Optimal. Leaf size=13 \[ -\frac {1}{b \sinh ^{-1}(a+b x)} \]
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Rubi [A] time = 0.07, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {5867, 5675} \[ -\frac {1}{b \sinh ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 5675
Rule 5867
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {1}{b \sinh ^{-1}(a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 13, normalized size = 1.00 \[ -\frac {1}{b \sinh ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 32, normalized size = 2.46 \[ -\frac {1}{b \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \]
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 \[ \int \frac {1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 14, normalized size = 1.08 \[ -\frac {1}{b \arcsinh \left (b x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.77, size = 150, normalized size = 11.54 \[ -\frac {b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b + b\right )} x + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} + a}{{\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} {\left (b^{2} x + a b\right )} + {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b + b\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 13, normalized size = 1.00 \[ -\frac {1}{b\,\mathrm {asinh}\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.88, size = 26, normalized size = 2.00 \[ \begin {cases} - \frac {1}{b \operatorname {asinh}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {a^{2} + 1} \operatorname {asinh}^{2}{\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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