Optimal. Leaf size=46 \[ \frac {(a+b x) \sinh ^{-1}(a+b x)}{b \sqrt {(a+b x)^2+1}}-\frac {\log \left ((a+b x)^2+1\right )}{2 b} \]
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Rubi [A] time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {5867, 5687, 260} \[ \frac {(a+b x) \sinh ^{-1}(a+b x)}{b \sqrt {(a+b x)^2+1}}-\frac {\log \left ((a+b x)^2+1\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 260
Rule 5687
Rule 5867
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a+b x)}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)}{\left (1+x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sinh ^{-1}(a+b x)}{b \sqrt {1+(a+b x)^2}}-\frac {\operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sinh ^{-1}(a+b x)}{b \sqrt {1+(a+b x)^2}}-\frac {\log \left (1+(a+b x)^2\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 62, normalized size = 1.35 \[ \frac {(a+b x) \sinh ^{-1}(a+b x)}{b \sqrt {a^2+2 a b x+b^2 x^2+1}}-\frac {\log \left (a^2+2 a b x+b^2 x^2+1\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 115, normalized size = 2.50 \[ \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + {\left (a^{2} + 1\right )} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 76, normalized size = 1.65 \[ \frac {{\left (x + \frac {a}{b}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.23, size = 131, normalized size = 2.85 \[ \frac {2 \arcsinh \left (b x +a \right )}{b}-\frac {\left (b^{2} x^{2}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, x b +2 a b x -\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a +a^{2}+1\right ) \arcsinh \left (b x +a \right )}{b \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}-\frac {\ln \left (1+\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 119, normalized size = 2.59 \[ -{\left (\frac {b^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {a b}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}\right )} \operatorname {arsinh}\left (b x + a\right ) - \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {asinh}\left (a+b\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (a + b x \right )}}{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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