Optimal. Leaf size=30 \[ a x+b x \text {csch}^{-1}(c x)+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c} \]
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Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6413, 272, 65,
214} \begin {gather*} a x+\frac {b \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c}+b x \text {csch}^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 6413
Rubi steps
\begin {align*} \int \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=a x+b \int \text {csch}^{-1}(c x) \, dx\\ &=a x+b x \text {csch}^{-1}(c x)+\frac {b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x} \, dx}{c}\\ &=a x+b x \text {csch}^{-1}(c x)-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c}\\ &=a x+b x \text {csch}^{-1}(c x)-(b c) \text {Subst}\left (\int \frac {1}{-c^2+c^2 x^2} \, dx,x,\sqrt {1+\frac {1}{c^2 x^2}}\right )\\ &=a x+b x \text {csch}^{-1}(c x)+\frac {b \tanh ^{-1}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(78\) vs. \(2(30)=60\).
time = 0.05, size = 78, normalized size = 2.60 \begin {gather*} a x+b x \text {csch}^{-1}(c x)-\frac {b \sqrt {1+c^2 x^2} \log \left (-\sqrt {c^2} x+\sqrt {1+c^2 x^2}\right )}{c \sqrt {c^2} \sqrt {1+\frac {1}{c^2 x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 36, normalized size = 1.20
method | result | size |
default | \(a x +b x \,\mathrm {arccsch}\left (c x \right )+\frac {b \ln \left (c x +c x \sqrt {1+\frac {1}{c^{2} x^{2}}}\right )}{c}\) | \(36\) |
derivativedivides | \(\frac {a c x +\mathrm {arccsch}\left (c x \right ) b c x +\ln \left (c x +c x \sqrt {1+\frac {1}{c^{2} x^{2}}}\right ) b}{c}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 49, normalized size = 1.63 \begin {gather*} a x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b}{2 \, c} \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. 143 vs.
\(2 (28) = 56\).
time = 0.38, size = 143, normalized size = 4.77 \begin {gather*} \frac {a c x + b c \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - b c \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - b \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + {\left (b c x - b c\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c} \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 \left (a + b \operatorname {acsch}{\left (c x \right )}\right )\, 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 = -0.03 \begin {gather*} \int a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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