Optimal. Leaf size=13 \[ -\frac {1}{b \sinh ^{-1}(a+b x)} \]
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Rubi [A]
time = 0.05, 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 = {5860, 5783}
\begin {gather*} -\frac {1}{b \sinh ^{-1}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 5783
Rule 5860
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 {\text {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 = 0.01, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{b \sinh ^{-1}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.04, size = 14, normalized size = 1.08
method | result | size |
default | \(-\frac {1}{b \arcsinh \left (b x +a \right )}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 150 vs.
\(2 (13) = 26\).
time = 0.28, size = 150, normalized size = 11.54 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs.
\(2 (13) = 26\).
time = 0.39, size = 32, normalized size = 2.46 \begin {gather*} -\frac {1}{b \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (10) = 20\).
time = 0.90, size = 26, normalized size = 2.00 \begin {gather*} \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}{\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\frac {1}{b\,\mathrm {asinh}\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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