Optimal. Leaf size=121 \[ -\frac {1}{5} c^2 d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {26 b d \sqrt {c x-1} \sqrt {c x+1}}{225 c^3}+\frac {1}{25} b c d x^4 \sqrt {c x-1} \sqrt {c x+1}-\frac {13 b d x^2 \sqrt {c x-1} \sqrt {c x+1}}{225 c} \]
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Rubi [A] time = 0.13, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {14, 5731, 12, 460, 100, 74} \[ -\frac {1}{5} c^2 d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {26 b d \sqrt {c x-1} \sqrt {c x+1}}{225 c^3}+\frac {1}{25} b c d x^4 \sqrt {c x-1} \sqrt {c x+1}-\frac {13 b d x^2 \sqrt {c x-1} \sqrt {c x+1}}{225 c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 74
Rule 100
Rule 460
Rule 5731
Rubi steps
\begin {align*} \int x^2 \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^2 d x^5 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d x^3 \left (5-3 c^2 x^2\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^2 d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{15} (b c d) \int \frac {x^3 \left (5-3 c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{25} b c d x^4 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^2 d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{75} (13 b c d) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {13 b d x^2 \sqrt {-1+c x} \sqrt {1+c x}}{225 c}+\frac {1}{25} b c d x^4 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^2 d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(13 b d) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{225 c}\\ &=-\frac {13 b d x^2 \sqrt {-1+c x} \sqrt {1+c x}}{225 c}+\frac {1}{25} b c d x^4 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^2 d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {(26 b d) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{225 c}\\ &=-\frac {26 b d \sqrt {-1+c x} \sqrt {1+c x}}{225 c^3}-\frac {13 b d x^2 \sqrt {-1+c x} \sqrt {1+c x}}{225 c}+\frac {1}{25} b c d x^4 \sqrt {-1+c x} \sqrt {1+c x}+\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{5} c^2 d x^5 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 89, normalized size = 0.74 \[ -\frac {d \left (15 a c^3 x^3 \left (3 c^2 x^2-5\right )+b \sqrt {c x-1} \sqrt {c x+1} \left (-9 c^4 x^4+13 c^2 x^2+26\right )+15 b c^3 x^3 \left (3 c^2 x^2-5\right ) \cosh ^{-1}(c x)\right )}{225 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 103, normalized size = 0.85 \[ -\frac {45 \, a c^{5} d x^{5} - 75 \, a c^{3} d x^{3} + 15 \, {\left (3 \, b c^{5} d x^{5} - 5 \, b c^{3} d x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (9 \, b c^{4} d x^{4} - 13 \, b c^{2} d x^{2} - 26 \, b d\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, c^{3}} \]
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 = 0.74 \[ \frac {-d a \left (\frac {1}{5} c^{5} x^{5}-\frac {1}{3} c^{3} x^{3}\right )-d b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{3} x^{3} \mathrm {arccosh}\left (c x \right )}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} x^{4}-13 c^{2} x^{2}-26\right )}{225}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 145, normalized size = 1.20 \[ -\frac {1}{5} \, a c^{2} d x^{5} - \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{2} d + \frac {1}{3} \, a d x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d \]
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 \[ \int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.28, size = 133, normalized size = 1.10 \[ \begin {cases} - \frac {a c^{2} d x^{5}}{5} + \frac {a d x^{3}}{3} - \frac {b c^{2} d x^{5} \operatorname {acosh}{\left (c x \right )}}{5} + \frac {b c d x^{4} \sqrt {c^{2} x^{2} - 1}}{25} + \frac {b d x^{3} \operatorname {acosh}{\left (c x \right )}}{3} - \frac {13 b d x^{2} \sqrt {c^{2} x^{2} - 1}}{225 c} - \frac {26 b d \sqrt {c^{2} x^{2} - 1}}{225 c^{3}} & \text {for}\: c \neq 0 \\\frac {d x^{3} \left (a + \frac {i \pi b}{2}\right )}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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