Optimal. Leaf size=94 \[ d x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (9 c^2 d+2 e\right )}{9 c^3}-\frac {b e x^2 \sqrt {c x-1} \sqrt {c x+1}}{9 c} \]
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Rubi [A] time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5705, 460, 74} \[ d x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (9 c^2 d+2 e\right )}{9 c^3}-\frac {b e x^2 \sqrt {c x-1} \sqrt {c x+1}}{9 c} \]
Antiderivative was successfully verified.
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Rule 74
Rule 460
Rule 5705
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {x \left (d+\frac {e x^2}{3}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c}+d x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{9} \left (b c \left (9 d+\frac {2 e}{c^2}\right )\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b \left (9 c^2 d+2 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3}-\frac {b e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c}+d x \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 76, normalized size = 0.81 \[ \frac {1}{9} \left (3 a x \left (3 d+e x^2\right )-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (c^2 \left (9 d+e x^2\right )+2 e\right )}{c^3}+3 b x \cosh ^{-1}(c x) \left (3 d+e x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 94, normalized size = 1.00 \[ \frac {3 \, a c^{3} e x^{3} + 9 \, a c^{3} d x + 3 \, {\left (b c^{3} e x^{3} + 3 \, b c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} e x^{2} + 9 \, b c^{2} d + 2 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{9 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.96 \[ \frac {\frac {a \left (\frac {1}{3} c^{3} x^{3} e +c^{3} d x \right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{3} x^{3} e}{3}+\mathrm {arccosh}\left (c x \right ) c^{3} d x -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} x^{2} e +9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 91, normalized size = 0.97 \[ \frac {1}{3} \, a e x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b e + a d x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d}{c} \]
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 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 116, normalized size = 1.23 \[ \begin {cases} a d x + \frac {a e x^{3}}{3} + b d x \operatorname {acosh}{\left (c x \right )} + \frac {b e x^{3} \operatorname {acosh}{\left (c x \right )}}{3} - \frac {b d \sqrt {c^{2} x^{2} - 1}}{c} - \frac {b e x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {2 b e \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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