Optimal. Leaf size=122 \[ -\frac {e \cosh ^{-1}(c x)^2}{4 c^2}-\frac {d^2 \cosh ^{-1}(c x)^2}{2 e}+\frac {\cosh ^{-1}(c x)^2 (d+e x)^2}{2 e}-\frac {2 d \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}-\frac {e x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{2 c}+2 d x+\frac {e x^2}{4} \]
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Rubi [A] time = 0.65, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5802, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac {e \cosh ^{-1}(c x)^2}{4 c^2}-\frac {d^2 \cosh ^{-1}(c x)^2}{2 e}+\frac {\cosh ^{-1}(c x)^2 (d+e x)^2}{2 e}-\frac {2 d \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}-\frac {e x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{2 c}+2 d x+\frac {e x^2}{4} \]
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) \cosh ^{-1}(c x)^2 \, dx &=\frac {(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e}-\frac {c \int \frac {(d+e x)^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e}\\ &=\frac {(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e}-\frac {c \int \left (\frac {d^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 d e x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^2 x^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{e}\\ &=\frac {(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e}-(2 c d) \int \frac {x \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {\left (c d^2\right ) \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e}-(c e) \int \frac {x^2 \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {2 d \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{2 c}-\frac {d^2 \cosh ^{-1}(c x)^2}{2 e}+\frac {(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e}+(2 d) \int 1 \, dx+\frac {1}{2} e \int x \, dx-\frac {e \int \frac {\cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c}\\ &=2 d x+\frac {e x^2}{4}-\frac {2 d \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{c}-\frac {e x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{2 c}-\frac {d^2 \cosh ^{-1}(c x)^2}{2 e}-\frac {e \cosh ^{-1}(c x)^2}{4 c^2}+\frac {(d+e x)^2 \cosh ^{-1}(c x)^2}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 105, normalized size = 0.86 \[ \frac {e \left (2 c^2 x^2-1\right ) \cosh ^{-1}(c x)^2}{4 c^2}+d x \cosh ^{-1}(c x)^2-\frac {2 d \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{c}-\frac {e x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{2 c}+2 d x+\frac {e x^2}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 98, normalized size = 0.80 \[ \frac {c^{2} e x^{2} + 8 \, c^{2} d x + {\left (2 \, c^{2} e x^{2} + 4 \, c^{2} d x - e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (c e x + 4 \, c d\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, c^{2}} \]
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 = 0.10, size = 100, normalized size = 0.82 \[ \frac {\frac {e \left (2 \mathrm {arccosh}\left (c x \right )^{2} c^{2} x^{2}-2 \,\mathrm {arccosh}\left (c x \right ) c x \sqrt {c x -1}\, \sqrt {c x +1}-\mathrm {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4 c}+d \left (\mathrm {arccosh}\left (c x \right )^{2} c x -2 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )}{c} \]
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 \[ \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} - \int \frac {{\left (c^{3} e x^{4} + 2 \, c^{3} d x^{3} - c e x^{2} - 2 \, c d x + {\left (c^{2} e x^{3} + 2 \, c^{2} d x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{c^{3} x^{3} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} - c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {acosh}\left (c\,x\right )}^2\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 110, normalized size = 0.90 \[ \begin {cases} d x \operatorname {acosh}^{2}{\left (c x \right )} + 2 d x + \frac {e x^{2} \operatorname {acosh}^{2}{\left (c x \right )}}{2} + \frac {e x^{2}}{4} - \frac {2 d \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {e x \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{2 c} - \frac {e \operatorname {acosh}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\- \frac {\pi ^{2} \left (d x + \frac {e x^{2}}{2}\right )}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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