Optimal. Leaf size=31 \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{2 b}+\frac {\log \left (\sinh ^{-1}(a+b x)\right )}{2 b} \]
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Rubi [A] time = 0.12, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5867, 5699, 3312, 3301} \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{2 b}+\frac {\log \left (\sinh ^{-1}(a+b x)\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 5699
Rule 5867
Rubi steps
\begin {align*} \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\sinh ^{-1}(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^2(x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\log \left (\sinh ^{-1}(a+b x)\right )}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b}\\ &=\frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )}{2 b}+\frac {\log \left (\sinh ^{-1}(a+b x)\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 24, normalized size = 0.77 \[ \frac {\text {Chi}\left (2 \sinh ^{-1}(a+b x)\right )+\log \left (\sinh ^{-1}(a+b x)\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )}, x\right ) \]
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 \[ \int \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 28, normalized size = 0.90 \[ \frac {\Chi \left (2 \arcsinh \left (b x +a \right )\right )}{2 b}+\frac {\ln \left (\arcsinh \left (b x +a \right )\right )}{2 b} \]
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 \[ \int \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )}\,{d x} \]
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 \[ \int \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{\mathrm {asinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{\operatorname {asinh}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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