, a b^{2} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{2}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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