Optimal. Leaf size=87 \[ -\frac {b^2}{4 x^2}-\frac {1}{2} a b c^2 \text {csch}^{-1}(c x)-\frac {1}{4} b^2 c^2 \text {csch}^{-1}(c x)^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2} \]
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Rubi [A]
time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6421, 5554,
3391} \begin {gather*} \frac {b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{2 x}-\frac {1}{2} a b c^2 \text {csch}^{-1}(c x)-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{4} b^2 c^2 \text {csch}^{-1}(c x)^2-\frac {b^2}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3391
Rule 5554
Rule 6421
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^3} \, dx &=-\left (c^2 \text {Subst}\left (\int (a+b x)^2 \cosh (x) \sinh (x) \, dx,x,\text {csch}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2}+\left (b c^2\right ) \text {Subst}\left (\int (a+b x) \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {b^2}{4 x^2}+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{2} \left (b c^2\right ) \text {Subst}\left (\int (a+b x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {b^2}{4 x^2}-\frac {1}{2} a b c^2 \text {csch}^{-1}(c x)-\frac {1}{4} b^2 c^2 \text {csch}^{-1}(c x)^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 100, normalized size = 1.15 \begin {gather*} -\frac {2 a^2+b^2-2 a b c \sqrt {1+\frac {1}{c^2 x^2}} x-2 b \left (-2 a+b c \sqrt {1+\frac {1}{c^2 x^2}} x\right ) \text {csch}^{-1}(c x)+b^2 \left (2+c^2 x^2\right ) \text {csch}^{-1}(c x)^2+2 a b c^2 x^2 \sinh ^{-1}\left (\frac {1}{c x}\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}}{x^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 163 vs.
\(2 (75) = 150\).
time = 0.40, size = 163, normalized size = 1.87 \begin {gather*} \frac {2 \, a b c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - {\left (b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, {\left (a b c^{2} x^{2} - b^{2} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, a b\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, x^{2}} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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