Optimal. Leaf size=49 \[ \frac {c \tanh ^{-1}\left (\sqrt {\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}+1}\right )}{b}+\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5892, 6314, 372, 266, 63, 207} \[ \frac {c \tanh ^{-1}\left (\sqrt {\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}+1}\right )}{b}+\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 266
Rule 372
Rule 5892
Rule 6314
Rubi steps
\begin {align*} \int \sinh ^{-1}\left (\frac {c}{a+b x}\right ) \, dx &=\int \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right ) \, dx\\ &=\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\int \frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right ) \sqrt {1+\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}}} \, dx\\ &=\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\frac {c \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {1}{x^2}} x} \, dx,x,\frac {a}{c}+\frac {b x}{c}\right )}{b}\\ &=\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}\right )}{2 b}\\ &=\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {c \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\frac {c^2}{(a+b x)^2}}\right )}{b}\\ &=\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\frac {c \tanh ^{-1}\left (\sqrt {1+\frac {c^2}{(a+b x)^2}}\right )}{b}\\ \end {align*}
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Mathematica [B] time = 0.13, size = 131, normalized size = 2.67 \[ \frac {(a+b x) \sqrt {\frac {a^2+2 a b x+b^2 x^2+c^2}{(a+b x)^2}} \left (c \tanh ^{-1}\left (\frac {a+b x}{\sqrt {a^2+2 a b x+b^2 x^2+c^2}}\right )+a \tanh ^{-1}\left (\frac {\sqrt {(a+b x)^2+c^2}}{c}\right )\right )}{b \sqrt {a^2+2 a b x+b^2 x^2+c^2}}+x \sinh ^{-1}\left (\frac {c}{a+b x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 242, normalized size = 4.94 \[ \frac {b x \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + c}{b x + a}\right ) + a \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + c\right ) - a \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - c\right ) - c \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right )}{b} \]
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 \[ \int \operatorname {arsinh}\left (\frac {c}{b x + a}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 46, normalized size = 0.94 \[ -\frac {c \left (-\frac {\arcsinh \left (\frac {c}{b x +a}\right ) \left (b x +a \right )}{c}-\arctanh \left (\frac {1}{\sqrt {\frac {c^{2}}{\left (b x +a \right )^{2}}+1}}\right )\right )}{b} \]
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 \[ -\frac {i \, c {\left (\log \left (\frac {i \, {\left (b^{2} x + a b\right )}}{b c} + 1\right ) - \log \left (-\frac {i \, {\left (b^{2} x + a b\right )}}{b c} + 1\right )\right )}}{2 \, b} + \frac {2 \, b x \log \left (c + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}}\right ) + a \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}\right ) - 2 \, {\left (b x + a\right )} \log \left (b x + a\right )}{2 \, b} + \int \frac {b^{2} c x^{2} + a b c x}{b^{2} c x^{2} + 2 \, a b c x + a^{2} c + c^{3} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 41, normalized size = 0.84 \[ \frac {c\,\mathrm {atanh}\left (\sqrt {\frac {c^2}{{\left (a+b\,x\right )}^2}+1}\right )}{b}+\frac {\mathrm {asinh}\left (\frac {c}{a+b\,x}\right )\,\left (a+b\,x\right )}{b} \]
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 \[ \int \operatorname {asinh}{\left (\frac {c}{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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