Optimal. Leaf size=242 \[ -\frac {9 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 x^2}-\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 x^4}+\frac {3}{32} c^4 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {9 b c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{32 x^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{4 x^4}+\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{16 x^4}+\frac {45}{256} b^3 c^4 \text {sech}^{-1}(c x)+\frac {45 b^3 c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{256 x^2}+\frac {3 b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{128 x^4} \]
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Rubi [A] time = 0.20, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6285, 5447, 3311, 32, 2635, 8} \[ -\frac {9 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 x^2}-\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 x^4}+\frac {9 b c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{32 x^2}+\frac {3}{32} c^4 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{4 x^4}+\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{16 x^4}+\frac {45 b^3 c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{256 x^2}+\frac {45}{256} b^3 c^4 \text {sech}^{-1}(c x)+\frac {3 b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{128 x^4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2635
Rule 3311
Rule 5447
Rule 6285
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^5} \, dx &=-\left (c^4 \operatorname {Subst}\left (\int (a+b x)^3 \cosh ^3(x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{4 x^4}+\frac {1}{4} \left (3 b c^4\right ) \operatorname {Subst}\left (\int (a+b x)^2 \cosh ^4(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 x^4}+\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{16 x^4}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{4 x^4}+\frac {1}{16} \left (9 b c^4\right ) \operatorname {Subst}\left (\int (a+b x)^2 \cosh ^2(x) \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{32} \left (3 b^3 c^4\right ) \operatorname {Subst}\left (\int \cosh ^4(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {3 b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{128 x^4}-\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 x^4}-\frac {9 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 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}{16 x^4}+\frac {9 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{32 x^2}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{4 x^4}+\frac {1}{32} \left (9 b c^4\right ) \operatorname {Subst}\left (\int (a+b x)^2 \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{128} \left (9 b^3 c^4\right ) \operatorname {Subst}\left (\int \cosh ^2(x) \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{32} \left (9 b^3 c^4\right ) \operatorname {Subst}\left (\int \cosh ^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)}{128 x^4}+\frac {45 b^3 c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{256 x^2}-\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 x^4}-\frac {9 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 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}{16 x^4}+\frac {9 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{32 x^2}+\frac {3}{32} c^4 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{4 x^4}+\frac {1}{256} \left (9 b^3 c^4\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{64} \left (9 b^3 c^4\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)}{128 x^4}+\frac {45 b^3 c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{256 x^2}+\frac {45}{256} b^3 c^4 \text {sech}^{-1}(c x)-\frac {3 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 x^4}-\frac {9 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{32 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}{16 x^4}+\frac {9 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{32 x^2}+\frac {3}{32} c^4 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 332, normalized size = 1.37 \[ \frac {-9 b c^4 x^4 \left (8 a^2+5 b^2\right ) \log (x)+9 b c^4 x^4 \left (8 a^2+5 b^2\right ) \log \left (c x \sqrt {\frac {1-c x}{c x+1}}+\sqrt {\frac {1-c x}{c x+1}}+1\right )+3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (8 a^2 \left (3 c^2 x^2+2\right )+b^2 \left (15 c^2 x^2+2\right )\right )-24 b \text {sech}^{-1}(c x) \left (8 a^2-2 a b \sqrt {\frac {1-c x}{c x+1}} \left (3 c^3 x^3+3 c^2 x^2+2 c x+2\right )+b^2 \left (3 c^2 x^2+1\right )\right )-8 a \left (8 a^2+3 b^2\right )-72 a b^2 c^2 x^2+24 b^2 \text {sech}^{-1}(c x)^2 \left (a \left (3 c^4 x^4-8\right )+b \sqrt {\frac {1-c x}{c x+1}} \left (3 c^3 x^3+3 c^2 x^2+2 c x+2\right )\right )+8 b^3 \left (3 c^4 x^4-8\right ) \text {sech}^{-1}(c x)^3}{256 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 351, normalized size = 1.45 \[ -\frac {72 \, a b^{2} c^{2} x^{2} - 8 \, {\left (3 \, b^{3} c^{4} x^{4} - 8 \, b^{3}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} + 64 \, a^{3} + 24 \, a b^{2} - 24 \, {\left (3 \, a b^{2} c^{4} x^{4} - 8 \, a b^{2} + {\left (3 \, b^{3} c^{3} x^{3} + 2 \, b^{3} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 3 \, {\left (3 \, {\left (8 \, a^{2} b + 5 \, b^{3}\right )} c^{4} x^{4} - 24 \, b^{3} c^{2} x^{2} - 64 \, a^{2} b - 8 \, b^{3} + 16 \, {\left (3 \, a b^{2} c^{3} x^{3} + 2 \, a b^{2} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 3 \, {\left (3 \, {\left (8 \, a^{2} b + 5 \, b^{3}\right )} c^{3} x^{3} + 2 \, {\left (8 \, a^{2} b + b^{3}\right )} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{256 \, x^{4}} \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.48, size = 485, normalized size = 2.00 \[ c^{4} \left (-\frac {a^{3}}{4 c^{4} x^{4}}+b^{3} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{3}}{4 c^{4} x^{4}}+\frac {3 \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{16 c^{3} x^{3}}+\frac {9 \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{32 c x}+\frac {3 \mathrm {arcsech}\left (c x \right )^{3}}{32}-\frac {3 \,\mathrm {arcsech}\left (c x \right )}{32 c^{4} x^{4}}+\frac {3 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{128 c^{3} x^{3}}+\frac {45 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{256 c x}+\frac {45 \,\mathrm {arcsech}\left (c x \right )}{256}-\frac {9 \,\mathrm {arcsech}\left (c x \right )}{32 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{4 c^{4} x^{4}}+\frac {\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{8 c^{3} x^{3}}+\frac {3 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{16 c x}+\frac {3 \mathrm {arcsech}\left (c x \right )^{2}}{32}-\frac {1}{32 c^{4} x^{4}}-\frac {3}{32 c^{2} x^{2}}\right )+3 a^{2} b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{4 c^{4} x^{4}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (3 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right ) c^{4} x^{4}+3 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}+2 \sqrt {-c^{2} x^{2}+1}\right )}{32 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}\right )\right ) \]
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 \[ \frac {3}{64} \, a^{2} b {\left (\frac {3 \, c^{5} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} + 1\right ) - 3 \, c^{5} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} - 1} - 1\right ) - \frac {2 \, {\left (3 \, c^{8} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 5 \, c^{6} x \sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}}{c^{4} x^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} - 2 \, c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 1}}{c} - \frac {16 \, \operatorname {arsech}\left (c x\right )}{x^{4}}\right )} - \frac {a^{3}}{4 \, x^{4}} + \int \frac {b^{3} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{3}}{x^{5}} + \frac {3 \, a b^{2} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^5} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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