Optimal. Leaf size=100 \[ \frac {2 b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^2}-\frac {4}{9} b c^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^3}+\frac {4 b^2 c^2}{9 x}-\frac {2 b^2}{27 x^3} \]
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Rubi [A] time = 0.11, antiderivative size = 100, 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 = {6286, 5446, 3310, 3296, 2637} \[ -\frac {4}{9} b c^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2 b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^3}+\frac {4 b^2 c^2}{9 x}-\frac {2 b^2}{27 x^3} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3310
Rule 5446
Rule 6286
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^4} \, dx &=-\left (c^3 \operatorname {Subst}\left (\int (a+b x)^2 \cosh (x) \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (2 b c^3\right ) \operatorname {Subst}\left (\int (a+b x) \sinh ^3(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {2 b^2}{27 x^3}+\frac {2 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^3}-\frac {1}{9} \left (4 b c^3\right ) \operatorname {Subst}\left (\int (a+b x) \sinh (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {2 b^2}{27 x^3}-\frac {4}{9} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{9} \left (4 b^2 c^3\right ) \operatorname {Subst}\left (\int \cosh (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {2 b^2}{27 x^3}+\frac {4 b^2 c^2}{9 x}-\frac {4}{9} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )+\frac {2 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 106, normalized size = 1.06 \[ \frac {-9 a^2+6 a b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (1-2 c^2 x^2\right )-6 b \text {csch}^{-1}(c x) \left (3 a+b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (2 c^2 x^2-1\right )\right )+2 b^2 \left (6 c^2 x^2-1\right )-9 b^2 \text {csch}^{-1}(c x)^2}{27 x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 178, normalized size = 1.78 \[ \frac {12 \, b^{2} c^{2} x^{2} - 9 \, b^{2} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 9 \, a^{2} - 2 \, b^{2} - 6 \, {\left (3 \, a b + {\left (2 \, b^{2} c^{3} x^{3} - b^{2} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 6 \, {\left (2 \, a b c^{3} x^{3} - a b c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{27 \, x^{3}} \]
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 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}}{x^{4}}\, dx \]
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 \[ \frac {2}{9} \, a b {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} - \frac {1}{3} \, b^{2} {\left (\frac {\log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{2}}{x^{3}} + 3 \, \int -\frac {3 \, c^{2} x^{2} \log \relax (c)^{2} + 3 \, {\left (c^{2} x^{2} + 1\right )} \log \relax (x)^{2} + 3 \, \log \relax (c)^{2} + 6 \, {\left (c^{2} x^{2} \log \relax (c) + \log \relax (c)\right )} \log \relax (x) - 2 \, {\left (3 \, c^{2} x^{2} \log \relax (c) + 3 \, {\left (c^{2} x^{2} + 1\right )} \log \relax (x) + {\left (c^{2} x^{2} {\left (3 \, \log \relax (c) - 1\right )} + 3 \, {\left (c^{2} x^{2} + 1\right )} \log \relax (x) + 3 \, \log \relax (c)\right )} \sqrt {c^{2} x^{2} + 1} + 3 \, \log \relax (c)\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + 3 \, {\left (c^{2} x^{2} \log \relax (c)^{2} + {\left (c^{2} x^{2} + 1\right )} \log \relax (x)^{2} + \log \relax (c)^{2} + 2 \, {\left (c^{2} x^{2} \log \relax (c) + \log \relax (c)\right )} \log \relax (x)\right )} \sqrt {c^{2} x^{2} + 1}}{3 \, {\left (c^{2} x^{6} + x^{4} + {\left (c^{2} x^{6} + x^{4}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac {a^{2}}{3 \, x^{3}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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