Optimal. Leaf size=150 \[ -\frac {e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac {2 b d \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {b e x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}+2 b^2 d x+\frac {1}{4} b^2 e x^2 \]
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Rubi [A] time = 0.76, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5802, 5822, 5676, 5718, 8, 5759, 30} \[ -\frac {e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac {2 b d \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {b e x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}+2 b^2 d x+\frac {1}{4} b^2 e x^2 \]
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) \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac {(b c) \int \frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e}\\ &=\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac {(b c) \int \left (\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 d e x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{e}\\ &=\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-(2 b c d) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {\left (b c d^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e}-(b c e) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}+\left (2 b^2 d\right ) \int 1 \, dx+\frac {1}{2} \left (b^2 e\right ) \int x \, dx-\frac {(b e) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c}\\ &=2 b^2 d x+\frac {1}{4} b^2 e x^2-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}+\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 174, normalized size = 1.16 \[ \frac {c \left (2 a^2 c x (2 d+e x)-2 a b \sqrt {c x-1} \sqrt {c x+1} (4 d+e x)+b^2 c x (8 d+e x)\right )-2 b c \cosh ^{-1}(c x) \left (b \sqrt {c x-1} \sqrt {c x+1} (4 d+e x)-2 a c x (2 d+e x)\right )-2 a b e \log \left (c x+\sqrt {c x-1} \sqrt {c x+1}\right )+b^2 \cosh ^{-1}(c x)^2 \left (4 c^2 d x+e \left (2 c^2 x^2-1\right )\right )}{4 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 185, normalized size = 1.23 \[ \frac {{\left (2 \, a^{2} + b^{2}\right )} c^{2} e x^{2} + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d x + {\left (2 \, b^{2} c^{2} e x^{2} + 4 \, b^{2} c^{2} d x - b^{2} e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 2 \, {\left (2 \, a b c^{2} e x^{2} + 4 \, a b c^{2} d x - a b e - {\left (b^{2} c e x + 4 \, b^{2} c d\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, {\left (a b c e x + 4 \, a b c d\right )} \sqrt {c^{2} x^{2} - 1}}{4 \, c^{2}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 245, normalized size = 1.63 \[ \frac {a^{2} x^{2} e}{2}+a^{2} d x +\frac {b^{2} \mathrm {arccosh}\left (c x \right )^{2} x^{2} e}{2}-\frac {b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x e}{2 c}+\frac {b^{2} e \,x^{2}}{4}-\frac {b^{2} \mathrm {arccosh}\left (c x \right )^{2} e}{4 c^{2}}+b^{2} \mathrm {arccosh}\left (c x \right )^{2} x d -\frac {2 b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, d}{c}+2 b^{2} d x +a b \,\mathrm {arccosh}\left (c x \right ) x^{2} e +2 a b \,\mathrm {arccosh}\left (c x \right ) x d -\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, e x}{2 c}-\frac {2 a b \sqrt {c x -1}\, \sqrt {c x +1}\, d}{c}-\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c^{2} \sqrt {c^{2} x^{2}-1}} \]
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 \[ b^{2} d x \operatorname {arcosh}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} e x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} a b e + \frac {1}{2} \, {\left (x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} - 2 \, \int \frac {{\left (c^{3} x^{4} + \sqrt {c x + 1} \sqrt {c x - 1} c^{2} x^{3} - c x^{2}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{c^{3} x^{3} + {\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} - c x}\,{d x}\right )} b^{2} e + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b d}{c} \]
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 \[ \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 240, normalized size = 1.60 \[ \begin {cases} a^{2} d x + \frac {a^{2} e x^{2}}{2} + 2 a b d x \operatorname {acosh}{\left (c x \right )} + a b e x^{2} \operatorname {acosh}{\left (c x \right )} - \frac {2 a b d \sqrt {c^{2} x^{2} - 1}}{c} - \frac {a b e x \sqrt {c^{2} x^{2} - 1}}{2 c} - \frac {a b e \operatorname {acosh}{\left (c x \right )}}{2 c^{2}} + b^{2} d x \operatorname {acosh}^{2}{\left (c x \right )} + 2 b^{2} d x + \frac {b^{2} e x^{2} \operatorname {acosh}^{2}{\left (c x \right )}}{2} + \frac {b^{2} e x^{2}}{4} - \frac {2 b^{2} d \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {b^{2} e x \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{2 c} - \frac {b^{2} e \operatorname {acosh}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right )^{2} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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