Optimal. Leaf size=215 \[ -\frac {4 e^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{9 c^3}-\frac {d e \cosh ^{-1}(c x)^2}{2 c^2}+\frac {4 e^2 x}{9 c^2}-\frac {d^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {2 d^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}-\frac {d e x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}+\frac {\cosh ^{-1}(c x)^2 (d+e x)^3}{3 e}-\frac {2 e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{9 c}+2 d^2 x+\frac {1}{2} d e x^2+\frac {2 e^2 x^3}{27} \]
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Rubi [A] time = 1.00, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5802, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac {d e \cosh ^{-1}(c x)^2}{2 c^2}+\frac {4 e^2 x}{9 c^2}-\frac {4 e^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{9 c^3}-\frac {d^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {2 d^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}-\frac {d e x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}+\frac {\cosh ^{-1}(c x)^2 (d+e x)^3}{3 e}-\frac {2 e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{9 c}+2 d^2 x+\frac {1}{2} d e x^2+\frac {2 e^2 x^3}{27} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5676
Rule 5718
Rule 5759
Rule 5802
Rule 5822
Rubi steps
\begin {align*} \int (d+e x)^2 \cosh ^{-1}(c x)^2 \, dx &=\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {(2 c) \int \frac {(d+e x)^3 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 e}\\ &=\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {(2 c) \int \left (\frac {d^3 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d^2 e x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d e^2 x^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^3 x^3 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{3 e}\\ &=\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}-\left (2 c d^2\right ) \int \frac {x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {\left (2 c d^3\right ) \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 e}-(2 c d e) \int \frac {x^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{3} \left (2 c e^2\right ) \int \frac {x^3 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {2 d^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {d e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {2 e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c}-\frac {d^3 \cosh ^{-1}(c x)^2}{3 e}+\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}+\left (2 d^2\right ) \int 1 \, dx+(d e) \int x \, dx-\frac {(d e) \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c}+\frac {1}{9} \left (2 e^2\right ) \int x^2 \, dx-\frac {\left (4 e^2\right ) \int \frac {x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c}\\ &=2 d^2 x+\frac {1}{2} d e x^2+\frac {2 e^2 x^3}{27}-\frac {2 d^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {4 e^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c^3}-\frac {d e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {2 e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c}-\frac {d^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {d e \cosh ^{-1}(c x)^2}{2 c^2}+\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}+\frac {\left (4 e^2\right ) \int 1 \, dx}{9 c^2}\\ &=2 d^2 x+\frac {4 e^2 x}{9 c^2}+\frac {1}{2} d e x^2+\frac {2 e^2 x^3}{27}-\frac {2 d^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {4 e^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c^3}-\frac {d e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {2 e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{9 c}-\frac {d^3 \cosh ^{-1}(c x)^2}{3 e}-\frac {d e \cosh ^{-1}(c x)^2}{2 c^2}+\frac {(d+e x)^3 \cosh ^{-1}(c x)^2}{3 e}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 131, normalized size = 0.61 \[ \frac {9 \cosh ^{-1}(c x)^2 \left (2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-3 c d e\right )+c x \left (c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )+24 e^2\right )-6 \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x) \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )}{54 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 163, normalized size = 0.76 \[ \frac {4 \, c^{3} e^{2} x^{3} + 27 \, c^{3} d e x^{2} + 9 \, {\left (2 \, c^{3} e^{2} x^{3} + 6 \, c^{3} d e x^{2} + 6 \, c^{3} d^{2} x - 3 \, c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (2 \, c^{2} e^{2} x^{2} + 9 \, c^{2} d e x + 18 \, c^{2} d^{2} + 4 \, e^{2}\right )} \sqrt {c^{2} x^{2} - 1} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 12 \, {\left (9 \, c^{3} d^{2} + 2 \, c e^{2}\right )} x}{54 \, c^{3}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 207, normalized size = 0.96 \[ \frac {18 \mathrm {arccosh}\left (c x \right )^{2} c^{3} x^{3} e^{2}+54 \mathrm {arccosh}\left (c x \right )^{2} c^{3} x^{2} d e +54 \mathrm {arccosh}\left (c x \right )^{2} c^{3} x \,d^{2}-12 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2} e^{2}-54 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x d e -108 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} d^{2}-27 \mathrm {arccosh}\left (c x \right )^{2} c d e -24 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}+4 e^{2} c^{3} x^{3}+27 c^{3} x^{2} d e +108 x \,c^{3} d^{2}+24 c x \,e^{2}}{54 c^{3}} \]
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 \[ \frac {1}{3} \, {\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} - \int \frac {2 \, {\left (c^{3} e^{2} x^{5} + 3 \, c^{3} d e x^{4} - 3 \, c d e x^{2} - 3 \, c d^{2} x + {\left (3 \, c^{3} d^{2} - c e^{2}\right )} x^{3} + {\left (c^{2} e^{2} x^{4} + 3 \, c^{2} d e x^{3} + 3 \, c^{2} d^{2} x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{3 \, {\left (c^{3} x^{3} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} - c x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {acosh}\left (c\,x\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.21, size = 223, normalized size = 1.04 \[ \begin {cases} d^{2} x \operatorname {acosh}^{2}{\left (c x \right )} + 2 d^{2} x + d e x^{2} \operatorname {acosh}^{2}{\left (c x \right )} + \frac {d e x^{2}}{2} + \frac {e^{2} x^{3} \operatorname {acosh}^{2}{\left (c x \right )}}{3} + \frac {2 e^{2} x^{3}}{27} - \frac {2 d^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {d e x \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {2 e^{2} x^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{9 c} - \frac {d e \operatorname {acosh}^{2}{\left (c x \right )}}{2 c^{2}} + \frac {4 e^{2} x}{9 c^{2}} - \frac {4 e^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\- \frac {\pi ^{2} \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right )}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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