Optimal. Leaf size=46 \[ 2 b^2 x-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+x \left (a+b \sinh ^{-1}(c x)\right )^2 \]
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Rubi [A]
time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5772, 5798, 8}
\begin {gather*} -\frac {2 b \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}+x \left (a+b \sinh ^{-1}(c x)\right )^2+2 b^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 5772
Rule 5798
Rubi steps
\begin {align*} \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=x \left (a+b \sinh ^{-1}(c x)\right )^2-(2 b c) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+x \left (a+b \sinh ^{-1}(c x)\right )^2+\left (2 b^2\right ) \int 1 \, dx\\ &=2 b^2 x-\frac {2 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+x \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 74, normalized size = 1.61 \begin {gather*} \left (a^2+2 b^2\right ) x-\frac {2 a b \sqrt {1+c^2 x^2}}{c}+\frac {2 b \left (a c x-b \sqrt {1+c^2 x^2}\right ) \sinh ^{-1}(c x)}{c}+b^2 x \sinh ^{-1}(c x)^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 72, normalized size = 1.57
method | result | size |
derivativedivides | \(\frac {a^{2} c x +b^{2} \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+2 a b \left (\arcsinh \left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c}\) | \(72\) |
default | \(\frac {a^{2} c x +b^{2} \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+2 a b \left (\arcsinh \left (c x \right ) c x -\sqrt {c^{2} x^{2}+1}\right )}{c}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 72, normalized size = 1.57 \begin {gather*} b^{2} x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, b^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b}{c} \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. 96 vs.
\(2 (44) = 88\).
time = 0.41, size = 96, normalized size = 2.09 \begin {gather*} \frac {b^{2} c x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + {\left (a^{2} + 2 \, b^{2}\right )} c x - 2 \, \sqrt {c^{2} x^{2} + 1} a b + 2 \, {\left (a b c x - \sqrt {c^{2} x^{2} + 1} b^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 82, normalized size = 1.78 \begin {gather*} \begin {cases} a^{2} x + 2 a b x \operatorname {asinh}{\left (c x \right )} - \frac {2 a b \sqrt {c^{2} x^{2} + 1}}{c} + b^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} x - \frac {2 b^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} & \text {for}\: c \neq 0 \\a^{2} x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs.
\(2 (44) = 88\).
time = 0.45, size = 111, normalized size = 2.41 \begin {gather*} 2 \, {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - \frac {\sqrt {c^{2} x^{2} + 1}}{c}\right )} a b + {\left (x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, c {\left (\frac {x}{c} - \frac {\sqrt {c^{2} x^{2} + 1} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2}}\right )}\right )} b^{2} + a^{2} x \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 {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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