Optimal. Leaf size=213 \[ -\frac {4 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x}-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}+\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}+\frac {14 b^3 c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{9 x}+\frac {2 b^3 \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{27 x^3} \]
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Rubi [A] time = 0.17, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6285, 5447, 3311, 3296, 2637, 2633} \[ -\frac {4 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x}-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}+\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}+\frac {14 b^3 c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{9 x}+\frac {2 b^3 \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{27 x^3} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 2637
Rule 3296
Rule 3311
Rule 5447
Rule 6285
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^4} \, dx &=-\left (c^3 \operatorname {Subst}\left (\int (a+b x)^3 \cosh ^2(x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}+\left (b c^3\right ) \operatorname {Subst}\left (\int (a+b x)^2 \cosh ^3(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}+\frac {1}{3} \left (2 b c^3\right ) \operatorname {Subst}\left (\int (a+b x)^2 \cosh (x) \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{9} \left (2 b^3 c^3\right ) \operatorname {Subst}\left (\int \cosh ^3(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}-\frac {1}{3} \left (4 b^2 c^3\right ) \operatorname {Subst}\left (\int (a+b x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{9} \left (2 i b^3 c^3\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\frac {i \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c x}\right )\\ &=\frac {2 b^3 c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{9 x}+\frac {2 b^3 \left (\frac {1-c x}{1+c x}\right )^{3/2} (1+c x)^3}{27 x^3}-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}-\frac {4 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}+\frac {1}{3} \left (4 b^3 c^3\right ) \operatorname {Subst}\left (\int \cosh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {14 b^3 c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{9 x}+\frac {2 b^3 \left (\frac {1-c x}{1+c x}\right )^{3/2} (1+c x)^3}{27 x^3}-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}-\frac {4 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 256, normalized size = 1.20 \[ \frac {-9 a^3-3 b \text {sech}^{-1}(c x) \left (9 a^2-6 a b \sqrt {\frac {1-c x}{c x+1}} \left (2 c^3 x^3+2 c^2 x^2+c x+1\right )+2 b^2 \left (6 c^2 x^2+1\right )\right )+9 a^2 b \sqrt {\frac {1-c x}{c x+1}} \left (2 c^3 x^3+2 c^2 x^2+c x+1\right )-6 a b^2 \left (6 c^2 x^2+1\right )+9 b^2 \text {sech}^{-1}(c x)^2 \left (b \sqrt {\frac {1-c x}{c x+1}} \left (2 c^3 x^3+2 c^2 x^2+c x+1\right )-3 a\right )+2 b^3 \sqrt {\frac {1-c x}{c x+1}} \left (20 c^3 x^3+20 c^2 x^2+c x+1\right )-9 b^3 \text {sech}^{-1}(c x)^3}{27 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 305, normalized size = 1.43 \[ -\frac {36 \, a b^{2} c^{2} x^{2} + 9 \, b^{3} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} + 9 \, a^{3} + 6 \, a b^{2} + 9 \, {\left (3 \, a b^{2} - {\left (2 \, b^{3} c^{3} x^{3} + b^{3} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, {\left (12 \, b^{3} c^{2} x^{2} + 9 \, a^{2} b + 2 \, b^{3} - 6 \, {\left (2 \, a b^{2} c^{3} x^{3} + a b^{2} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, {\left (9 \, a^{2} b + 20 \, b^{3}\right )} c^{3} x^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{27 \, x^{3}} \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.51, size = 387, normalized size = 1.82 \[ c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {2 \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{3}+\frac {\mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{3 c^{2} x^{2}}-\frac {4 \,\mathrm {arcsech}\left (c x \right )}{3 c x}+\frac {40 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27}-\frac {2 \,\mathrm {arcsech}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {4 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9 c^{2} x^{2}}-\frac {4}{9 c x}-\frac {2}{27 c^{3} x^{3}}\right )+3 a^{2} b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\right ) \]
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 \[ \frac {1}{3} \, a^{2} b {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {3 \, \operatorname {arsech}\left (c x\right )}{x^{3}}\right )} - \frac {a^{3}}{3 \, x^{3}} + \int \frac {b^{3} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{3}}{x^{4}} + \frac {3 \, a b^{2} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{2}}{x^{4}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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