Optimal. Leaf size=150 \[ 2 b^2 d x+\frac {1}{4} b^2 e x^2-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{2 c}-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2}+\frac {(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 e} \]
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Rubi [A]
time = 0.50, 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 = {5963, 5975,
5893, 5915, 8, 5939, 30} \begin {gather*} -\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 \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5893
Rule 5915
Rule 5939
Rule 5963
Rule 5975
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 = 0.19, size = 174, normalized size = 1.16 \begin {gather*} \frac {c \left (2 a^2 c x (2 d+e x)-2 a b \sqrt {-1+c x} \sqrt {1+c x} (4 d+e x)+b^2 c x (8 d+e x)\right )-2 b c \left (-2 a c x (2 d+e x)+b \sqrt {-1+c x} \sqrt {1+c x} (4 d+e x)\right ) \cosh ^{-1}(c x)+b^2 \left (4 c^2 d x+e \left (-1+2 c^2 x^2\right )\right ) \cosh ^{-1}(c x)^2-2 a b e \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.45, size = 240, normalized size = 1.60
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (\mathrm {arccosh}\left (c x \right )^{2} x c -2 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {e \left (-2 \sqrt {c x -1}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, x c +2 \mathrm {arccosh}\left (c x \right )^{2} x^{2} c^{2}-\mathrm {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4}\right )}{c}+2 a b \,\mathrm {arccosh}\left (c x \right ) d c x +a b c \,\mathrm {arccosh}\left (c x \right ) e \,x^{2}-2 a b \sqrt {c x -1}\, \sqrt {c x +1}\, d -\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, e x}{2}-\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c \sqrt {c^{2} x^{2}-1}}}{c}\) | \(240\) |
default | \(\frac {\frac {a^{2} \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (\mathrm {arccosh}\left (c x \right )^{2} x c -2 \,\mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {e \left (-2 \sqrt {c x -1}\, \mathrm {arccosh}\left (c x \right ) \sqrt {c x +1}\, x c +2 \mathrm {arccosh}\left (c x \right )^{2} x^{2} c^{2}-\mathrm {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4}\right )}{c}+2 a b \,\mathrm {arccosh}\left (c x \right ) d c x +a b c \,\mathrm {arccosh}\left (c x \right ) e \,x^{2}-2 a b \sqrt {c x -1}\, \sqrt {c x +1}\, d -\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, e x}{2}-\frac {a b \sqrt {c x -1}\, \sqrt {c x +1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 c \sqrt {c^{2} x^{2}-1}}}{c}\) | \(240\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 263, normalized size = 1.75 \begin {gather*} \frac {{\left (2 \, a^{2} + b^{2}\right )} c^{2} x^{2} \cosh \left (1\right ) + {\left (2 \, a^{2} + b^{2}\right )} c^{2} x^{2} \sinh \left (1\right ) + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d x + {\left (4 \, b^{2} c^{2} d x + {\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} \cosh \left (1\right ) + {\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 2 \, {\left (4 \, a b c^{2} d x + {\left (2 \, a b c^{2} x^{2} - a b\right )} \cosh \left (1\right ) + {\left (2 \, a b c^{2} x^{2} - a b\right )} \sinh \left (1\right ) - {\left (b^{2} c x \cosh \left (1\right ) + b^{2} c x \sinh \left (1\right ) + 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 x \cosh \left (1\right ) + a b c x \sinh \left (1\right ) + 4 \, a b c d\right )} \sqrt {c^{2} x^{2} - 1}}{4 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.20, size = 240, normalized size = 1.60 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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