Optimal. Leaf size=45 \[ 2 x-\frac {2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b} \]
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Rubi [A]
time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5858, 5772,
5798, 8} \begin {gather*} \frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b}-\frac {2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b}+2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 5772
Rule 5798
Rule 5858
Rubi steps
\begin {align*} \int \sinh ^{-1}(a+b x)^2 \, dx &=\frac {\text {Subst}\left (\int \sinh ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b}-\frac {2 \text {Subst}\left (\int \frac {x \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b}+\frac {2 \text {Subst}(\int 1 \, dx,x,a+b x)}{b}\\ &=2 x-\frac {2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{b}+\frac {(a+b x) \sinh ^{-1}(a+b x)^2}{b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 1.04 \begin {gather*} \frac {2 (a+b x)-2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)+(a+b x) \sinh ^{-1}(a+b x)^2}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.86, size = 46, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {\arcsinh \left (b x +a \right )^{2} \left (b x +a \right )-2 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a}{b}\) | \(46\) |
default | \(\frac {\arcsinh \left (b x +a \right )^{2} \left (b x +a \right )-2 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a}{b}\) | \(46\) |
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 \begin {gather*} \text {Failed to integrate} \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. 88 vs.
\(2 (43) = 86\).
time = 0.39, size = 88, normalized size = 1.96 \begin {gather*} \frac {{\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 2 \, b x - 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 63, normalized size = 1.40 \begin {gather*} \begin {cases} \frac {a \operatorname {asinh}^{2}{\left (a + b x \right )}}{b} + x \operatorname {asinh}^{2}{\left (a + b x \right )} + 2 x - \frac {2 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {asinh}\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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