Optimal. Leaf size=204 \[ \frac {9 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^2}-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^4}+\frac {3}{32} c^4 \left (a+b \text {csch}^{-1}(c x)\right )^3+\frac {3 b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{32 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 x^4}+\frac {45}{256} b^3 c^4 \text {csch}^{-1}(c x)+\frac {3 b^3 c \sqrt {\frac {1}{c^2 x^2}+1}}{128 x^3}-\frac {45 b^3 c^3 \sqrt {\frac {1}{c^2 x^2}+1}}{256 x} \]
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Rubi [A] time = 0.18, antiderivative size = 204, 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 = {6286, 5446, 3311, 32, 2635, 8} \[ \frac {9 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^2}-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^4}-\frac {9 b c^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{32 x}+\frac {3 b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{16 x^3}+\frac {3}{32} c^4 \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 x^4}-\frac {45 b^3 c^3 \sqrt {\frac {1}{c^2 x^2}+1}}{256 x}+\frac {3 b^3 c \sqrt {\frac {1}{c^2 x^2}+1}}{128 x^3}+\frac {45}{256} b^3 c^4 \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2635
Rule 3311
Rule 5446
Rule 6286
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^5} \, dx &=-\left (c^4 \operatorname {Subst}\left (\int (a+b x)^3 \cosh (x) \sinh ^3(x) \, dx,x,\text {csch}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {csch}^{-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 \sinh ^4(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^4}+\frac {3 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{16 x^3}-\frac {\left (a+b \text {csch}^{-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 \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )+\frac {1}{32} \left (3 b^3 c^4\right ) \operatorname {Subst}\left (\int \sinh ^4(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {3 b^3 c \sqrt {1+\frac {1}{c^2 x^2}}}{128 x^3}-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^2}+\frac {3 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{32 x}-\frac {\left (a+b \text {csch}^{-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 {csch}^{-1}(c x)\right )-\frac {1}{128} \left (9 b^3 c^4\right ) \operatorname {Subst}\left (\int \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )-\frac {1}{32} \left (9 b^3 c^4\right ) \operatorname {Subst}\left (\int \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {3 b^3 c \sqrt {1+\frac {1}{c^2 x^2}}}{128 x^3}-\frac {45 b^3 c^3 \sqrt {1+\frac {1}{c^2 x^2}}}{256 x}-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^2}+\frac {3 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{32 x}+\frac {3}{32} c^4 \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {\left (a+b \text {csch}^{-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 {csch}^{-1}(c x)\right )+\frac {1}{64} \left (9 b^3 c^4\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {3 b^3 c \sqrt {1+\frac {1}{c^2 x^2}}}{128 x^3}-\frac {45 b^3 c^3 \sqrt {1+\frac {1}{c^2 x^2}}}{256 x}+\frac {45}{256} b^3 c^4 \text {csch}^{-1}(c x)-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^2}+\frac {3 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{32 x}+\frac {3}{32} c^4 \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 277, normalized size = 1.36 \[ \frac {-64 a^3+9 b c^4 x^4 \left (8 a^2+5 b^2\right ) \sinh ^{-1}\left (\frac {1}{c x}\right )-24 b \text {csch}^{-1}(c x) \left (8 a^2+2 a b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (3 c^2 x^2-2\right )+b^2 \left (1-3 c^2 x^2\right )\right )+48 a^2 b c x \sqrt {\frac {1}{c^2 x^2}+1}-72 a^2 b c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1}+72 a b^2 c^2 x^2+24 b^2 \text {csch}^{-1}(c x)^2 \left (a \left (3 c^4 x^4-8\right )+b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (2-3 c^2 x^2\right )\right )-24 a b^2+8 b^3 \left (3 c^4 x^4-8\right ) \text {csch}^{-1}(c x)^3+6 b^3 c x \sqrt {\frac {1}{c^2 x^2}+1}-45 b^3 c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1}}{256 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 346, normalized size = 1.70 \[ \frac {72 \, a b^{2} c^{2} x^{2} + 8 \, {\left (3 \, b^{3} c^{4} x^{4} - 8 \, b^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 64 \, a^{3} - 24 \, a b^{2} + 24 \, {\left (3 \, a b^{2} c^{4} x^{4} - 8 \, a b^{2} - {\left (3 \, b^{3} c^{3} x^{3} - 2 \, b^{3} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, {\left (3 \, {\left (8 \, a^{2} b + 5 \, b^{3}\right )} c^{4} x^{4} + 24 \, b^{3} c^{2} x^{2} - 64 \, a^{2} b - 8 \, b^{3} - 16 \, {\left (3 \, a b^{2} c^{3} x^{3} - 2 \, a b^{2} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 3 \, {\left (3 \, {\left (8 \, a^{2} b + 5 \, b^{3}\right )} c^{3} x^{3} - 2 \, {\left (8 \, a^{2} b + b^{3}\right )} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{256 \, x^{4}} \]
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 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{3}}{x^{5}}\, dx \]
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 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^5} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{3}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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