Optimal. Leaf size=30 \[ a x+\frac {b \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c}+b x \text {csch}^{-1}(c x) \]
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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 = {6278, 266, 63, 208} \[ a x+\frac {b \tanh ^{-1}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c}+b x \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 6278
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 \operatorname {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) \operatorname {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 [A] time = 0.05, size = 44, normalized size = 1.47 \[ a x+\frac {b x \sqrt {\frac {1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}+b x \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 143, normalized size = 4.77 \[ \frac {a c x + b c \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - b c \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - b \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + {\left (b c x - b c\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c} \]
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 \[ \int b \operatorname {arcsch}\left (c x\right ) + a\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 1.20 \[ a x +b x \,\mathrm {arccsch}\left (c x \right )+\frac {b \ln \left (c x +c x \sqrt {1+\frac {1}{c^{2} x^{2}}}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 49, normalized size = 1.63 \[ a x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b}{2 \, c} \]
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 \[ \int a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right ) \,d x \]
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 \[ \int \left (a + b \operatorname {acsch}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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