Optimal. Leaf size=84 \[ -\frac {\left (9 a^2 c+2 d\right ) \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}-\frac {d x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+c x \cosh ^{-1}(a x)+\frac {1}{3} d x^3 \cosh ^{-1}(a x) \]
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Rubi [A]
time = 0.05, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5908, 471, 75}
\begin {gather*} -\frac {\sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 c+2 d\right )}{9 a^3}+c x \cosh ^{-1}(a x)+\frac {1}{3} d x^3 \cosh ^{-1}(a x)-\frac {d x^2 \sqrt {a x-1} \sqrt {a x+1}}{9 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rule 471
Rule 5908
Rubi steps
\begin {align*} \int \left (c+d x^2\right ) \cosh ^{-1}(a x) \, dx &=c x \cosh ^{-1}(a x)+\frac {1}{3} d x^3 \cosh ^{-1}(a x)-a \int \frac {x \left (c+\frac {d x^2}{3}\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {d x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+c x \cosh ^{-1}(a x)+\frac {1}{3} d x^3 \cosh ^{-1}(a x)+\frac {1}{9} \left (a \left (-9 c-\frac {2 d}{a^2}\right )\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {\left (9 a^2 c+2 d\right ) \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}-\frac {d x^2 \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+c x \cosh ^{-1}(a x)+\frac {1}{3} d x^3 \cosh ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 60, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (2 d+a^2 \left (9 c+d x^2\right )\right )}{9 a^3}+\left (c x+\frac {d x^3}{3}\right ) \cosh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.69, size = 62, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {\mathrm {arccosh}\left (a x \right ) c a x +\frac {a \,\mathrm {arccosh}\left (a x \right ) d \,x^{3}}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (d \,a^{2} x^{2}+9 a^{2} c +2 d \right )}{9 a^{2}}}{a}\) | \(62\) |
default | \(\frac {\mathrm {arccosh}\left (a x \right ) c a x +\frac {a \,\mathrm {arccosh}\left (a x \right ) d \,x^{3}}{3}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (d \,a^{2} x^{2}+9 a^{2} c +2 d \right )}{9 a^{2}}}{a}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 74, normalized size = 0.88 \begin {gather*} -\frac {1}{9} \, {\left (\frac {\sqrt {a^{2} x^{2} - 1} d x^{2}}{a^{2}} + \frac {9 \, \sqrt {a^{2} x^{2} - 1} c}{a^{2}} + \frac {2 \, \sqrt {a^{2} x^{2} - 1} d}{a^{4}}\right )} a + \frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \operatorname {arcosh}\left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 71, normalized size = 0.85 \begin {gather*} \frac {3 \, {\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (a^{2} d x^{2} + 9 \, a^{2} c + 2 \, d\right )} \sqrt {a^{2} x^{2} - 1}}{9 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.14, size = 90, normalized size = 1.07 \begin {gather*} \begin {cases} c x \operatorname {acosh}{\left (a x \right )} + \frac {d x^{3} \operatorname {acosh}{\left (a x \right )}}{3} - \frac {c \sqrt {a^{2} x^{2} - 1}}{a} - \frac {d x^{2} \sqrt {a^{2} x^{2} - 1}}{9 a} - \frac {2 d \sqrt {a^{2} x^{2} - 1}}{9 a^{3}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c x + \frac {d x^{3}}{3}\right )}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 70, normalized size = 0.83 \begin {gather*} \frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d}{9 \, a^{3}} - \frac {\sqrt {a^{2} x^{2} - 1} {\left (3 \, a^{2} c + d\right )}}{3 \, a^{3}} \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 = -0.01 \begin {gather*} \int \mathrm {acosh}\left (a\,x\right )\,\left (d\,x^2+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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